Planes, Spheres and Pseudospheres

by Greg Egan

• Symmetries and the 24-cell | Average Shadows | Ball Bearings in a Hypersphere | Black Hole Passing | Borehole Oscillators | Born Rigid Motion | Catacaustics | Resonant Modes of a Conical Cavity | The Doppler Shift Triangle Law | Conic Section Orbits | Double-Crossing Orbits | Dihedral Angles | The Ellipse and the Atom | Find The Fake Coin | The Finite Fall | General Relativity in 2+1 dimensions | Howell’s Moving Orbits | Klein's Quartic Curve | Paparazzo vs Bodyguards | Light Mill applet | Littlewood applet | LRL made easy | The Mermin-Peres Magic Square Game | There Are No Corkscrew Orbits | Peeling the Octonions | Ordered Sums | Optimised Origami | Pick Two Points | Weak-field GR near a planar mass | Polar Orbits Around Binary Stars | Planes, Spheres and Pseudospheres | Reptends and Reciprocals | The Rindler Horizon | Rotating Elastic Rings, Disks and Hoops | Constant Weight Rollercoasters | Steffen’s Polyhedron | Superpermutations | Symmetric Waves | Symplectic applet | The Tell-Tale Board | Exact values of regular 10j symbols
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A sheet of paper or a piece of fabric can lie perfectly flat on a table, but it can also be rolled up into a cylinder or a cone, and twisted in certain ways, without being stretched or sheared. Figures drawn on the material when it lies flat retain their shape, in the sense that distances measured within the material itself are unchanged by the way it happens to sit in three-dimensional space.

The geometry of ordinary paper is flat, Euclidean geometry, but what about the equivalent for spherical geometry (which would be something like a piece of orange peel), or for hyperbolic geometry?

The aim of this page is to investigate the possibilities for all three cases: given a piece of an idealised two-dimensional material whose geometry is Euclidean, spherical, or hyperbolic, what shapes can it adopt in three-dimensional Euclidean space?

• Assumptions
• Gaussian Curvature
• Intrinsically Flat Surfaces
  • Planes
  • Cylinders
  • Cones
  • Tangent surfaces
  • Ruled surfaces
  • Rectifying developables
  • Möbius strips
    • Möbius strips and Klein’s bottles in four dimensions
• Intrinsically Curved Surfaces
  • Surfaces of Revolution
  • Surfaces Built From Helices
    • Dini’s surface
    • Other Surfaces Built From Helices
  • The sine-Gordon Equation
    • Exact solutions of the sine-Gordon Equation
  • Choose your own geodesic
    • Geodesics with inflection points
    • A pseudospherical Möbius strip
• Appendix: Computing Gaussian Curvature
  • Gaussian curvature of an embedded surface
  • Gaussian curvature from a metric
• References

Assumptions

• Our idealised material is assumed to be infinitesimally thin, and perfectly flexible. For example, a flat sheet could be rolled into a cylinder whose radius is as small as we like, so long as it’s not zero. But we will not allow the material to contain sharp folds, where the normal to the surface undergoes a discontinuous jump from one side of the fold to the other.
• The material is assumed to be perfectly inextensible. So however much we bend it, it doesn’t stretch or shear: all distances along curves drawn on the material, and all angles between such curves where they cross, remain unchanged.
• We will sometimes talk about solutions where the material might intersect itself in three-dimensional space, because the most mathematically simple statements about general solutions won’t rule that out.
• We will assume that all the mathematical functions describing the way the material is embedded in three-dimensional space are “sufficiently smooth”: whenever we need to take a derivative of something, that derivative exists and is continuous, except possibly at the boundary of the material.

Gaussian Curvature

There are three kinds of two-dimensional space that have constant Gaussian curvature, K:

• the Euclidean plane, with K = 0
• spheres, with K > 0
• hyperbolic planes, with K < 0.

[There are also spaces that consist of pieces of these spaces, with boundaries, or pieces where all or some of the boundaries have been joined together.]

Gaussian curvature, K, is a number that quantifies the manner and extent in which the geometry of a two-dimensional space departs from Euclidean geometry. It can be defined in many different ways, but one approach is to look at the geometry of a circle: the set of points that lie at a fixed distance from a given point, where we need to measure this along the geodesic curves that minimise the distance between points.

On the surface of a sphere with radius a, we measure distances along great circles, and a circle of radius r centred on the pole in spherical coordinates will lie at an angle of θ = r/a radians from the pole. A straightforward calculation gives the circumference and area of such a circle:

Circumference(r) = 2 π a sin(r/a)
Area(r) = 2 π a² (1 – cos(r/a))

In general, a two-dimensional space might have different curvature at different points, but we can still characterise the local geometry by examining the limiting behaviour as the radius r of the circle becomes very small. If we take the first two non-zero terms of the Taylor series for the formulas above, we get:

Circumference(r) ≈ 2 π r – (π/(3 a²)) r³
Area(r) ≈ π r² – (π/(12 a²)) r⁴

We can quantify the extent to which these formulas deviate from the usual Euclidean formulas for circumference and area, 2 π r and π r², by defining the Gaussian curvature K to be either:

K = (3/π) lim_{r → 0} (2 π r – Circumference(r)) / r³

or:

K = (12/π) lim_{r → 0} (π r ² – Area(r)) / r⁴

For the case of a sphere of radius a, both definitions yield:

K = 1 / a²

Another way of characterising Gaussian curvature is via the notion of angular excess. For any triangle in the plane whose sides are straight line segments, the sum of the three interior angles at the vertices is exactly π. On a sphere of radius a, the sum of the three angles for a triangle whose sides are geodesics depends on the area of the triangle, A. Specifically:

Sum-of-angles(A) = π + A/a² = π + K A

We won’t prove this, but it is easy to check for some simple cases. For example, a triangle with one vertex at the north pole, and two that are 90 degrees apart on the equator, will have an area one-eighth that of the whole sphere, or 4πa²/8 = πa²/2, and since all three angles are right-angles, the sum of its angles will be 3π/2, in agreement with the formula. The local value of the curvature K can be obtained in the limiting case of a small triangle:

K = lim_A→0 (Sum-of-angles(A) – π) / A

We have used a sphere to demonstrate how geometry is affected by positive Gaussian curvature, but exactly the same mathematical relationships hold in surfaces with negative curvature. So when K < 0, the circumference and area of a circle are larger than those of a circle of the same radius in the plane, and the sum of the three angles of a triangle whose sides are geodesics will be less than π.

The description we are given of a two-dimensional space will usually come in either of two forms. We might be told how the space has been embedded as a surface in three-dimensional Euclidean space, like the sphere we have just described. Or we might be given a description of the surface’s intrinsic geometry, in terms of a metric.

The appendix to this web page explains how the Gaussian curvature is calculated in both these cases. For a more comprehensive treatment, there are many good textbooks, such as Lipschutz,^[1] that give a detailed account of “the classical theory of surfaces,” which deals with the geometry of two-dimensional spaces via surfaces embedded in three-dimensional space.

Intrinsically Flat Surfaces

There are four kinds of surfaces embedded in three-dimensional Euclidean space that are intrinsically flat, with a constant Gaussian curvature of zero:

• planes
• cylinders
• cones
• tangent surfaces

Some of these categories can be seen as either special cases or limiting cases of others; for example, cylinders are like cones whose apex is moved infinitely far away.

Planes

That planes are flat is not just obvious, it’s the case where, if our measure of intrinsic curvature gave any other result than zero, we would throw it away and look for a new one. To check that we don’t need to do that, note that we can parameterise any plane with the formula:

x(u, v) = P + u a + v b

where P is a point on the plane and a and b are two fixed, linearly independent vectors that are parallel to the plane. All three second derivatives of x(u, v) with respect to the parameters are zero, so the second fundamental form, S, of the surface, defined in the appendix, will be zero everywhere, making its determinant zero, while the first fundamental form, g, will have a non-zero determinant because a and b are linearly independent. So the Gaussian curvature, K = det(S) / det(g), will be zero.

Cylinders

Given a curve c(s) that lies in a plane, the cylinder generated by c(s) is the surface traced out by all the lines perpendicular to the plane that pass through points on c(s). If the curve is closed, the cylinder will be finite in one direction, and if the curve is self-intersecting, the cylinder will be a self-intersecting surface, but if the curve is infinitely long and never intersects itself, the cylinder will be infinite in all directions, and will be an embedding of the entire Euclidean plane.

Suppose the parameter s for the curve is its arc length, and let p be a unit-length normal to the plane of the curve (we will reserve n for the normal vector to the cylinder itself). Then we can parameterise the cylinder as:

x(u, v) = c(u) + v p

Following the methods described in the appendix, we have:

x_u = c'(u)
x_v = p
n = x_u × x_v / |x_u × x_v| = c'(u) × p
x_uu = c''(u)
x_uv = 0
x_vv = 0

g(u, v)	=	xu · xu xu · xv xu · xv xv · xv	=	1 0 0 1
S(u, v)	=	xuu · n xuv · n xuv · n xvv · n	=	c''(u) · n 0 0 0

K = det(S) / det(g) = 0 / 1 = 0

There is an easy way to construct an isometry (a one-to-one distance-preserving function) between these cylinders and all or part of the Euclidean plane: we map the point with coordinates (u, v) on the cylinder to the point (x, y) in the usual Cartesian coordinates on the plane, setting:

x = u
y = v

The metric in u, v coordinates is exactly the same as the Euclidean metric, so this map preserves distances and angles measured within the embedded surface.

If the curve c(s) never intersects itself, in the case of an infinitely long curve we get an isometry between the entire Euclidean plane and the cylinder. For finite curves, or if we choose to limit the range of the v coordinate, the isometry is with a piece of the plane.

If the cylinder is generated by a closed curve, the topology is no longer that of a piece of the plane, but that of a piece with two opposite edges joined together. Locally, the distinction is not important, and the Gaussian curvature remains zero everywhere, but obviously a circular cylinder is not globally isometric to a rectangle, or a strip of finite width, because some points on the opposite edges end up closer to each other in the cylinder.

Cones

We will define a cone as the surface traced out by all the lines that pass through a curve c(s) and some fixed point A, the apex of the cone, which does not lie on the curve. Unlike the cylinder, we will not require c(s) to be planar, but we will impose the condition that the tangent to the curve never points directly towards the apex. We can parameterise this surface as:

x(u, v) = A + v (c(u) – A)

The original curve is given by v = 1, and the apex by v = 0. We then have, for v ≠ 0, and assuming c(s) is parameterised by arc length:

x_u = v c'(u)
x_v = c(u) – A
n = x_u × x_v / |x_u × x_v| = v c'(u) × (c(u) – A) / |v c'(u) × (c(u) – A)|
x_uu = v c''(u)
x_uv = c'(u)
x_vv = 0

g(u, v)	=	xu · xu xu · xv xu · xv xv · xv	=	v2 v c'(u) · (c(u) – A) v c'(u) · (c(u) – A) |c(u) – A|2
S(u, v)	=	xuu · n xuv · n xuv · n xvv · n	=	v c''(u) · n 0 0 0

K = det(S) / det(g) = 0 / (v² (|c(u) – A|² – (c'(u) · (c(u) – A))²)) = 0

In general, the apex, v = 0, will not contain a neighbourhood that looks like part of a plane, so it needs to be excluded, and the surfaces with v < 0 and v > 0 are not connected to each other. Depending on the exact shape of the curve and the relative location of the apex, cones might or might not be self-intersecting elsewhere.

The most familiar example of a cone is a right circular cone, generated by a circle and an apex that lies directly above the centre of the circle. The portion of the plane that is embedded as a right circular cone is always a wedge with an angle less than 2π. However, cones generated by sufficiently meandering curves can come from wedges of any angle, including angles greater than or equal to 2π.

If the wedge angle is exactly 2π, as in the second illustration at the start of this section, then the cone might or might not count as an embedding of the entire Euclidean plane, depending on how strict we want to be about the smoothness of the embedding at the apex. If the wedge angle is greater than 2π, as in the third illustration, then the cone cannot be an embedding of just one copy of the Euclidean plane (in the illustration, an extra piece that comes from a second copy of the plane is coloured green). But it will still be a perfectly good intrinsically flat space, so long as we exclude the apex.

To construct an isometry between a cone described in the (u, v) coordinates we used in the definition, and part of the plane (or maybe parts of more than one plane), it is easiest if we work in polar coordinates (r, θ). An isometry is then given by:

r = v |c(u) – A|
θ = ∫₀^u (1/|c(t) – A|) √[1 – ((c(t) – A) · c'(t))² / |c(t) – A|²] dt

The integral for θ amounts to adding up the distance we travel along the curve perpendicular to a line towards the apex, divided by the distance to the apex, to give us the angle we have travelled “around” the apex to reach the coordinate value u.

For example, if c(s) is a circle of radius R and the apex lies at a distance H directly above the centre of the circle, we have:

c(s) = (R cos(s/R), R sin(s/R), 0)
A = (0, 0, H)

|c(s) – A| = |(R cos(s/R), R sin(s/R), –H)| = √(R² + H²)

r = v |c(u) – A| = v √(R² + H²)
θ = ∫₀^u (1/|c(t) – A|) √[1 – ((c(t) – A) · c'(t))² / |c(t) – A|²] dt
= (1/√(R² + H²)) ∫₀^u √[1 – ((R cos(t/R), R sin(t/R), –H) · (–sin(t/R), cos(t/R), 0))² / (R² + H²)] dt
= u/√(R² + H²)

The u coordinate will range from 0 to 2πR, so θ will range from 0 to 2πR/√(R² + H²). This confirms that any right circular cone has a wedge angle of less than 2π. But for other cones, if θ exceeds 2π we will need to obtain the remainder of the surface from one or more extra copies of the plane.

Tangent surfaces

Suppose we have a curve c(s), parameterised by arc length s, that has no points of inflection: that is, no points where the unit-length tangent to the curve, c'(s), has a rate of change of zero as we move along the curve. This is the same as saying that for all s, c''(s) ≠ 0.

The tangent surface of c(s) is the surface consisting of all lines that are tangents to the curve. [In some literature, this is called the tangent developable; the term “developable” refers to any intrinsically flat surface.] We can parameterise the tangent surface as:

x(u, v) = c(u) + v c'(u)

Because c(s) is parameterised by arc length, c'(s) has a constant length of 1, and its rate of change, c''(s), is always orthogonal to it, since it can only change direction, not length. We then have, for v ≠ 0:

x_u = c'(u) + v c''(u)
x_v = c'(u)
n = x_u × x_v / |x_u × x_v| = v c''(u) × c'(u) / |v c''(u) × c'(u)|
x_uu = c''(u) + v c'''(u)
x_uv = c''(u)
x_vv = 0

g(u, v)	=	xu · xu xu · xv xu · xv xv · xv	=	1 + v2 |c''(u)|2 1 1 1
S(u, v)	=	xuu · n xuv · n xuv · n xvv · n	=	v c'''(u) · n 0 0 0

K = det(S) / det(g) = 0 / (v² |c''(u)|²) = 0

It might not be immediately obvious from our definitions, but we do not obtain a single, smooth surface if we allow the v coordinate to take on both positive and negative values. The first figure at the start of this section makes this clear; the yellow and red surfaces meet along the generating curve, but there is a cusp where they come together, rather than a smooth transition.

