by Greg Egan

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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
  • The Sine-Gordon Equation
• Appendix: Computing Gaussian Curvature
  • Gaussian curvature of an embedded surface
  • Gaussian curvature from a metric
• References

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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	=
S(u, v)	=	xuu · n xuv · n xuv · n xvv · n	=

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	=

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	=
S(u, v)	=	xuu · n xuv · n xuv · n xvv · n	=

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)	=		=
S(u, v)	=		=	–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. It is given by:

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
y = v (κ(s)/√[τ(s)² + κ(s)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 + (1/3) 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

=

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.

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	=
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 while 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:

If 0 < χ ≤ 1		0 < ρ < a χ
If χ > 1		a √[χ2 – 1] < ρ < a χ

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:

If 0 < ξ ≤ 1		0 < ρ < a ξ
If ξ > 1		a √[ξ2 – 1] < 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:

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:

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
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

If 0 < χ ≤ 1		L = (π/2) a
If χ > 1		L = (π/2 – arctan(√[χ2 – 1])) a

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

If 0 < χ < 1		A = 2 π a2 χ
If χ ≥ 1		A = 2 π a2

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):

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 √