To find an isometry between a tangent surface and part of the plane, we will focus on the particular example of the tangent surface to a helix, h(s), with radius a and pitch controlled by b:

We define a parameter c = √(a² + b²)

h(s) = (a cos(s/c), a sin(s/c), b s/c)
h'(s) = (–a/c sin(s/c), a/c cos(s/c), b/c)
h''(s) = (–a/c² cos(s/c), –a/c² sin(s/c), 0)
h'''(s) = (a/c³ sin(s/c), –a/c³ cos(s/c), 0)

x(u, v) = h(u) + v h'(u) = (a cos(u/c), a sin(u/c), b u/c) + v (–a/c sin(u/c), a/c cos(u/c), b/c)

x_u = h'(u) + v h''(u) = (–a/c sin(u/c), a/c cos(u/c), b/c) + v (–a/c² cos(u/c), –a/c² sin(u/c), 0)
x_v = h'(u) = (–a/c sin(u/c), a/c cos(u/c), b/c)
n = v h''(u) × h'(u) / |v h''(u) × h'(u)| = sign(v) (–b/c sin(u/c), b/c cos(u/c), –a/c)
x_uu = h''(u) + v h'''(u) = (–a/c² cos(u/c), –a/c² sin(u/c), 0) + v (a/c³ sin(u/c), –a/c³ cos(u/c), 0)
x_uv = h''(u) = (–a/c² cos(u/c), –a/c² sin(u/c), 0)
x_vv = 0

g(u, v)	=	1 + v2 |h''(u)|2 1 1 1	=	1 + a2 v2 / c4 1 1 1
S(u, v)	=	v h'''(u) · n 0 0 0	=	–sign(v) a b v / c4 0 0 0

To construct an isometry, we will first identify a convenient set of geodesics on the tangent surface, which will need to correspond to straight lines in the plane. To start, we will note that any straight line (in the sense of being straight in three-dimensional Euclidean space) in any smoothly embedded surface is a geodesic.^[4] So the lines of fixed u and varying v in a tangent surface are all geodesics.

To find other geodesics, we will make use of the fact that the metric for the helix tangent surface is completely independent of the u coordinate. It follows that changing u by a fixed amount everywhere maps the surface into itself isometrically; that is, it is a symmetry of the surface, which corresponds to a screwlike motion, where we rotate the surface around the axis of the helix while also moving everything along the axis.

A further consequence of this is that, along any geodesic G(s), the dot product between the tangent to the geodesic, G'(s), and the vector field that corresponds to a uniform change in the u coordinate, which is just (1,0) in (u, v) coordinates, is constant. This result is known as Killing’s theorem, and it is described in a bit more detail here [this article is associated with my novel Incandescence, but there is no need to know anything about that book to follow the discussion of the geometry].

So, Killing’s theorem gives us, for a geodesic:

G(s) = (u_G(s), v_G(s))

with tangent vector:

G'(s) = (u_G'(s), v_G'(s))

a conserved quantity C along the geodesic:

G'(s)^T g(G(s)) (1,0) = C
(1 + a² v_G(s)² / c⁴) u_G'(s) + v_G'(s) = C

We can check this for the geodesics we already know about, the lines of constant u and varying v:

u_G(s) = u₀
v_G(s) = s

u_G'(s) = 0
v_G'(s) = 1

(1 + a² v_G(s)² / c⁴) u_G'(s) + v_G'(s) = 1

If the geodesic is parameterised by arc length, its tangent will have a length of 1 everywhere, and we can use that to solve for u_G'(s) in terms of v_G(s) and v_G'(s).

G'(s)^T g(G(s)) G'(s) = 1
(1 + a² v_G(s)² / c⁴) u_G'(s)² + 2 u_G'(s) v_G'(s) + v_G'(s)² = 1
u_G'(s) = (–v_G'(s) ± √[1 + (a²/c⁴) v_G(s)² (1 – v_G'(s)²)]) / (1 + a² v_G(s)² / c⁴)

With this result, we have for the conserved quantity along the geodesic:

(1 + a² v_G(s)² / c⁴) u_G'(s) + v_G'(s) = C
± √[1 + (a²/c⁴) v_G(s)² (1 – v_G'(s)²)] = C
1 + (a²/c⁴) v_G(s)² (1 – v_G'(s)²) = C²

Suppose we want to find a geodesic that meets one of our straight-line geodesics of constant u at some point (u₀, v₀), and the two geodesics are orthogonal to each other where they meet, at s = 0. Then we will want the tangent to the geodesic at s = 0 to be:

G'(0) = (u_G'(0), v_G'(0)) = (c² / (a v₀), –c² / (a v₀))

since this tangent has unit length, and it is orthogonal to (0,1), the tangent to the straight-line geodesic:

G'(0)^T g(u₀, v₀) G'(0) = 1
G'(0)^T g(u₀, v₀) (0,1) = 0

The initial value this gives us for v_G'(0) lets us compute the conserved quantity at the start of the geodesic:

C² = 1 + (a²/c⁴) v_G(s)² (1 – v_G'(s)²) = (a²/c⁴) v₀²

and the differential equation we need to solve becomes:

1 + (a²/c⁴) v_G(s)² (1 – v_G'(s)²) = (a²/c⁴) v₀²

with the initial conditions:

v_G(0) = v₀
v_G'(0) = –c² / (a v₀)

This is solved by:

v_G(s) = √[s² – 2 (c²/a) s + v₀²]

If we look at the points in the (s, v₀) plane where v_G(s) = 0, it turns out they lie on a circle, with radius c²/a, and centre (c²/a, 0). This circle in the (s, v₀) plane corresponds to the helix in the tangent surface, and v_G(s) is precisely the distance from (s, v₀) to this circle, measured along a tangent to the circle.

A line of constant v₀ in the (s, v₀) plane with |v₀| < c²/a will intersect the circle, and for values of s that lie inside the circle, v_G(s) becomes imaginary. In other words, these geodesics of constant v₀ reach the boundary of the tangent surface, the helix, and come to an end there. But for |v₀| > c²/a these geodesics never hit the boundary. Similarly, geodesics of constant s and varying v₀ will only hit the boundary and come to an end if 0 < s < 2c²/a.

If we substitute this solution for v_G(s) back into our original equation for the conserved quantity, before we eliminated u_G'(s), we can now solve that equation for u_G'(s) and integrate it to find u_G(s). The solution we obtain is:

u_G(s) = u₀ + 2 (c²/a) arctan(s / (v₀ + v_G(s)))

How does u_G(s) relate to the circle in the (s, v₀) plane that corresponds to the helix? For points that lie on that circle, it must equal u₀ plus the arc length along the circle, measured from the origin. This means that throughout the (s, v₀) plane, the range of u values is limited to the interval u₀ ± π (c²/a), and a single copy of the plane can only correspond to that portion of the tangent surface.

So the isometry we have found maps regions of the helix tangent surface where the u coordinate changes by 2π (c²/a) to copies of the plane with a disk of radius c²/a cut out of it. The second figure at the start of this section uses different colours to show the different copies of the plane that are mapped to the surface.

Note that the curvature of the generating helix, as a curve in three-dimensional Euclidean space:

κ = |h''(s)| = a/c²

is precisely the curvature (the inverse of the radius, c²/a) of the corresponding circle in the plane. This is certainly not true in general for the images of curves under this isometry; for example, most straight lines in the plane do not map to straight lines in three-dimensional space. However, it is not a coincidence either; the generating curve of any tangent surface always has the same curvature in three-dimensional Euclidean space as it has intrinsically as a curve in the tangent surface, or equivalently, as the corresponding curve under an isometry with the Euclidean plane.

Ruled surfaces

All four kinds of intrinsically flat surfaces we have described are examples of ruled surfaces: surfaces that are built entirely from a family of straight lines. But not every ruled surface is intrinsically flat. If we compute the Gaussian curvature of an arbitrary ruled surface, what further conditions are required to ensure that K = 0?

Suppose we have a curve, c(s), parameterised by arc length, and at each point on the curve there is a unit-length vector L(s) that gives the direction of the straight line passing through that point. The resulting surface is parameterised as:

x(u, v) = c(u) + v L(u)

We perform the usual calculations to find the Gaussian curvature:

x_u = c'(u) + v L'(u)
x_v = L(u)
n = x_u × x_v / |x_u × x_v| = (c'(u) + v L'(u)) × L(u) / |(c'(u) + v L'(u)) × L(u)|
= (c'(u) + v L'(u)) × L(u) / √(|c'(u) + v L'(u)|² – (c'(u) · L(u))²)
x_uu = c''(u) + v L''(u)
x_uv = L'(u)
x_vv = 0

g(u, v)	=	xu · xu xu · xv xu · xv xv · xv	=	|c'(u) + v L'(u)|2 c'(u) · L(u) c'(u) · L(u) 1
S(u, v)	=	xuu · n xuv · n xuv · n xvv · n	=	(c''(u) + v L''(u)) · n L'(u) · n L'(u) · n 0

K = det(S) / det(g) = –[L'(u) · c'(u) × L(u) / (|c'(u) + v L'(u)|² – (c'(u) · L(u))²)]²

This expression for the curvature tells us that it can never be positive, since it is the opposite of a squared real-valued quantity. What’s more, it cannot be equal to a constant negative value, since the only way to remove the dependence on v would involve either setting L'(u) = 0, or otherwise making the numerator zero, both of which would produce flat surfaces.

But the ruled surface will be intrinsically flat if, and only if:

L'(u) · c'(u) × L(u) = 0

This scalar triple product will obviously be zero if one or more of the three vectors are zero. But it will also be zero if any two of the vectors are parallel, or if all three vectors are coplanar.

• For a generalised cylinder (which includes the plane as a special case), the directions of the lines are constant, so L'(u) = 0.
• For a cone, L(u) is a scalar multiple of c(u) – A, where A is the apex, so by the product rule for derivatives, L'(u) lies in the plane spanned by L(u) and c'(u).
• For a tangent surface, L(u) = c'(u).

Are there are any other possibilities besides these four cases? We can splice different kinds of surface together, e.g. we could join a cone and a cylinder, but locally, is there any other way to meet this condition on the ruled surface that we haven’t already considered?

The answer is no; this list is complete! Here is a proof, taken from Lipschutz.^[5]

The one remaining question is whether there is any way to have an intrinsically flat surface that is not a ruled surface. The answer to that is no;^[6] we won’t reproduce the proof here, but it formalises the intuitive idea that if, at every point, one of the principal curvatures of the surface is zero, then if you keep moving across the surface in the corresponding direction, you must be following a straight line in three-dimensional space.

Rectifying developables

Given a smooth, non-self-intersecting curve c(s) in three-dimensional Euclidean space, is it possible to find an intrinsically flat surface in which that curve is a geodesic? Intuition suggests that the answer should be yes: a strip of (perfectly flexible) paper, narrow enough that it won’t bump into itself in any tight places on the curve, could be positioned so that its centreline coincides with the curve.

The surface that meets this condition is known as the rectifying developable for the curve.^[7] We know from the previous section that it will be a ruled surface, so we can parameterise it as:

x(u, v) = c(u) + v L(u)

for some L(u) yet to be determined, which we will write initially as:

L(u) = A(u) c'(u) + B(u) c''(u) + D(u) c'(u) × c''(u)

A normal vector to the surface at any point on the curve c(u) will be given by:

N(u) = L(u) × c'(u)
= B(u) c''(u) × c'(u) + D(u) (c'(u) × c''(u)) × c'(u)
= B(u) c''(u) × c'(u) + D(u) c''(u)

Along a geodesic, the tangent to the curve, c'(u), its derivative, c''(u), and the normal vector to the surface, N(u), all lie in the same plane.^[3] So to make c(u) a geodesic, we must have:

N(u) · (c'(u) × c''(u)) = 0

which requires (assuming c''(u) is nonzero) that B(u) = 0. Then the condition for this ruled surface to be intrinsically flat is:

L'(u) · c'(u) × L(u) = 0
[A'(u) c'(u) + A(u) c''(u) + D'(u) c'(u) × c''(u) + D(u) c'(u) × c'''(u)] · [c'(u) × (A(u) c'(u) + D(u) c'(u) × c''(u))] = 0
[A'(u) c'(u) + A(u) c''(u) + D'(u) c'(u) × c''(u) + D(u) c'(u) × c'''(u)] · [–D(u) c''(u)] = 0
A(u) |c''(u)|² + D(u) (c'(u) × c'''(u)) · c''(u) = 0
A(u) κ²(u) – D(u) κ²(u) τ(u) = 0

Here κ(u) is the length of c''(u), which measures the curvature of c(u), and τ(u) is the torsion of the curve, which measures the rate at which the plane of the curve is changing its orientation as we move along the curve. There are several different formulas that all yield this quantity, but for our purposes the most convenient formula is:

τ(s) = (c'(s) × c''(s)) · c'''(s) / κ(s)²

If we want L(u) to be a unit-length vector, this condition will be satisfied by:

L(u) = (τ(u) c'(u) + c'(u) × c''(u)) / √[τ(u)² + κ(u)²]

The numerator here is a vector associated with the curve, known as the Darboux vector, which describes the “angular velocity” of the orthogonal set of axes given by the unit-length tangent to the curve, c'(u), its derivative, c''(u), and their cross product, as we move along the curve. (It is common practice to also normalise the last two vectors to unit length, giving an orthonormal frame known as the Frenet-Serret frame.)

So, if we take a smooth, non-self-intersecting curve and construct a ruled surface whose lines lie in the direction of the Darboux vector at each point on the curve, the curve will be a geodesic for that surface.

If c(u) is a straight line, the Darboux vector will be undefined, but we can use any plane that contains that line as a surface with the required properties.

The next simplest case would be a planar curve. This will have zero torsion, so L(u) will be a fixed vector perpendicular to the plane of the curve, giving us a cylinder. Similarly, if we choose a curve that is a geodesic on a cone, the rectifying developable will be that cone.

But generically, the surface we get from a curve will be a tangent surface, and we can explicitly identify the curve whose tangents are the same set of straight lines as we get from the Darboux vectors of the original curve. If we define:

T(s) = c(s) + [κ(s)/(τ(s) κ'(s) – τ'(s) κ(s))] (τ(s) c'(s) + c'(s) × c''(s))

then it turns out that T'(s) is parallel to the Darboux vector for c(s), so the tangent surface for T(s) is the rectifying developable for c(s). Note that the parameter s here is an arc length parameter for c(s) only, and not for T(s).

T(s) will constitute an edge to the surface, just as any curve does for its own tangent surface. The distance measured along the ruling line from the point c(s) to the edge of the surface will be:

Distance to edge along ruling
= | κ(s) √[τ(s)² + κ(s)²] / (τ(s) κ'(s) – τ'(s) κ(s)) |

If we want to know the orthogonal distance, measured within the surface, between a point on the edge, T(s), and the generating curve (which is a geodesic for the surface), we can make use of the fact that the sine of the angle between the ruling lines and the curve is κ(s)/√[τ(s)² + κ(s)²], from which we obtain:

Orthogonal distance from edge to prescribed geodesic
= | κ(s)² / (τ(s) κ'(s) – τ'(s) κ(s)) |
= | 1 / d(τ(s)/κ(s))/ds |

We can also use that formula for the the sine of the angle between the ruling lines and the curve to write an isometry between the rectifying developable and the plane, in Cartesian coordinates (x, y):

x = u + v (τ(u)/√[τ(u)² + κ(u)²])
y = v (κ(u)/√[τ(u)² + κ(u)²])

If the curvature vector c''(u) is zero at an isolated point u₀, the unit-length normal vector c''(u)/|c''(u)|, which is undefined at u₀, can either have the same, or opposite, limiting values on either side of u₀. If the torsion also goes to zero at the same point, the normalised Darboux vector we are using for L(u) can also switch direction. This doesn’t affect the surface itself (since the ruling lines, which follow the Darboux vector in both directions, are unchanged) but in order to make the (u, v) coordinates continuous across the change, we can multiply L(u) by a sign that is chosen in order to extend continuity as much as possible.

If the torsion does not go to zero at a point where the curvature does, then that marks a point where c(s) intersects the edge curve T(s).

If we construct the rectifying developable for a smooth closed curve on which the curvature is never zero, then the surface will be topologically a cylinder: a piece of the plane whose opposite edges join up without a twist. But if there are an odd number of points of zero curvature and torsion where the direction of the normalised curvature and Darboux vectors change discontinuously, the surface will be a Möbius strip.

Möbius strips

We can embed a Möbius strip isometrically in three dimensions by wrapping the strip around three cylinders, as in the first image above. In this case, the centreline is a piecewise function that splices together three helices and three line segments, and the surface is spliced together from three cylindrical regions and three planar regions.

We can also embed a Möbius strip as the rectifying developable of a single smooth curve, as in the second image above. The third image shows what is happening with the Frenet-Serret frame. The red vector would change direction at the inflection point, I, if we simply defined it as c''(u)/|c''(u)|, but we multiply it with a sign that makes it continuous as it crosses that point. However, as a result of that single change of sign, it no longer agrees with its original direction when it comes full circle; it needs to traverse the loop twice before that happens. This is accompanied by a similar change in the Darboux vector, so the ruling lines reverse orientation after a single circuit around the loop, making the surface a Möbius strip.

The function we have chosen for the y coordinate of the curve might look a bit strange and arbitrary at first glance:

y(t) = cos(t) + (2/5) cos(2t) + (1/15) cos(3t)

In fact, this is the simplest trigonometric function with a Taylor series at π that is a constant plus a sixth-order term:

y(π + ε) ≈ –2/3 + (1/30) ε⁶

This produces an inflection point at π with suitable behaviour for the curvature and torsion: both go to zero, but the torsion must go to zero at least as rapidly as the curvature, in order for the width of the strip we can fit around the centreline, |1 / d(τ(s)/κ(s))/ds|, to remain non-zero. Because we have chosen to make the curve symmetrical under a 180° rotation around the y-axis, the torsion must be an even function of ε, so the lowest order it can have is quadratic.

κ(π + ε) ≈ (4/(5√5)) |ε|
τ(π + ε) ≈ –(2/9) ε²

Möbius strips and Klein’s bottles in four dimensions

Although this web page is focused on surfaces embedded in three-dimensional space, we will note that embeddings in four dimensions allow for more possibilities. Intrinsically flat surfaces in four dimensions need not be ruled surfaces, and some topologies that cannot be embedded in three dimensions because the surface will unavoidably intersect itself can be embedded in four dimensions.

Consider the following surface embedded in four dimensions, parameterised as:

x(u, v) = (cos u cos v, sin u cos v, 2 cos(u/2) sin v, 2 sin(u/2) sin v)

In four dimensions, we can still use the methods described in the appendix to find the metric:

x_u = (–sin u cos v, cos u cos v, –sin(u/2) sin v, cos(u/2) sin v)
x_v = (–cos u sin v, –sin u sin v, 2 cos(u/2) cos v, 2 sin(u/2) cos v)

g(u, v)

=

xu · xu xu · xv xu · xv xv · xv

=

1 0 0 1+3 cos2 v

While we could proceed to compute the Gaussian curvature from the metric with the whole apparatus of Christoffel symbols and the Riemann curvature tensor described in the second appendix, we don’t actually need to do all that work to conclude that the surface is intrinsically flat! The metric is already almost in the form of the two-dimensional Euclidean metric, except for the g₂₂ component, which is a function of the second parameter v, rather than being 1.

But if we changed to a new parameter in place of v that measured arc length along each curve of varying v and constant u, then g₂₂ would become 1, and the metric would be precisely the Euclidean metric, since this change would have no effect on the other components.

What is the topology of this surface? If we allow u to range from 0 to 2π, and v to range from –V to V for some value of V greater than 0 and strictly less than π, then we can examine how the two ends, u = 0 and u = 2π, of the coordinate rectangle are related in the embedding:

x(0, v) = (cos v, 0, 2 sin v, 0)
x(2π, v) = (cos v, 0, –2 sin v, 0)
x(2π, –v) = (cos v, 0, 2 sin v, 0)

We see that the two ends coincide, but with the value of the v coordinate negated, which amounts to a 180° twist. This is precisely the topology of a Möbius strip.

What if we set V = π, widening the strip so that v ranges from –π to π? We then have the other two sides of the coordinate rectangle meeting:

x(u, –π) = (–cos u, –sin u, 0, 0)
x(u, π) = (–cos u, –sin u, 0, 0)

Since these sides meet up with no change in orientation, this shows that the surface has closed up into a Klein’s bottle. Of course, it remains intrinsically flat, and there are no self-intersections (other than the required meeting of the borders of the coordinate rectangle). The “tube” of the Klein’s bottle is generated by taking an ellipse, given by the curve for varying v at u = 0:

x(0, v) = (cos v, 0, 2 sin v, 0)

and applying a continuous family of four-dimensional rotations to it, by an angle of u in the xy plane and an angle of u/2 in the zw plane, where we are calling the four coordinates (x, y, z, w).

The animation at the start of this section shows a single, rigid Möbius strip with this embedding, but it is projected down to three dimensions using the basis:

{(0, 0, cos α, sin α), (1, 0, 0, 0), (0, 1, 0, 0)}

with α cyling from 0 to 2π over the course of the animation. To be clear, these projections to three-dimensional space are not intrinsically flat.

Intrinsically Curved Surfaces

Surfaces of Revolution

We will start our exploration of surfaces of constant non-zero Gaussian curvature by looking for examples that can be produced by taking a planar curve and rotating it around an axis. This will certainly not encompass all the surfaces we want to catalogue, eventually, but the advantage of starting here is to find some examples while keeping the mathematics relatively simple, as these surfaces of revolution can be described with a single function of one variable.

One way to parameterise a surface of revolution is to describe the z coordinate as a function of the radial cylindrical coordinate, ρ, i.e. the distance from the axis. While we could reverse this, and describe the radius as a function of z, or, to allow for completely general curves, describe both ρ and z as functions of some parameter, the approach we have chosen will make our calculations easier, and the results simpler to express.

So, we will parameterise our surface as:

x(φ, ρ) = (ρ cos φ, ρ sin φ, f(ρ))

where the right-hand side here are ordinary Cartesian coordinates, x, y, z. We have departed from our earlier practice of using (u, v) as the names for the two surface coordinates, in favour of (φ, ρ), which are more suggestive of the geometry, at least for readers who have been exposed to the conventional choice of symbols for cylindrical coordinate systems.

Following the usual recipe for computing the Gaussian curvature of an embedded surface (spelled out in the appendix), we have:

x_φ = (–ρ sin φ, ρ cos φ, 0)
x_ρ = (cos φ, sin φ, f '(ρ))
n = x_φ × x_ρ / |x_φ × x_ρ| = (f '(ρ) cos φ, f '(ρ) sin φ, –1) / √[1 + f '(ρ)²]
x_φφ = (–ρ cos φ, –ρ sin φ, 0)
x_φρ = (–sin φ, cos φ, 0)
x_ρρ = (0, 0, f ''(ρ))

g(φ, ρ)	=	xφ · xφ xφ · xρ xφ · xρ xρ · xρ	=	ρ2 0 0 1 + f '(ρ)2
S(φ, ρ)	=	xφφ · n xφρ · n xφρ · n xρρ · n	=	–ρ f '(ρ) / √[1 + f '(ρ)2] 0 0 –f ''(ρ) / √[1 + f '(ρ)2]

K = det(S) / det(g) = f '(ρ) f ''(ρ) / [ρ (1 + f '(ρ)²)²)]

To produce a surface of constant Gaussian curvature, K, we need to solve the differential equation here for f(ρ). As a first step, we can rewrite this as:

–1/(2 ρ) d(1/(1 + f '(ρ)²))/dρ = K

This is straightfoward to solve for f '(ρ):

d(1/(1 + f '(ρ)²))/dρ = –2 K ρ
1/(1 + f '(ρ)²) = –K ρ² + c
f '(ρ) = ±√[1/(–K ρ² + c) – 1]

Here c is a constant of integration. The values that c can take such that the quantity inside the square root remains non-negative for some non-empty range of values for ρ will depend on the sign of K. The condition we need to satisfy is:

0 < –K ρ² + c < 1

That is to say, a parabola centred on ρ = 0 that either points down (if K > 0) or up (if K < 0), and which has a maximum or minimum value of c at ρ = 0, must lie between 0 and 1 for some non-empty range of values for ρ.

Suppose K = 1/a², i.e. the Gaussian curvature takes the positive value that corresponds to a sphere of radius a. Then c must be greater than zero, or the downwards-pointing parabola will never take on any positive values. It will be convenient to set c = χ². The range of values for ρ will then be:

Here f '(ρ) → ∞ as ρ → a χ, and f '(ρ) = 0 at ρ = a √[χ² – 1].

For negative curvature, say K = –1/a², we must have c less than 1, or the upwards-pointing parabola will never take on any values less than 1. In this case, we will set c = 1 – ξ², and the range of values for ρ will be:

We now have f '(ρ) → ∞ as ρ → a √[ξ² – 1], and f '(ρ) = 0 at ρ = a ξ.

We can integrate our result for f '(ρ) to find f(ρ). For the special case c = 1, which is only compatible with positive curvature, so we will set K = 1/a²:

Sphere

K = 1/a²    c = 1

f '(ρ) = ±ρ / √[a² – ρ²]

f(ρ) = z_eq ∓ √[a² – ρ²]

This is just the formula for two quadrants of a circle of radius a, and the resulting surface of revolution is a sphere of radius a, with the z coordinate of its equator given by the constant of integration z_eq.

For the special case c = 0, which is only compatible with negative curvature:

Tractroid

K = –1/a²    c = 0

f '(ρ) = ±√[a² – ρ²] / ρ

f(ρ) = z_eq ± (√[a² – ρ²] – a arccosh(a/ρ))

This function is called a tractrix, and its surface of revolution, a tractroid, is the most famous example of a pseudosphere: a surface of constant negative curvature.

If c ≠ 1 and c ≠ 0, we have:

Generic surface of revolution of constant Gaussian curvature K

f(ρ) = z_eq ± √[(1 – c)/K] (E(arcsin(√[K/c] ρ) | c/(c – 1)) – e₀)

If K > 0		e0 = E(c/(c – 1))
If K < 0		e0 = E(arcsin(√[(c – 1)/c]) | c/(c – 1))

The function E is an elliptic integral of the second kind, defined as:

E(s | m) = ∫₀^s √[1 – m sin² t] dt = ∫₀^{sin s} √[1 – m u²] / √[1 – u²] du
E(m) = E(½π | m)

The image on the right shows cross-sections through the surfaces of revolution of constant Gaussian curvature K = 1/a², for a variety of equatorial radii. Here, c = χ², and χ is the maximum value of ρ/a for each curve.

These curves all meet the horizontal axis with vertical tangents, so they can be smoothly extended by reflection in this axis; for example, the curve for χ = 1 will yield a complete sphere. However, the points at the top of the curve will (in all other cases besides the sphere) need to be excluded from the surface, with the “poles” for χ < 1 having a local topology like the tip of a cone, and the tops of the curves for χ > 1 giving rise to circular boundaries.

We can compute the lengths of these curves:

L = ∫_ρ1^ρ₂ √[1 + f '(ρ)²] dρ
= ∫_ρ1^ρ₂ 1/√[χ² – (ρ/a)²] dρ
= a arcsin(ρ/(a χ)) |_ρ1^ρ₂

So for χ ≤ 1, this is the same as the meridian from the equator to the pole on a sphere of radius a, while for χ > 1 it is shorter.

What about the surface area?

A = 2 π ∫_ρ1^ρ₂ ρ √[1 + f '(ρ)²] dρ
= 2 π ∫_ρ1^ρ₂ ρ/√[χ² – (ρ/a)²] dρ
= –2 π a √[a² χ² – ρ²] |_ρ1^ρ₂

For χ < 1, this is less than the hemisphere of a sphere of radius a, while for χ ≥ 1 it is exactly the same.

Now let’s compute geodesics on these surfaces. To start, note that the metric in (φ, ρ) coordinates can be found from our initial calculations, once we substitute the result for f '(ρ):

Because the metric is independent of the φ coordinate, we can find a family of geodesics by imposing the requirement that each such curve has a conserved quantity: the dot product of the unit-length tangent to the geodesic and the vector field that corresponds to a uniform increase in the φ coordinate. We won’t go through the derivation, but the results are easily checked. We claim that these curves are geodesics (for c ≠ 0):

Geodesics on generic surface of revolution

φ_G(s) = φ₀ + arctan((√[c/K] / ρ₀) tan(s √K)) / √c
ρ_G(s) = √[ρ₀² + (c/K – ρ₀²) sin²(s √K)]

To verify this, we compute the coordinates of the tangent to the geodesic:

φ_G'(s) = ρ₀ K / (ρ₀² K + (c – ρ₀² K) sin²(s √K))
ρ_G'(s) = (c – ρ₀² K) sin(s √K) / √[c tan²(s √K) + ρ₀² K]

Along these geodesics, the metric becomes:

and we have:

(φ_G'(s), ρ_G'(s))^T g(φ_G(s), ρ_G(s)) (φ_G'(s), ρ_G'(s)) = 1
(φ_G'(s), ρ_G'(s))^T g(φ_G(s), ρ_G(s)) (1, 0) = ρ₀

This confirms that the tangent has a length of 1, and that its dot product with the φ coordinate vector field is a conserved quantity, the constant ρ₀, that parameterises the family of geodesics.

For positive curvature we can write the geodesics as:

Geodesics on positive-curvature surface of revolution

K = 1/a²    c = χ²

Generic case (with s measured from point where ρ = ρ₀):
φ_G(s) = φ₀ + arctan((aχ/ρ₀) tan(s/a)) / χ
ρ_G(s) = √[ρ₀² + (a² χ² – ρ₀²) sin²(s/a)]

Meridians (with s measured from the equator):
φ_G(s) = φ₀
ρ_G(s) = a χ cos(s/a)

At s = 0, we have (for both positive and negative curvature):

φ_G(0) = φ₀
ρ_G(0) = ρ₀
φ_G'(0) = 1/ρ₀
ρ_G'(0) = 0

So at s = 0 the geodesic is tangent to a circle of latitude of radius ρ₀, and this is the minimum value of the ρ coordinate for the geodesic. However, for surfaces with χ > 1, if we choose a value for ρ₀ that is less than the minimum value for ρ on the entire surface, a √[χ² – 1], we still obtain a valid geodesic, but it only exists for values of s large enough to give valid values for ρ.

For the positive curvature case, we have:

φ_G((π/2) a) = φ₀ + π/(2 χ)
ρ_G((π/2) a) = a χ

This tells us that after we travel a distance, (π/2) a, that is the same as the equator-to-pole distance on a sphere of radius a, we arrive at the equator (the maximum value for ρ). The metric in (φ, ρ) coordinates is singular at the equator, but we can calculate the angle between the geodesic and the equator as a limit:

cosine of angle between geodesic and circle of latitude
= dot product between tangent to geodesic and unit vector in the φ direction
= (φ_G'(s), ρ_G'(s))^T g(φ_G(s), ρ_G(s)) (1/ρ_G(s), 0)
= ρ₀ / ρ_G(s)

cosine of angle between geodesic and equator (ρ = a χ)
= ρ₀ / (a χ)

By varying ρ₀, we can construct a family of geodesics that all pass through the same point on the equator at various angles.

The animation above shows surfaces of revolution, all with the same constant positive curvature, as the parameter χ cycles between 0.5 and 1.5. The two views show the same surface from the front and the back.

The Earth has been mapped to these surfaces by a simple isometry: when χ is less than 1, part of the equator of the Earth, and the full length of each meridian crossing it, is mapped to the equator and meridians on the surface; when χ is greater than 1, the full equator of the Earth, and part of the equator of a second copy of the Earth, and part of each meridian crossing them, is mapped to the equator and meridians on the surface. There is generally a discontinuity in the map along the midline of the view of the surface from behind, but this is not a boundary or singularity in the surface itself. As we mentioned earlier, the poles for the surfaces with χ less than 1 are singular points, in a similar way to the tip of a cone, and need to be excluded. For χ greater than 1 there are two circular boundaries where ρ hits a non-zero minimum value and the surface cannot be extended.

The red curves are geodesics, fanning out from a point on the equator and then reconverging. The geodesics are only drawn until they have gone halfway around the axis, or until they have reconverged, whichever is further. Regardless of the value of χ, the geodesics always reconverge after the same distance, π a, but for χ greater than 1 some of the geodesics hit the boundary of the surface and terminate there. Also, the distance π a will no longer correspond to half the length of the equator, and the second reconvergence, at a distance of 2π a, will no longer take place at the point where we started.

The equator is always a geodesic, but in general geodesics will only form closed loops of finite length when χ is rational, since each portion of a geodesic that lies north or south of the equator will span a longitude of π/χ. And even for rational values of χ, it might take a very large number of crossings of the equator before these changes in longitude sum to an integer multiple of 2π. This leads to the conclusion that, in general, the shortest geodesic path from a point P (not on the equator) that wraps around the surface and returns to P will not be part of a finite loop, which in turn means it must return to P with its tangent pointing in a different direction than when it began. The angle between the tangent to the geodesic and the circle of latitude that P lies on must have the same cosine whenever they cross, so if they are not the same angles, they must be opposites.

So, while the local geometry of these surfaces is exactly like that of a sphere, the global properties are not the same. Insects confined to a surface like this, but able to travel distances comparable to the dimensions of the whole surface, could certainly distinguish it from a sphere by suitable “experiments” with geodesics.

Now let’s consider the case of negative curvature.

The image on the right shows cross-sections through the surfaces of revolution of constant Gaussian curvature K = –1/a², for various equatorial radii. We have set c = 1 – ξ², and ξ is the maximum value of ρ/a for each curve.

All of these curves meet the horizontal axis with a slope of zero, so unlike the case with positive curvature, the surfaces cannot be smoothly extended by reflection here. For ξ < 1, the surfaces will have a conical tip at the point of maximum z; for ξ = 1, the tractrix extends to infinity in the z direction. For ξ > 1, the curves all have a vertical tangent at the minimum radius, ρ = a √[ξ² – 1], so they can be extended smoothly there by reflection in a horizontal line at the maximum z value.

The lengths of these curves are:

L = ∫_ρ1^ρ₂ √[1 + f '(ρ)²] dρ
= ∫_ρ1^ρ₂ 1/√[1 – ξ² + (ρ/a)²] dρ
= a arctanh(ρ/√[ρ² + a²(1 – ξ²)]) |_ρ1^ρ₂

The area of the surface of revolution is:

A = 2 π ∫_ρ1^ρ₂ ρ √[1 + f '(ρ)²] dρ
= 2 π ∫_ρ1^ρ₂ ρ/√[1 – ξ² + (ρ/a)²] dρ
= 2 π a √[ρ² + a² (1 – ξ²)] |_ρ1^ρ₂

For ξ < 1 this is less than the hemisphere of a sphere of radius a, while for ξ ≥ 1 it is exactly the same.

For negative curvature we can write the geodesics as:

Geodesics on negative-curvature surface of revolution

K = –1/a²    c = 1 – ξ²

Generic case (with s measured from point where ρ = ρ₀):
φ_G(s) = φ₀ + arctan(√[1 – ξ²] (a/ρ₀) tanh(s/a)) / √[1 – ξ²]
ρ_G(s) = √[ρ₀² + (ρ₀² + a² (1 – ξ²)) sinh²(s/a)]

Asymptotes to waist (for ξ > 1 only, with s measured from the equator):
φ_G(s) = φ₀ + (½log(1 + (ξ² – 1) exp(2s/a)) – log(ξ)) / √[ξ² – 1]
ρ_G(s) = a √[ξ² – 1 + exp(–2s/a)]

Meridians:
φ_G(s) = φ₀

• for ξ < 1 (with s measured from the pole):
  ρ_G(s) = a √[1 – ξ²] sinh(s/a)
• for ξ > 1 (with s measured from the “waist”):
  ρ_G(s) = a √[ξ² – 1] cosh(s/a)

We can also compute the special case c = 0, ξ = 1, for geodesics on the tractroid:

Geodesics on tractroid

K = –1/a²    c = 0

Generic case (with s measured from point where ρ = ρ₀):
φ_G(s) = φ₀ + (a/ρ₀) tanh(s/a)
ρ_G(s) = ρ₀ cosh(s/a)

Meridians (with s measured from the equator):
φ_G(s) = φ₀
ρ_G(s) = a exp(–s/a)

Polar geodesic coordinate systems

If we pick a point on the tractroid, say (ρ, φ) = (ρ_C, φ_C),
we can construct a set of geodesics that all pass through
that point, and whose tangents at that point are parameterised
by an angle θ:

φ_G(s) = φ_C + a cos(θ) sinh(s/a) / [ρ_C (cosh(s/a) – sin(θ) sinh(s/a))]
ρ_G(s) = ρ_C (cosh(s/a) – sin(θ) sinh(s/a))

The arc length parameter s is measured from the chosen point,
and the angle θ is measured from a horizontal tangent.

As with the positively curved surfaces, in the generic case these geodesics are tangent, at s = 0, to a circle of latitude of radius ρ₀, and this is the minimum value of the ρ coordinate for the geodesic. If ξ > 1 and we choose ρ₀ to be less than the minimum value for ρ on the entire surface, a √[ξ² – 1], we still obtain a valid geodesic, but it only exists for values of s large enough to give valid values for ρ.

However, unlike the positive curvature case, all geodesics on a given surface do not take the same distance from their point of minimum ρ before they reach the equator. And the equator itself is not a geodesic; its geodesic curvature (the curvature in terms of the intrinsic geometry, relative to the geodesics, which are considered to be straight) is the same as its curvature in three-dimensional Euclidean space: 1/(a ξ), the reciprocal of the radius of the equatorial circle.

For ξ ≤ 1, most geodesics will start and end on the equator. Meridians will start on the equator and end on the pole, or for ξ = 1 will continue to infinity.

For ξ > 1, the surface has a “waist,” a circle of minimum ρ where ρ = a √[ξ² – 1]. This is clearly a geodesic by symmetry, and if we insert this value for ρ₀ into the formula for the generic ρ_G(s), it becomes constant. Geodesics with ρ₀ < a √[ξ² – 1] will cross the waist, and have endpoints on the two distinct copies of the equator. Geodesics with ρ₀ > a √[ξ² – 1] will never cross the waist, and will start and end on the same copy of the equator. Geodesics with ρ₀ = a √[ξ² – 1] that do not start on the waist are asymptotic to it.

As before, the cosine of the angle between a geodesic and any circle of latitude will be equal to ρ₀ / ρ_G(s), so the cosine of the angle between a geodesic and the equator is ρ₀ / (a ξ). For ξ > 1 the geodesics that cross the waist will be those that make angles greater than arccos √[1 – 1/ξ²] with the equator. If a geodesic leaves the equator at precisely this angle, it will approach the waist asymptotically, never reaching it but also never returning to the equator.

The animation above shows front and back views of surfaces of revolution with K = –1, with the parameter ξ ranging from 1.10125 to 1.70728. These values have no special significance in terms of the properties of the surface, but they were chosen so that the particular tiling of the hyperbolic plane used to illustrate the surface wraps around neatly at the two endpoints. In this tiling, seven equilateral triangles meet at a point, and the equilateral triangles are further subdivided into six right-angled triangles. This tiling can’t exist in the Euclidean plane, where the angles at each vertex of an equilateral triangle are always π/3 radians, but in the hyperbolic plane the sum of the angles of a triangle will be less than π by an amount that depends on the area of the triangle, as we discussed earlier. For these equilateral triangles with angles of 2π/7 at each vertex, for a total of 6π/7, the area is π/7.

The total area of each of the surfaces in the animation is 4π. The isometries used to map the hyperbolic plane to these surfaces take a fixed geodesic in the hyperbolic plane to the circle around the waist of the surface, and then map geodesics orthogonal to the first one to meridians on the surface.

The images on the left and below show some examples of these surfaces rotating, to give a clearer sense of how the negatively curved geometry fits into three-dimensional Euclidean space.

The animation above shows surfaces with K = –1 and the parameter ξ ranging from (√13)/7 ≈ 0.515 to (√45)/7 ≈ 0.958. Each surface corresponds to a circular wedge of the hyperbolic plane, with a wedge angle of 2 π √[1 – ξ²], a radius of arctanh(ξ), and a total surface area of 2 π (1 – √[1 – ξ²]). The wedge angles here range from 6 × 2π/7 to 2 × 2π/7, as can be seen by the numbers of triangles from the tiling that fit around the pole.

The tractroid, with ξ = 1, can be viewed as a limiting case of this family of conelike surfaces, where the height of the cone goes to infinity while the wedge angle goes to zero, in such a way that the total area of the surface remains finite at 2π a². Because the distance from the pole to the equator goes to infinity, any real-world example of the tractroid will just be a portion of the whole surface, in the vicinity of the equator rather than the pole.

The equator of the tractroid is not a geodesic; it corresponds to a finite portion of an infinite curve in the hyperbolic plane known as a horocycle, which can be thought of as a kind of circle whose centre lies, not in the hyperbolic plane itself, but “at infinity.” In hyperbolic geometry, a point like this is known as an ideal point. In the Poincaré disk model of the hyperbolic plane, the plane is represented by the interior of a disk, while the ideal points lie on the boundary of the disk, and horocycles are represented by circles that are tangent to the boundary circle.

The meridians of the tractroid are geodesics, meeting the horocycle of the equator orthogonally, and converging on a common ideal point. Because there are no pairs of geodesics from the edges in our tiling that converge on an ideal point, there is necessarily a visible “seam” in the pattern along one meridian of the tractroid in the image on the right.

The image on the left shows the outlines of regions of the hyperbolic plane, in the Poincaré disk model, that are mapped to the negatively curved surfaces we have illustrated.

The dashed white shapes are the two extremes of the wedges mapped to the cone-like surfaces with ξ < 1.

The solid white line shows the region mapped to the tractroid. The large arc on the left, corresponding to the equator of the tractroid, is part of a horocycle; the small arc on the right, corresponding to the circle of latitude where we chose to truncate the tractroid, is part of another horocycle.

The black shapes are three regions mapped to the hourglass-like surfaces with ξ > 1. The curves that cross the x-axis are geodesics, corresponding to meridians of the hourglass, while the upper and lower boundaries are hypercycles (curves at a fixed distance from a geodesic, in this case the x-axis), corresponding to the circles of latitude of maximum ρ. The x-axis itself corresponds to the waist of the hourglass.

Surfaces Built From Helices

In the previous section, we constructed surfaces of revolution composed of circles of varying radius, ρ, with their z coordinate (their location along the axis of revolution) given by a function, f(ρ), which satisfied a differential equation that ensured constant Gaussian curvature.

One way to generalise this construction, while still working with just one function of a single variable, is to replace all the circles with helices of a specified pitch, controlled by a parameter b that is a constant for each surface. The parameterisation of the surface then takes the form:

x(φ, ρ) = (ρ cos φ, ρ sin φ, f(ρ) + b φ)

The surfaces of revolution correspond to b = 0. Computing the Gaussian curvature:

x_φ = (–ρ sin φ, ρ cos φ, b)
x_ρ = (cos φ, sin φ, f '(ρ))
n = x_φ × x_ρ / |x_φ × x_ρ| = (ρ f '(ρ) cos φ – b sin φ, b cos φ + ρ f '(ρ) sin φ, –ρ) / √[b² + ρ² (1 + f '(ρ)²)]
x_φφ = (–ρ cos φ, –ρ sin φ, 0)
x_φρ = (–sin φ, cos φ, 0)
x_ρρ = (0, 0, f ''(ρ))

g(φ, ρ)	=	xφ · xφ xφ · xρ xφ · xρ xρ · xρ	=	ρ2 + b2 b f '(ρ) b f '(ρ) 1 + f '(ρ)2
S(φ, ρ)	=	xφφ · n xφρ · n xφρ · n xρρ · n	=	–ρ2 f '(ρ) / √[b2 + ρ2 (1 + f '(ρ)2)] b / √[b2 + ρ2 (1 + f '(ρ)2)] b / √[b2 + ρ2 (1 + f '(ρ)2)] –ρ f ''(ρ) / √[b2 + ρ2 (1 + f '(ρ)2)]

K = det(S) / det(g) = (ρ³ f '(ρ) f ''(ρ) – b²) / (b² + ρ² (1 + f '(ρ)²))²)

We can rewrite this differential equation in f as:

q(ρ) = (b² + ρ² (1 + f '(ρ)²))^–1
–½ ρ q'(ρ) – q(ρ) = K

which is solved by:

q(ρ) = c / ρ² – K
f '(ρ) = ±√[1/(–K ρ² + c) – (1 + b²/ρ²)]

Here c is a constant of integration, and the result for f '(ρ) agrees with that for a surface of revolution if we set b = 0.

For the quantity inside the square root to be finite and non-negative we need:

0 < –K ρ² + c < ρ² / (ρ² + b²)
and ρ > 0

The right-hand inequality in the first line above becomes an equality when ρ² is a root of the equation:

K (ρ²)² + (K b² – c + 1) ρ² – b² c = 0

In everything that follows, we will assume that b > 0. For the case of positive curvature, we will set K = 1/a² and c = χ², since we need c > 0 in order to have a non-empty range for ρ:

½(d + √[d² + 4 a² b² χ²]) < ρ² < (a χ)²
where d = a² (χ² – 1) – b²

So we have the same maximum value for ρ as for the positive-curvature surfaces of revolution, but now we always have a non-zero minimum value for ρ, regardless of the value of χ. The derivative f '(ρ) approaches infinity at the outer limit for ρ, and zero at the inner limit.

For negative curvature, we will set K = –1/a² and c = 1 – ξ², since we need c < 1 in order to have a non-empty range for ρ:

If 0 < ξ ≤ 1 and b < a (1 – √[1 – ξ2])		½(d – √[d2 + 4 a2 b2 (ξ2 – 1)]) < ρ2 < ½(d + √[d2 + 4 a2 b2 (ξ2 – 1)])
If ξ > 1		a2 (ξ2 – 1) < ρ2 < ½(d + √[d2 + 4 a2 b2 (ξ2 – 1)])

where d = a² ξ² – b²

In the first case, f '(ρ) is zero at two positive roots for ρ², and the upper bound we have placed on b is needed in order for those roots to be real and positive. In the second case, f '(ρ) approaches infinity at the inner limit for ρ, and zero at the outer limit.

In general we will have a non-zero minimum value for ρ, but if ξ = 1 the limits become:

If ξ = 1 and b < a

0 < ρ2 < a2 – b2

Dini’s surface

We will integrate f '(ρ) first for the special case ξ = 1 (i.e. c = 0). This yields a tractroid when b = 0, and in fact the integral is identical apart from changing the constant a in the previous formula to √[a² – b²]:

Dini’s surface

K = –1/a²    c = 0    b < a    ρ_m = √[a² – b²]

f '(ρ) = ±√[ρ_m² – ρ²] / ρ

f(ρ) = z_ρm ± (√[ρ_m² – ρ²] – ρ_m arccosh(ρ_m/ρ))

The result we obtain by substituting this solution into our parameterisation of helical surfaces is known as Dini’s surface.

Note that the parameterisations for this family of surfaces in the literature can look very different from the one we have used. For example, MathWorld gives the parameterisation:

(x, y, z) = (a cos u sin v, a sin u sin v, a [cos v + log(tan ½v)] + b u)

which has a Gaussian curvature of –1/(a² + b²) rather than –1/a², and while u here is the same as our φ, the coordinate v is analogous to the colatitude θ in spherical polar coordinates, and our ρ coordinate is proportional to sin v. But if we replaced a here with √[a² – b²] this would describe exactly the same surface as ours.

Because the metric for all the surfaces built from helices is independent of the φ coordinate, as it was for the surfaces of revolution, we again have a conserved quantity along each geodesic: the dot product between the unit-length tangent to the geodesic and the vector (1,0), which represents a uniform change of φ. The difference is that, this time, rather than pointing around in a circle, the vector for a change in φ points along the helices from which the surface is built.

For Dini’s surface, the formulas for ρ in the geodesics take a very similar form to that of geodesics on the tractroid, but the formulas for φ are considerably more complicated. We don’t really have “meridians” anymore, but the geodesics can be grouped into three classes: those which have a minimum ρ value, ρ₀ > 0, those where ρ is only bounded below by zero, and are of finite length, and those which are of infinite length and approach ρ = 0 asymptotically.

Geodesics on Dini’s surface

K = –1/a²    c = 0    b < a    ρ_m = √[a² – b²]

Geodesics with lower bound ρ₀ on ρ (with s measured from the point where ρ = ρ₀):
ρ_G(s) = ρ₀ cosh(s/a)
φ_G(ρ) = φ₀ + (ρ_m/b) log( (ρ_m + √[ρ_m² – ρ²]) (ρ_m – √[ρ_m² – ρ₀²]) / (ρ ρ₀))
   + ½ (a/b) log(
      (a – √[ρ_m² – ρ²]) (a + √[ρ_m² – ρ₀²]) (ρ₀² + b (b – √[b² + ρ₀²])) (ρ (ρ – √[ρ² – ρ₀²]) + (b (b + √[b² + ρ₀²]))) /
      [(a + √[ρ_m² – ρ²]) (a – √[ρ_m² – ρ₀²]) (ρ₀² + b (b + √[b² + ρ₀²])) (ρ (ρ – √[ρ² – ρ₀²]) + (b (b – √[b² + ρ₀²])))] )
The angle between the geodesic and the helix ρ = ρ_m is acos(√[b² + ρ₀²]/a).

Geodesics where ρ goes to zero in a finite distance (with s measured from the point where ρ = ρ_m):
ρ_G(s) = ρ_m cosh(s/a) – √[a² – γ²] sinh(s/a)
φ_G(ρ) = φ₀ + (ρ_m/b) arctanh(√[1 – (ρ/ρ_m)²]) – (a/b) arccoth(a/√[ρ_m² – ρ²])
   – ½ (a/b) log(
      (ρ_m √[a² – γ²] + b γ – a²) (b (b + γ) + ρ (ρ – √[ρ² + b² – γ²])) /
      [(ρ_m √[a² – γ²] – b γ – a²) (b (b – γ) + ρ (ρ – √[ρ² + b² – γ²]))] )
Here γ is a parameter, 0 < γ < b. The angle between the geodesic and the helix ρ = ρ_m is acos(γ/a).

Geodesics where ρ goes to zero asymptotically (with s measured from the point where ρ = ρ_m):
ρ_G(s) = ρ_m exp(–s/a)
φ_G(ρ) = φ₀ + (a/b) log(ρ_m(a – √[ρ_m² – ρ²])/(a ρ)) + (ρ_m/b) log((ρ_m + √[ρ_m² – ρ²])/ρ)
The angle between the geodesic and the helix ρ = ρ_m is acos(b/a).

It might be a bit surprising, at first, to realise that there are geodesics on Dini’s surface that travel between the outer helix, with ρ = ρ_m, and points where ρ = 0, in a finite distance. On the tractroid, the equator lies at one end of an infinitely long funnel that takes an infinite distance to taper down to zero width. But although there is a contribution to the z coordinate of Dini’s surface that grows infinitely negative as ρ approaches zero, these finite-length geodesics counterbalance that with the contribution from φ approaching positive infinity, so they terminate at a finite value for z.

We can measure the width of Dini’s surface by the length of the orthogonal geodesics between the outer helix ρ = ρ_m and the line ρ = 0; these are the geodesics where we set γ = 0, and their length is a arccosh(a/b). The surface can be continued infinitely far in the positive and negative φ directions, along the helices.

We can construct an isometry between Dini’s surface and an infinitely long strip within the hyperbolic plane, bounded on one side by a geodesic, and on the other by a hypercycle, a curve that lies at a fixed distance from that geodesic. The hypercycle corresponds to the outer helix of Dini’s surface, and the geodesic corresponds to the line ρ = 0. In the Poincaré disk model, a hypercycle is represented by a circular arc that meets the boundary of the disk at the same points as its associated geodesic, while the geodesic itself is either a diameter of the disk, or a circular arc that meets the boundary orthogonally.

The image on the right shows a finite piece of Dini’s surface, with a = 1, b = 1/4 and –3π ≤ φ ≤ 3π along the outer helix. We have cut the surface along two geodesics orthogonal to the outer helix, but to be clear, these are not lines of constant φ. The isometry that we’ve used here maps the x-axis of the Poincaré disk model to the line ρ = 0, while the outer helix on the surface corresponds to a circular arc in the upper half of the Poincaré disk that passes through the points (–1, 0), (0, √[(1–b)/(1+b)]) and (1, 0). The tiling is the same one we used for all the negatively-curved surfaces in the previous section.

The image on the left shows how the shape of Dini’s surface changes as we vary b, which cycles here between 1/12 and 1/4. For a given range of φ (measured along the outer helix), the surface area is equal to Δφ √[1–b²], so as the pitch of the helices decreases, the area increases. The precise shape of the corresponding region in the hyperbolic plane changes too; while the length of the arc of the hypercycle remains equal to Δφ, the length of the portion of the geodesic along the x-axis at the bottom of the region is Δφ b, while the length of the geodesics joining those two boundaries is arccosh(1/b).

In the limit as b approaches zero, each loop of the outer helix of Dini’s surface approaches the same arc of a horocycle as the equator of the tractroid. But so long as b does not actually reach zero, we can continue to include as many loops as we wish without the surface self-intersecting.

It’s impossible to embed the entire hyperbolic plane in three-dimensional Euclidean space, but the parameters for Dini’s surface can be chosen so that it contains a disk of any finite radius R, while its Gaussian curvature remains fixed at –1. If we set:

a = 1
b = 1/cosh(2R)
Δφ = 2 cosh(2R) arccosh(√[1 + tanh²(R)])

then a disk of radius R will fit inside the surface, with the outer helix, the centreline, and the orthogonal geodesics that leave the outer helix at the minimum and maximum values of φ, all being tangent to the disk.

Other Surfaces Built From Helices

We will now consider the general case of a surface of constant Gaussian curvature constructed from helices, where c ≠ 0 and b ≠ 0. Although it is possible to write down a closed form for the function f(ρ) in terms of elliptic integrals, the expression is so complicated that it is not very enlightening, and it is generally more practical to use direct numerical integration of f '(ρ) to construct the solution.

The two plots on the right give examples of f(ρ) for positive and negative curvature, respectively. The green component of each curve’s colour is determined by the value of the parameter we use to set c, while the red component depends on b.

For positive curvature (top right), the maximum value of ρ/a is χ, so it is independent of b, and each family of curves where χ is fixed and b varies meet the horizontal axis at the same point. Increasing b results in a larger lower bound on ρ, shrinking the overall range of values for ρ, and also a lower maximum value for f(ρ).

All of these curves are vertical (i.e. f '(ρ) is infinite) at their maximum value of ρ, so they can be smoothly extended by reflection in the horizontal axis.

For negative curvature (bottom right), the curves with ξ ≥ 1 all share a common minimum value for ρ/a of √[ξ² – 1], while the maximum value for ρ/a starts at ξ when b = 0 but then falls as b increases. The curves with ξ > 1 become vertical at their minium ρ value, so as with the hourglass-shaped surfaces of rotation we can smoothly extend them by reflection in a horizontal line through that point.

For curves with ξ < 1 both the minimum and maximum values of ρ depend on b as well as ξ, but there is also a ξ-dependent upper bound on the allowed values for b/a of:

b/a < 1 – √[1 – ξ²]

For ξ = 0.8 this bound is 0.4, so there are three curves plotted, for b/a equal to 0, 1/5 and 1/3. For the lower values of ξ, only b = 0 is plotted.

No curves with ξ < 1 are vertical at either endpoint, so they cannot be extended by reflection.

In all cases, increasing b shrinks the range of values for ρ, and in almost all cases it also lowers the maximum value for f(ρ). The only exception is when ξ = 1, which is Dini’s surface, where f(ρ) goes to infinity as ρ approaches zero.

To set up isometries between these helical surfaces and the sphere or hyperbolic plane, it is helpful to know the geodesic curvature, κ_g, of the helices. This can be computed as the scalar triple product between the unit-length tangent to the helix, t(s), its rate of change with respect to arc length along the helix, t'(s), and the unit normal to the surface^[3]. This turns out to be:

κ_g = √[(c – K ρ²) / (b² + ρ²)]

Note that when the numerator here is zero, the derivative f '(ρ) of the helix function is infinite, which occurs at the maximum value of ρ for all surfaces of positive curvature, and the minimum value of ρ for surfaces of negative curvature with ξ>1. So in those cases, the widest or narrowest helix on the surface is a geodesic.

The image above shows parts of the surfaces of constant positive curvature with a = 1, χ = 1, and b ranging from 0.01 to 0.5, as seen from two opposite viewpoints. In each case the surface consists of portions of three copies of the sphere, with the full length of the equator of all three spheres joined together to form the “equatorial helix” (the helix of maximum ρ), while the range of latitude varies depending on the value of b.

For low values of b, the surface is self-intersecting. The critical value for b occurs when:

|f(ρ_max) – f(ρ_min)| = π b

For a = 1, χ = 1, this can be found numerically to be b_crit ≈ 0.2629.

The red curves in the image are geodesics fanning out from a single point on the equatorial helix, some of which hit the boundary of the surface along helices corresponding to circles of latitude, while others reconverge in the usual manner, after integer multiples of the distance π a. Because of the symmmetry of the situation, the bundle of geodesics that avoid hitting the boundary the first time will continue to do so, and on the idealised version of the surface that continues indefinitely, they will reconverge an infinite number of times at different points. For the lowest value of b at which the surface is non-self-intersecting, a fan of geodesics about 116° wide centred on the equatorial helix will reconverge without hitting the boundary.

The image above shows parts of the surfaces of constant negative curvature with a = 1, ξ = √[3/2] (i.e. c = 1 – ξ² = – ½), and b ranging from 0.01 to 0.5, as seen from two opposite viewpoints. Here we have held the total length of the centreline of the ribbon constant, but the width of the ribbon in each case is the maximum possible, which becomes slightly less as b increases.

Finally, if K is negative and 0 < c < 1, we get a helical version of the cone-shaped surfaces we obtained for the same parameters when b = 0. The image above shows parts of the surfaces of constant negative curvature with a = 1, ξ = √[1/2] (i.e. c = 1 – ξ² = ½), and b ranging from 0.01 to 0.292893, as seen from two opposite viewpoints. This ribbon corresponds to an annulus in the hyperbolic plane, and although we can extend its length indefinitely, we will need multiple copies of the hyperbolic plane (illustrated here by changing the shading of the triangular tiling), as any isometry will identify different portions of the ribbon with the same annulus. Here, we show the ribbon obtained from exactly three copies of the annulus.

As with the other helical surfaces, increasing b shrinks the width of the ribbon, but in this case there is a finite upper bound on b of a(1 – √c), at which the width reaches zero.

The sine-Gordon Equation

Suppose we have a surface with coordinates u, v that have been chosen so that the metric for the surface takes the form:

g(u, v)

=

1 cos(ω(u, v)) cos(ω(u, v)) 1

where ω(u, v) is some as yet unspecified function of the coordinates, and for now we just require that it obeys 0 < ω(u, v) < π in the region of the surface of interest to us. Geometrically, ω(u, v) measures the angle between the curves of the coordinate grid, where we hold u constant and vary v, or vice versa. The diagonal components of the metric being 1 means that the coordinate values along these grid lines are actual distances.

If we compute the Gaussian curvature of the surface, using the method described in the appendix, this turns out to be:

K(u, v) = –∂_u,vω(u, v) / sin(ω(u, v))

If the surface has a constant Gaussian curvature K, this gives us partial differential equation that ω(u, v) must satisfy:

∂_u,vω(u, v) = –K sin(ω(u, v))

The equation for K = –1:

The sine-Gordon equation

∂_u,vω(u, v) = sin(ω(u, v))

is known as the sine-Gordon equation. (The current name for this equation is a pun on another equation, the Klein–Gordon equation, which was actually named much later. The sine-Gordon equation was first studied in the 1860s by Edmond Bour, in relation to surfaces of constant negative curvature, whereas the Klein–Gordon equation was developed in 1926, and describes the wave function of a relativistic particle with zero spin.)

Before searching directly for solutions of the sine-Gordon equation, we should check if any solutions are hiding in the surfaces of constant curvature we have already studied. We haven’t seen any metrics of the form shown above, but there is a way of taking a metric in coordinates x, y that is diagonal and depends only on one coordinate, say y, and obtaining new coordinates in which the metric has diagonal components of 1.

Suppose we are able to find a function F(s) that satisfies the following equation, where g(y) is the metric on the surface as a function of the y coordinate:

(1, F '(s))^T g(F(s)) (1, F '(s)) = L²

Because the metric is diagonal, the left-hand side, when multiplied out, only involves F '(s)², so it will also be true that:

(1, –F '(s))^T g(F(s)) (1, –F '(s)) = L²

Now, define new coordinates on the surface, u, v, such that:

x(u, v) = (u + v)/L
y(u, v) = F((u – v)/L)

This gives us:

∂_u (x(u, v), y(u, v)) = (1 / L, F '((u – v)/L) / L)
∂_v (x(u, v), y(u, v)) = (1 / L, –F '((u – v)/L) / L)

The metric then has diagonal components in the new coordinates of:

g_uu = 1
g_vv = 1

We can apply this method to the metric for the tractroid in (φ, ρ) coordinates, which depends only on the ρ coordinate:

g(ρ)

=

ρ2 0 0 1 + f '(ρ)2

=

ρ2 0 0 a2 / ρ2

This gives us the equation for F(s):

F(s)² + a² F '(s)² / F(s)² = L²

If we set L = 1 and a = 1, this is solved by:

F(s) = sech(s + C)
F '(s) = –sech(s + C) tanh(s + C)

We then have :

ρ = F(u – v)
g_uv = (1, F '(u – v))^T g(F(u – v)) (1, –F '(u – v))
= F(u – v)² – F '(u – v)² / F(u – v)²
= 2 sech(u – v + C)² – 1

ω(u, v) = acos(2 sech(u – v + C)² – 1)

This function ω(u, v) is a solution for the sine-Gordon equation:

sin(ω(u, v)) = √[1 – cos(ω(u, v))²]
= √[4 sech(u – v + C)² – 4 sech(u – v + C)⁴]
= 2 sech(u – v + C) √[1 – sech(u – v + C)²]
= 2 sech(u – v + C) tanh(|u – v + C|)

∂_u ω(u, v) = –4 sech(u – v + C)² tanh(u – v + C) /√[1 – cos(ω(u, v))²]
= –2 sign(u – v + C) sech(u – v + C)

∂_u,v ω(u, v) = 2 sech(u – v + C) tanh(|u – v + C|)
= sin(ω(u, v))

We can take our original parameterisation of the tractroid in three-dimensional space and rewrite it in terms of these new coordinates. For the sake of simplicity, we will continue to set a = 1, and we will set C = 0.

x(u, v) = (sech(u–v) cos(u+v), sech(u–v) sin(u+v), tanh(u–v) – (u–v))

x_u = (–sech(u–v) (sin(u+v) + cos(u+v) tanh(u–v)), sech(u–v) (cos(u+v) – sin(u+v) tanh(u–v)), –tanh(u–v)²)
x_v = (sech(u–v) (–sin(u+v) + cos(u+v) tanh(u–v)), sech(u–v) (cos(u+v) + sin(u+v) tanh(u–v)), tanh(u–v)²)

n = x_u × x_v / |x_ω × x_v| = sign(u–v) (cos(u+v) tanh(u–v), sin(u+v) tanh(u–v), –sech(u–v))

x_uu = (–2 cos(u+v) sech(u–v)³ + 2 sin(u+v) sech(u–v) tanh(u–v), –2 sin(u+v) sech(u–v)³ – 2 cos(u+v) sech(u–v) tanh(u–v), –2 sech(u–v)² tanh(u–v))
x_uv = (–2 cos(u+v) sech(u–v) tanh(u–v)², –2 sin(u+v) sech(u–v) tanh(u–v)², 2 sech(u–v)² tanh(u–v))
x_vv = (–2 cos(u+v) sech(u–v)³ – 2 sin(u+v) sech(u–v) tanh(u–v), –2 sin(u+v) sech(u–v)³ + 2 cos(u+v) sech(u–v) tanh(u–v), –2 sech(u–v)² tanh(u–v))

g(u, v)	=	xu · xu xu · xv xu · xv xv · xv	=	1 2 sech(u–v)2 – 1 2 sech(u–v)2 – 1 1
S(u, v)	=	xuu · n xuv · n xuv · n xvv · n	=	0 –2 sech(u–v) tanh(|u–v|) –2 sech(u–v) tanh(|u–v|) 0

K = det(S) / det(g) = –1

The metric here is just what we would expect, but the second fundamental form, S(u, v), tells us something new and interesting: the tangents to the u and v coordinate grid lines give directions in which the surface has zero curvature:

(1,0)^T S(u, v) (1,0) = 0
(0,1)^T S(u, v) (0,1) = 0

If we slice the surface at any point P with a plane that contains the normal to the surface and the tangent to either of these grid lines, the intersection of the plane with the surface will have a curvature of zero at P. Curves like these grid lines are known as asymptotic curves or asymptotic lines,^[8] because their tangents point along the asymptotes of the hyperbolas that appear in plots of the kind shown at the start of the appendix, in the case of negative Gaussian curvature. The directions of maximum positive and negative curvature lie midway between the two asymptotes.

Any surface with negative Gaussian curvature will have two families of asymptotic curves, since at every interior point on the surface there must be two asymptotic directions, where the curvature of a normal section changes from positive to negative. Starting at any point, we can keep following one of these directions, integrating the vector field to trace out a curve.

The images on the right show parts of a tractroid, with a grid using the u, v coordinates derived above, and of Dini’s surface with the same kind of coordinates. Note that at the boundary, the metric in these coordinates becomes degenerate, with both grid lines pointing in the same direction.

For Dini’s surface, the parameterisation is:

x(u, v) = (√[1–b²] sech(U(u,v)) cos(u+v), √[1–b²] sech(U(u,v)) sin(u+v), √[1–b²] tanh(U(u,v)) – (u–v))

where U(u,v) = √[(1+b)/(1–b)] u – √[(1–b)/(1+b)] v

This parameterisation has the same properties as the u, v coordinates on the tractroid, and it is not a coincidence that the grid lines for a coordinate system where the metric has diagonal components of 1 are also asymptotic curves in both of these embeddings of the hyperbolic plane. But how can we show that these two properties will occur together more generally?

Suppose we have a surface embedded in three-dimensional space, with a coordinate system that gives it a metric with 1s on the diagonal:

g(u, v)

=

xu · xu xu · xv xu · xv xv · xv

=

1 cos(ω(u, v)) cos(ω(u, v)) 1

while the second fundamental form is:

S(u, v)

=

xuu · n xuv · n xuv · n xvv · n

=

L(u, v) M(u, v) M(u, v) N(u, v)

where the functions L(u, v), M(u, v), N(u, v) are yet to be determined.

By looking at the components of the metric and taking derivatives with respect to u and v, we obtain:

x_u · x_u = 1
x_uu · x_u = 0
x_uv · x_u = 0

x_v · x_v = 1
x_uv · x_v = 0
x_vv · x_v = 0

x_u · x_v = cos(ω(u, v))
x_uu · x_v = –sin(ω(u, v)) ∂_uω(u, v)
x_vv · x_u = –sin(ω(u, v)) ∂_vω(u, v)

From these dot products, and the definition of the second fundamental form’s components, we have:

x_uu = (cos(ω(u, v)) x_u – x_v) ∂_uω(u, v) / sin(ω(u, v)) + L(u, v) n
x_vv = (cos(ω(u, v)) x_v – x_u) ∂_vω(u, v) / sin(ω(u, v)) + N(u, v) n
x_uv = M(u, v) n

Suppose we now calculate x_uuv · n and x_uvv · n. Starting from the equations above, in each case we have two different ways to perform the calculation that must agree, because the order in which we take partial derivatives makes no difference.

(∂_v x_uu) · n = (∂_u x_uv) · n
(cos(ω(u, v)) M(u, v) – N(u, v)) ∂_uω(u, v) / sin(ω(u, v)) + ∂_vL(u, v) = ∂_uM(u, v)

(∂_u x_vv) · n = (∂_v x_uv) · n
(cos(ω(u, v)) M(u, v) – L(u, v)) ∂_vω(u, v) / sin(ω(u, v)) + ∂_uN(u, v) = ∂_vM(u, v)

These two equations are known (in the general case, without specialising to the form of the metric we have imposed), as the Mainardi-Codazzi equations^[9]. They must be satisfied by the components of the metric and the second fundamental form in order for them to be compatible, arising from the same surface.

A family of solutions that satisfies both these equations is:

L(u, v) = 0
N(u, v) = 0
M(u, v) = α sin(ω(u, v))

where α can take any value. But if we want the Gaussian curvature of the surface to be a constant, K, we must have:

K = det(S) / det(g) = –M(u, v)² / sin(ω(u, v))² = –α²

So the curvature must be negative, and for K = –1 we must have M(u, v) = ±sin(ω(u, v)). And we have already seen that ω(u, v) must satisfy the sine-Gordon equation.

We have not proved that there are no other solutions of the Mainardi-Codazzi equations, but this calculation shows that the two properties of the coordinate grid we noted in our example are compatible. This suggests that we ought to be able to find embeddings of the hyperbolic plane in three-dimensional space by the following strategy:

• Find solutions ω(u, v) of the sine-Gordon equation with 0 < ω(u, v) < π.
• Construct a surface in three dimensions whose u, v coordinate grid lines are asymptotic curves, making an angle of ω(u, v) when they cross.

In order to perform the second step, one approach would be to find a way to construct the coordinate grid lines as curves in three-dimensional space. If we know the curvature, κ(s), and torsion, τ(s), of any curve as a function of its arc length parameter s, then the Frenet-Serret formulas let us construct the curve, up to an overall rotation and translation.

The magnitude of the curvature κ is the magnitude of the derivative of a unit-length tangent to the curve, with respect to arc length. Our coordinates u, v are already arc length parameters for the coordinate grid lines, so the curvatures in question are:

|κ₁| = |x_uu| = |(cos(ω(u, v)) x_u – x_v) ∂_uω(u, v) / sin(ω(u, v))|
= |∂_uω(u, v)|

|κ₂| = |x_vv| = |(cos(ω(u, v)) x_v – x_u) ∂_vω(u, v) / sin(ω(u, v))|
= |∂_vω(u, v)|

The magnitude of the torsion of a curve is the magnitude of the derivative of a unit normal to the plane spanned by the curve’s tangent and the tangent’s derivative, with respect to arc length. In our case, the unit normal to that plane is just the unit normal to the surface itself, since we have:

x_u · n = x_uu · n = 0
x_v · n = x_vv · n = 0

Taking the derivative with respect to each coordinate of the first zero dot product in each case:

∂_u (x_u · n) = 0
x_uu · n + x_u · ∂_un = 0
x_u · ∂_un = 0

∂_v (x_u · n) = 0
x_uv · n + x_u · ∂_vn = 0
x_u · ∂_vn = ±sin(ω(u, v))

∂_u (x_v · n) = 0
x_uv · n + x_v · ∂_un = 0
x_v · ∂_un = ±sin(ω(u, v))

∂_v (x_v · n) = 0
x_vv · n + x_v · ∂_vn = 0
x_v · ∂_vn = 0

Knowing these dot products, along with the fact that since n is a unit vector its derivatives are orthogonal to n itself, tells us enough about the derivatives to find the magnitude of the torsions:

∂_un = ±(cos(ω(u, v)) x_u – x_v) / sin(ω(u, v))
|τ₁| = |∂_un| = 1

∂_vn = ±(cos(ω(u, v)) x_v – x_u) / sin(ω(u, v))
|τ₂| = |∂_vn| = 1

(In fact the torsion of any asymptotic curve satisfies τ² = –K, on any surface of negative curvature, regardless of whether or not K is constant.^[10])

The sign of the torsion is chosen by convention to be positive for a right-handed helix and negative for a left-handed helix, and it turns out that we always have one family of asymptotic curves with each sign. When constructing our surface, we just have to give one family of curves positive torsion and the other negative torsion; if we swap the assignment, that will simply change the surface into its mirror image (which might or might not be a different shape than the original). The sign of the curvature depends on which direction along the curve we associate with an increase, rather than a decrease, of the coordinate value, so we can assign it arbitrarily.

Exact solutions of the sine-Gordon Equation

We have already seen one exact solution of the sine-Gordon equation:

Solution of the sine-Gordon equation from the tractroid

ω(u, v) = acos(2 sech(u – v)² – 1)

What about other surfaces of revolution with K = –1? If we find the asymptotic curves by integrating the unit-length vector fields that give a normal curvature of zero, we are led to the family of solutions:

Solutions of the sine-Gordon equation from surfaces of revolution
Parameter c < 1, c ≠ 0

ω(u, v) = acos(2 cd(√[c/(1–c)] (u – v), (c–1)/c)² – 1)

Here cd is one of the twelve Jacobi elliptic functions.

A similar calculation for Dini’s surface yields:

Solutions of the sine-Gordon equation from Dini’s surface
Parameter –1 < b < 1

ω(u, v) = acos(2 sech(√[(1+b)/(1–b)] u – √[(1–b)/(1+b)] v)² – 1)

For general surfaces of constant negative curvature built from helices:

Solutions of the sine-Gordon equation from surfaces built from helices
Parameters c < 1, c ≠ 0, –1 < b < 1

ω(u, v) = acos(2 cd(√[c/(1–c)] (√[(1+b)/(1–b)] u – √[(1–b)/(1+b)] v), (c–1)/c)² – 1)

In going from the first two solutions to the third and fourth, we see an example of a method that can be used to get a whole family of new solutions to the sine-Gordon equation from any one solution. If ω(u, v) is a solution:

∂_u,v ω(u, v) = sin(ω(u, v))

Define ω_(α)(u, v) = ω(α u, v/α)

Then ∂_u,v ω_(α)(u, v)
= ∂_u,v ω(α u, v/α)
= (α)(1/α) sin(ω(α u, v/α))
= sin(ω_(α)(u, v))

The third of our surface-based solutions above can be obtained from the first, and the fourth from the second, by using this method with α = √[(1+b)/(1–b)].

This method is very simple as we have described it, but it is also worth noting that it arises from the relativistic invariance of the sine-Gordon equation, when viewed as a wave equation in one dimension of space and one of time! The sine-Gordon equation can be rewritten as:

∂_t,t ω(x, t) – ∂_x,x ω(x, t) + sin(ω(x, t)) = 0

where instead of the variables u, v we use x, t, with:

x = u + v
t = u – v
u = ½(x + t)
v = ½(x – t)

Here x and t are space-time coordinates, and the form of the equation is much closer to that of more familiar wave equations. If we apply a “boost” to our coordinate system, to change to the coordinates used by an observer with a speed of b times the speed of light as measured in the original coordinates, and then switch back to u, v coordinates, we get a new solution of the sine-Gordon equation, in precisely the way we did when changing from surfaces of revolution to surfaces generated by helices with parameter b. In fact, we could apply a boost directly in the u, v coordinates if we treat them as “null,” or light-ray, coordinates. You might recognise √[(1+b)/(1–b)] as the factor by which the frequency of light changes due to the Doppler shift, for an observer approaching a light source at a fraction of b times the speed of light, while its reciprocal √[(1–b)/(1+b)] is the corresponding factor for light coming from the opposite direction.

There is another method that lets us generate new solutions; this one, which is a bit more mysterious, is known as a Bäcklund transform. Suppose, as before, that we have an existing solution, ω(u, v). We then look for new solutions ω_(β)(u, v) that satisfy the following equations:

Bäcklund transform for sine-Gordon equation
If ω(u, v) is an existing solution, then ω_(β)(u, v) is a new family of solutions.

∂_u ½(ω_(β) – ω) = β sin(½(ω_(β) + ω))
∂_v ½(ω_(β) + ω) = (1/β) sin(½(ω_(β) – ω))

Differentiating the first equation with respect to v yields:

∂_{u, v} ω_(β) = ∂_{u, v} ω + β cos(½(ω_(β) + ω)) ∂_v (ω_(β) + ω)
= sin(ω) + 2 cos(½(ω_(β) + ω)) sin(½(ω_(β) – ω))
= sin(ω) + sin(ω_(β)) – sin(ω)
= sin(ω_(β))

The simplest solution we can start with is the trivial solution, ω = 0. The Bäcklund transform conditions are then:

∂_u ω_(β) = 2 β sin(½ω_(β))
∂_v ω_(β) = (2 / β) sin(½ω_(β))

We can make progress on this by treating ω_(β) as an independent variable, and integrating the reciprocal of the common term in the derivatives above:

∫ (1 / sin(½ω_(β))) dω_(β) = 2 log(tan(ω_(β)/4))

This suggests we look for solutions of the form:

ω_(β) = ±4 arctan(exp(A u + B v))

which we can make into precisely what we need by setting A = β and B = 1/β:

Bäcklund transform of trivial solution ω = 0
Parameter β can take any non-zero value

ω_(β) = ±4 arctan(exp(β u + v/β))

If we compute the cosine of this, it turns out to be:

cos(ω_(β)) = 1 – 2 sech(β u + v/β)²

This is essentially the same as the solution we obtained from Dini’s surface, if we set β = √[(1+b)/(1–b)], and replace ω(u, v) with π – ω(u, –v).

This solution is often written in the arctan form, in space-time coordinates x, t, and with β = ±√[(1–V)/(1+V)], which gives us:

1-soliton solution of the sine-Gordon equation
Parameter –1 < V < 1

This is the solution from Dini’s surface, expressed in space-time coordinates.

ω_V (x, t) = ±4 arctan(exp(±(x – V t) / √[1 – V²]))

This is called the 1-soliton solution of the sine-Gordon equation, because it describes a travelling wave with a single fixed shape moving with a speed of V. Note that V can take on any value, positive or negative, with a magnitude less than 1; in the sine-Gordon equation, 1 corresponds to the speed of light. The value of ω goes from 0 for large negative values of ±(x – V t), to either 2π or –2π at large positive values. If we are thinking of ω as an angle, then 0, 2π and –2π are all the same, but ω either increases steadily across the whole wave, or decreases steadily, so it embodies a kind of twist.

When ω is an integer multiple of π, the two asymptotic curves of the associated surface become parallel, and the surface itself has a cusp where the curvature is undefined. So the line in the 1-soliton solution where ω = π corresponds to the boundary of Dini’s surface, and the whole x-t plane corresponds to two distinct copies of the surface. There is nothing to stop us gluing the two copies together along a shared boundary in three-dimensional space, but that will not give us a smooth isometry with a larger, single region of the hyperbolic plane.

Amazingly, it turns out that we can find solutions to the sine-Gordon equation from repeated applications of the Bäcklund transform without explicitly performing any further integrations. In 1892, the famous differential geometer Luigi Bianchi showed that Bäcklund transforms with different values of the parameter β commute, and this allowed him to construct the following solution:^[11]

Bianchi solutions of the sine-Gordon equation
If ω is an existing solution, and ω_(β1) and ω_(β2) are derived from ω by single Bäcklund transforms,
then ω_{(β1, β2)} is the result of two successive Bäcklund transforms.

ω_{(β1, β2)} = ω + 4 arctan([(β₂ + β₁) / (β₂ – β₁)] tan(¼(ω_(β2) – ω_(β1))))

A direct computation, relying on ω being a solution of the sine-Gordon equation, and ω_(β1) and ω_(β2) being Bäcklund transforms of ω with parameters β₁ and β₂, confirms that ω_{(β1, β2)} is a solution:

∂_u,vω_{(β1, β2)} – sin(ω_{(β1, β2)}) = 0

We can iterate this process as many times as we wish. For example, suppose we pick three parameters, β₁, β₂, β₃. The Bianchi solutions ω_{(β1, β2)} and ω_{(β2, β3)} correspond to applying Bäcklund transforms with the parameters β₁ and β₃ to the same initial solution ω_(β2), so we can obtain a thrice-transformed solution:

ω_{(β1, β2, β3)} = ω_(β2) + 4 arctan([(β₃ + β₁) / (β₃ – β₁)] tan(¼(ω_{(β2, β3)} – ω_{(β1, β2)})))

The solutions we obtain by the Bianchi method that are reached by two Bäcklund transforms from the trivial solution are known as 2-soliton solutions. Some examples of these are given below, along with contour plots of the solution in space-time coordinates.

2-soliton KK collision
β₁ = –√[(1–V)/(1+V)]
β₂ = √[(1+V)/(1–V)]
Plus signs for amplitudes of both solutions

ω_{KK, V} (x, t) = 4 arctan(V sinh(x / √[1–V²]) / cosh(V t / √[1–V²]))

2-soliton KA collision
β₁ = √[(1–V)/(1+V)]
β₂ = –√[(1+V)/(1–V)]
Minus sign for amplitude of 1st solution, plus sign for 2nd solution

ω_{KA, V} (x, t) = 4 arctan(V cosh(x / √[1–V²]) / sinh(V t / √[1–V²]))

We can use complex values for β if they are complex conjugates of each other, resulting in real-valued periodic solutions known as “breathers.”^[12]

Stationary “breather”
β₁ = exp(i α)
β₂ = exp(–i α)
Plus signs for amplitudes of both solutions

ω_{SB, α}(x, t) = 4 arctan(cot(α) sin(sin(α) t) / cosh(cos(α) x))

Bäcklund and Bianchi came across these transformations in the context of the following geometric problem. Suppose you have a surface S of constant negative curvature, and you try to construct a second such surface, Σ, by choosing a line segment at every point of S, tangent to S, of a fixed length, with its far end tracing out the new surface, Σ. By choosing the direction of the line segment at each point in the right way, can you make Σ into another surface of constant negative curvature?

The answer is yes! If the original surface is parameterised by x(u, v) with u, v measuring arc length along asymptotic curves, there is a family of new surfaces:^[13]

New surface of constant curvature K = –1 via Bäcklund transform
ω is angle between asymptotic curves on original surface x(u, v).
ω_(β), derived from ω by a Bäcklund transform, is angle between asymptotic curves on new surface x_(β)(u, v).

x_(β)(u, v) = x(u, v) + 2β / [(1+β²) sin ω(u, v)] (sin(½(ω(u, v) – ω_(β)(u, v))) x_u + sin(½(ω(u, v) + ω_(β)(u, v))) x_v)

On the new surface, the coordinates u, v again measure arc length along asymptotic curves.

In general, we will not have closed form expressions for geodesics on these surfaces, so to establish an isometry between such a surface and the hyperbolic plane, we can either find suitable geodesics on the surface by numerical methods, and identify them with known geodesics on the hyperbolic plane, or we can rely on the u, v coordinate grid on the surface, and use numerical methods to construct matching curves in the hyperbolic plane, with the same curvature and the same angles between them. We calculated the curvature, |κ₁| and |κ₂|, of the u, v curves previously, as curves in flat three-dimensional space, but for asymptotic curves this is the same as their geodesic curvature, which is measured relative to the geodesics of the surface.

The image above shows the u, v coordinate grid that we obtain on the hyperbolic plane by matching the curvature of the grid lines to the 2-soliton KK solution for V = 0.5, with –π ≤ u, v ≤ π. (We are using the Poincaré disk model, but we have omitted the background tiling that we showed in previous images of the Poincaré disk, to avoid visual clutter.) The grid lines where u varies are solid, and those where v varies are dashed, and the different colours correspond to regions of the solution where ω lies between different integer multiples of π. As we cross the cusps where the u and v curves become parallel, ω crosses an integer multiple of π, and the grid wraps back onto the same region of the hyperbolic plane. So, this gives us four separate one-to-one isometries, mapping each of the four regions of the surface between these cusps to corresponding regions of four different copies of the hyperbolic plane.

The image above left shows parts of the surfaces of constant negative curvature associated with the 2-soliton KK collision solution of the sine-Gordon equation, for values of the velocity parameter V cycling between 0.25 and 0.75, seen from two opposite viewpoints. For each surface, we have plotted the portion –π ≤ u, v ≤ π. There are four distinct regions separated by cusps corresponding to curves in the solution where ω is an integer multiple of π, and the different regions are indicated by changing the hues used in the tiling. The image above right shows the four individual regions, separated out into two pairs, for V = 0.5.

The image above left shows parts of the surfaces of constant negative curvature associated with the 2-soliton KA collision solution of the sine-Gordon equation, for values of the velocity parameter V cycling between 0.25 and 0.75, seen from two opposite viewpoints. For each surface, we have plotted the portion –π ≤ u, v ≤ π. As with the surface for the KK collision, there are four distinct regions separated by cusps corresponding to integer multiple of π, the different regions are indicated by changing the hues used in the tiling, and the image above right shows the individual regions for V = 0.5.

The image above left shows part of the surface associated with a stationary breather solution of the sine-Gordon equation, with sin(α) = 3/4; the image above right shows a single region between cusps. Choosing sin(α) = (q–1)/q will always produce a shape with 2(q–1)-fold symmetry like this.

Any rational value for sin(α) will generate a surface that closes up eventually. Suppose sin(α) = p/q, where p and q are integers and we have cancelled any common factors. Then it can be shown that the angle around the axis between successive cusps is given by:

Δφ = (1/sin(α) – 1) π = (q – p)/p π

For p = q – 1 we have Δφ = π/(q – 1), which divides 2π, but in general Δφ will only divide some multiple of 2π.

Note that:

p Δφ = π (q – p)
2 p Δφ = 2 π (q – p)

are both integer multiples of π, and the first will be an integer multiple of 2π if and only if q – p is even. So the surface will close up after wrapping around the axis q – p times if q – p is odd, and ½(q – p) times if it is even.

The images above show a single cusp-to-cusp portion of the surface associated with a stationary breather, as sin(α) ranges between 3/4 and 1/3.

For sin(α) = 1/3 we have Δφ = 2π, so the surface wraps completely around the axis.

When α falls below π/4, the breather solution reaches values that cross ±π, which introduces a new cusp inside each portion of the surface. These cusps correspond to the roughly elliptical white contours that can be seen in the second plot in the breather solutions shown earlier.

Choose your own geodesic

Earlier, we showed that given any smooth, non-self-intersecting curve c(s) in three-dimensional Euclidean space, there is an intrinsically flat surface that contains this curve, and for which this curve is a geodesic. Intuitively, what we are doing is taking a strip of paper and positioning a straight line drawn on it to coincide with the three-dimensional curve c(s).

Can we do the same thing for surfaces with constant non-zero Gaussian curvature? The intrinsically flat surface we found was generated by a family of lines, and I am not aware of an equally simple construction for K ≠ 0, but we can set up a system of partial differential equations whose solution will give us the surface we want.

For our purposes, it will be convenient to choose a coordinate system for the surface where u corresponds to longitude on a sphere and v corresponds to latitude (measuring both in radians), or analogous coordinates on the hyperbolic plane. The curve where u varies and v is fixed at zero, the “equator,” will be a geodesic. The curves where v varies and u is fixed (at any value) will correspond to meridians, i.e. geodesics that meet the equator orthogonally. The curves where u varies and v is fixed at a non-zero value will correspond to circles of latitude, which are not geodesics. A simple calculation shows that the geodesic curvature of a circle of lattitude on a sphere is κ_g = √|K| tan v, while the corresponding curve on the hyperbolic plane (i.e. a curve that lies at a fixed distance v/√|K| from some geodesic) has κ_g = √|K| tanh v.

We know the components of the metric in these coordinates, and we can write both cases in a similar form:

For K > 0, define s(v) = sin v, c(v) = cos v
For K < 0, define s(v) = sinh v, c(v) = cosh v

x_u · x_u = c(v)² / |K|
x_v · x_v = 1 / |K|
x_u · x_v = 0

Differentiating each of these three equations with respect to u and v, we obtain:

x_u · x_uu = 0
x_u · x_uv = –c(v) s(v) / K

x_v · x_uv = 0
x_v · x_vv = 0

x_v · x_uu = –x_u · x_uv = c(v) s(v) / K
x_u · x_vv = –x_v · x_uv = 0

These dot products are enough to let us write x_u in terms of derivatives that are purely with respect to v, or x_v in terms of derivatives that are purely with respect to u.

PDEs for surface of constant Gaussian curvature K

For K > 0, define s(v) = sin v, c(v) = cos v
For K < 0, define s(v) = sinh v, c(v) = cosh v

If supplied curve x(0, v) is like a meridian, from which we generate curves of varying u like circles of latitude:

x_u = ±c(v) / |x_vv| (x_vv × x_v)

If supplied curve x(u, 0) is like the equator, from which we generate curves of varying v like meridians:

x_v = c(v) s(v) / [K |x_uu|²] x_uu ± √[1 – c(v)² s(v)² / (|K| |x_uu|²)] / (|x_uu| c(v)) (x_uu × x_u)

The first equation will apply if we wish to treat our initial curve as a “meridian” with fixed longitude u and varying latitude v, which we then extend by computing the curves of varying u, corresponding to circles of latitude. The second equation will apply if we treat the initial curve as the “equator” with v = 0 and varying longitude u, which we then extend by computing the geodesics of varying v, corresponding to meridians.

The quantity:

c(v) s(v) / (|x_uu| √|K|)

shows up, squared, inside the square root in the second PDE. This can be understood geometrically as the ratio between the geodesic curvature of a circle of latitude:

κ_g = √|K| s(v) / c(v)

and its curvature as a space curve:

κ = |K| |x_uu| / c(v)²

When the absolute value of the ratio, |κ_g/κ|, exceeds 1, it becomes impossible for the curve to be embedded in a surface in such a way that it has the correct geodesic curvature. At this point, the quantity inside the square root in the second PDE becomes negative, making it clear that the integration cannot continue, and we have reached the edge of the surface. It’s easy to check that this is what happens at the endpoints of the meridians of the surfaces of revolution we examined earlier, if we choose u to be the angular coordinate, and the curves of constant v, as space curves, are all just circles. For example, for the hourglass-like surfaces of constant negative curvature, described by a parameter ξ that is greater than 1, the length (measured from the waist) of the meridians is:

L = a arctanh(1/ξ)

and at the boundary, the radius of the circles of latitude is:

ρ_max = a ξ

So at the boundary, we have:

κ_g = √|K| sinh(v) / cosh(v)
= tanh(v) / a
= tanh(L / a) / a
= 1 / (a ξ)

κ = 1 / ρ_max = 1 / (a ξ)

However, it is also possible for |κ_g/κ| to be less than 1 while the extrinsic curvature of the surface goes to infinity, also marking a boundary.

The image above left shows a family of surfaces of constant positive Gaussian curvature that contain an ellipse as a geodesic. The length-to-width ratio of the ellipse ranges from 1.05 to 1.5, but the overall scale is adjusted to keep the perimeter constant at 2π, the same as the equator of a unit sphere. In effect, these surfaces are the maximal portions of an idealised world globe, made of a material that can be bent but not stretched or sheared, that allow the equator to be deformed into the ellipses in question. As the aspect ratio of the ellipse becomes more extreme, the surface encompasses less of the original globe, with the range of latitude shrinking most along the meridians where the curvature of the equator is reduced the most, i.e. in the region where the ellipse is narrowest.

The image above right shows a similar family of surfaces where the curve chosen for the geodesic corresponding to the equator is non-planar:

(r cos t, r sin t, h cos 2t)

with the ratio h/r ranging from 0.05 to 0.5, while again the overall scale is adjusted to keep the total length of the curve equal to 2π. So these surfaces correspond to the maximal portions of an idealised globe that allow the equator to be deformed into this non-planar curve.

Geodesics with inflection points

The partial differential equations for surfaces of constant curvature containing a specified curve as a geodesic need to be treated carefully when the curve contains inflection points, i.e. points where its curvature as a space curve is zero, since the quantity |x_vv| or |x_uu|, which appears in the denominator of terms in these equations, goes to zero at such points. Of course the equations can still make sense if the numerators also go to zero sufficiently rapidly, so the ratio can be interpreted as a finite limit as the inflection point. But we can say a couple of things independently of the equations, just by thinking about the geometry.

First, no surface with positive Gaussian curvature everywhere can contain a smooth curve with an inflection point. This follows from the fact that the normal curvature (as discussed in the appendix) must range between two positive values as we rotate the plane normal to the surface, with which we slice it to produce the normal sections. The curvature of these normal sections is equal to the length of the projection, onto the normal to the surface, of the curvature vector of any curve passing through the same point in the same direction,^[14] and so a curve with zero curvature would allow for zero normal curvature, which is impossible.

For a surface with negative Gaussian curvature, the situation is very different. A hyperboloid can contain straight lines, as we have seen, and Dini’s surface contains a single straight line, the z-axis, that is a geodesic. Every non-meridian geodesic on the tractroid (such as the blue lines in the image on the top right) contains two points of inflection (marked in black). So, if the curve we nominate to be a geodesic for a surface of constant negative curvature contains inflection points, that need not be an obstacle to the surface existing in some finite region around those points.

Let’s look at that last example a bit more closely. For the tractroid with a = 1, along geodesics of the form:

ρ(s) = ρ₀ cosh(s), where s ranges from –arccosh(1/ ρ₀) to arccosh(1/ ρ₀)
φ(s) = tanh(s)/ρ₀
(x, y, z) = (ρ cos(φ), ρ sin(φ), arccosh(1/ρ) – √[1 – ρ²])

the squared curvature of the geodesic as a space curve is:

κ(s)² = (1 – ρ₀² cosh(s)⁴)² / [ρ₀² cosh(s)⁶ (1 – ρ₀² cosh(s)²)]

So there are inflection points at:

s = ±arccosh(1/√ρ₀)
(where ρ = √ρ₀
and φ = ±√[1 – ρ₀]/ρ₀)

Any geodesic like this will be intersected orthogonally at any given value of s by another geodesic (red curves in the image on the middle right), with parameter t:

ρ_s(t) = ρ₀ sinh(s) cosh(t), where t ranges from –arccosh(1/ [ρ₀ sinh(s)]) to arccosh(1/ [ρ₀ sinh(s)])
φ_s(t) = [tanh(t)/sinh(s) + coth(s)] / ρ₀

The two geodesics intersect orthogonally at:

t = –arccosh(coth(s))
cosh(t) = coth(s)
tanh(t) = sinh(s) tanh(s) – cosh(s)

If we set s = u and t = v – arccosh(coth(u)), this puts a new set of coordinates on the tractroid, with the properties discussed at the start of this section, and which satisfy the partial differential equation that we derived for x_v. In the image on the middle right, the blue curve is the original geodesic v = 0 [the “equator”], the cyan curves are non-geodesic curves of constant v [“circles of latitude”], and the red curves are geodesics of constant u [“meridians”].

We have seen that an inflection point in a curve need not be an obstacle to that curve being a geodesic of a surface of constant negative Gaussia curvature. However, there are further restrictions that we need to consider.

If a geodesic has an inflection point, i.e. a point of zero curvature, then its tangent must be pointing in one of the asymptotic directions^[8] of the surface at that point: a direction such that all curves have a normal curvature of zero. So, the geodesic will share its tangent with an asymptotic curve (the black curve in the image on the bottom right).

The asymptotic curve at this point has its curvature vector orthogonal to the normal to the surface, whereas the geodesic, though it has a zero curvature vector at the point itself, has a principal normal vector (the unit vector parallel or antiparallel to the curvature vector, extended by continuity across inflection points) that is parallel or antiparallel to the normal to the surface. So, the instantaneous planes in which the two curves lie are momentarily locked together, at a right angle to each other. This means the torsion, the rate at which those planes rotate, must be the same (or opposite) at this point.

But we saw earlier that the torsion of any asymptotic curve on a surface of constant Gaussian curvature of –1 is τ = ±1, and more generally τ² = –K.^[10] So, in order for a curve with an inflection point to be a geodesic of a surface of constant Gaussian curvature K, its torsion at that point must satisfy the same equation (or the inflection point must be of a different kind, with not only the curvature, but also its derivative, being zero).

Another way to see this is to explicitly calculate two of the three independent components of the second fundamental form along the supplied curve, in the case where we treat that curve as an “equator”, and assuming K = –1.

On the equator, v = 0, we have:

n = ±x_uu / |x_uu|
x_uu · n = ±|x_uu|
= ±κ(u), the curvature of x(u, 0)

x_v = x_u × x_uu / |x_uu|
x_uv = ∂_u x_v
= x_u × x_uuu / |x_uu| – [(x_uuu · x_uu) / |x_uu|³] x_u × x_uu
x_uv · n = ± ((x_u × x_uuu) · x_uu) / |x_uu|²
= ±τ(u), the torsion of x(u, 0)

This is enough to let us write the second fundamental form, on the equator, as:

S(u, 0)

=

xuu · n xuv · n xuv · n xvv · n

=

±κ(u) ±τ(u) ±τ(u) ∓(1 – τ(u)2) / κ(u)

Here, we have filled in the lower right corner of the matrix by the requirement that K = det(S) / det(g) = –1, and the fact that det(g) = 1 in our chosen coordinates. This shows us that the torsion τ(u) must be ±1 at any point where the curvature κ(u) is zero, in order for the lower right entry of the second fundamental form to have a finite limit (assuming that the derivative of κ(u) there is non-zero).

But this restriction on the torsion of the curve at the inflection points is still not enough to guarantee that we can construct a surface of constant negative curvature. If we compute the third derivative, x_vvv, it contains terms that, near the inflection point u₀, include factors of the form:

[κ'(u₀) τ''(u₀) – κ''(u₀) τ''(u₀)] / (u – u₀)

The numerator here must vanish in order for the v-curve through the inflection point to have a finite third derivative. This amounts to the requirement that the curvature and torsion have series expansions:

κ(u₀ + ε) = κ₁ ε + κ₂ ε² + ...
τ(u₀ + ε) = ±1 + α (κ₁ ε + κ₂ ε²) + ...

for some constant α. This is not hard to arrange. But for higher derivatives, there are further, increasingly complex conditions that must be met.

How can we construct a curve that meets this infinite set of restrictions at an inflection point? Obviously the tractroid geodesics that we discussed earlier, and any other geodesic with an inflection point that we find on a known surface of constant negative curvature, will meet all the conditions, but we want to be able to construct new surfaces, not just reconstruct known cases.

One approach would be to start by specifying a curve, M, that has a torsion of ±1 but a non-zero curvature at some point P, treating it as a “meridian”, and using the first of our PDEs to construct a surface whose “equator” E meets M at P. The curve E will then have an inflection point at P. This follows from the same calculations we used to construct the second fundamental form, where the lower right entry can now be seen as the curvature of M at P, and the condition on the determinant becomes:

det(S) = ±κ_M(P) κ_E(P) – τ_M(P)² = –1

Since τ_M(P) = ±1 and κ_M(P) ≠ 0, this can only be satisfied by κ_E(P) = 0, so E must have an inflection point at P.

A pseudospherical Möbius strip

The simplest way to construct a Möbius strip is to choose a circle as the centreline, then sweep out a surface by taking a line segment orthogonal to the circle at some point and move it around the circle, while rotating it by half the angle through which it has moved. As it comes back to its starting point, it will have been rotated by 180 degrees, putting the required twist into the surface. If the circle has radius R, and the coordinates on the surface are θ for the angle around the circle and s for the distance along the line segment, the surface has Cartesian coordinates:

x(θ, s) = (R + s cos(½θ)) cos(θ)
y(θ, s) = (R + s cos(½θ)) sin(θ)
z(θ, s) = s sin(½θ)

The image on the right shows a Möbius strip like this. The surface has been coloured according to its Gaussian curvature at each point, with red for the most negative curvature, shading to purple for values closer to zero, but still negative. So this surface is not flat, but nor does it have a constant negative curvature. And the centreline here is not a geodesic, because the curvature vector for the circle always points radially inwards, which is generally not a direction normal to the surface.

Now, suppose we want to construct a Möbius strip whose surface has a constant Gaussian curvature of K = –1, and whose centreline is a geodesic of the surface.

We previously showed how to construct a Möbius strip that was intrinsically flat, with its centreline as a geodesic. It would be nice if we could take the same centreline:

(x(t), y(t), z(t)) = (3 sin(t), cos(t) + (2/5) cos(2t) + (1/15) cos(3t), (3/4) sin(2t))

and apply one of our partial differential equations for surfaces of constant Gaussian curvature to extend it into a surface with K = –1. Unfortunately, this is impossible! This centreline contains an inflection point (a point where the curvature, κ, is zero) at t = π, and it was one of the requirements of the construction that the torsion, τ, must go to zero at any such point at least as rapidly as the curvature. But as we saw in the previous section, the torsion at any inflection point on a geodesic of a surface of constant Gaussian curvature K must satisfy τ² = –K.

And as we also discussed in that section, there are actually an infinite number of conditions that must be satisfied at the inflection point in order for all the derivatives of the orthogonal coordinate curve to be finite. These conditions are increasingly complex, and I am not aware of any way to reframe them as a single equation the original curve could satisfy that would guarantee they were all met. Alternatively, if we tried to work backwards by specifying an orthogonal curve with finite derivatives and a torsion of 1, we would obtain a candidate centreline curve with an inflection point, but there would be no easy way to guarantee that it formed a loop.

In lieu of any perfect solution, we can construct a curve numerically that meets the conditions required for some chosen number of derivatives of the orthogonal coordinate curve to be finite. The surface itself can then be approximated with a Taylor series in each of the three Cartesian coordinates.

The image on the right has a centreline that is a sum of trigonometric functions:

x(t) = sin(t)

y(t) = 0.715336 cos(t) – 0.222511 cos(2t) + 0.0194119 cos(3t) + 0.000532444 sin(2t)

z(t) = 0.145095 sin(2t) + 0.00409208 sin(3t) – 0.0094447 sin(4t) + 0.00238777 sin(5t) – 0.00024958 sin(6t)

These coefficients have been chosen to produce an inflection point at t = 0, and also to meet the conditions required for the sixth-order Taylor series for the orthogonal coordinate curves to have finite derivatives everywhere on the centreline.

Appendix: Computing Gaussian Curvature

Gaussian curvature of an embedded surface

We assume that the surfaces we’re dealing with are described by mathematical functions whose second derivatives exist, and are continuous, everywhere, except perhaps along their boundaries. So at any interior point P on the surface we can always approximate its shape in a small region using a Taylor series. If we choose the origin of our coordinates to lie on the point P, and choose the z-axis to be orthogonal to the surface, the Taylor series up to quadratic terms in x and y can be written in the form:

z(x, y) = ½ A x² + B x y + ½ C y²