In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the Gauss-Bonnet theorem.

1. Triangles in Hyperbolic Space

Suppose we are given 3 points in hyperbolic space {\mathbb{H}^n}. A triangle with these points as vertices is a set of three geodesic segments with these three points as endpoints. The fact that there is a unique triangle requires a (brief) proof. Consider the hyperboloid model: three points on the hyperboloid determine a unique 3-dimensional real subspace of {\mathbb{R}^{n+1}} which contains these three points plus the origin. Intersecting this subspace with the hyperboloid gives a copy of {\mathbb{H}^2}, so we only have to check there is a unique triangle in {\mathbb{H}^2}. For this, consider the Klein model: triangles are euclidean triangles, so there is only one with a given three vertices.

In hyperbolic space, it is still true that knowing enough side lengths and/or angles of a triangles determines it. For example, knowing two side lengths and the angle between them determines the triangle. Similarly, knowing all the angles determines it. However, not every set of angles can be realized (in euclidean space, for example, the angles must add to {\pi}), and the inequalities which must be satisfied are more complicated for hyperbolic space.

2. Ideal Triangles and Area Theorems

We can think about moving one (or more) of the points of a hyperbolic triangle off to infinity (the boundary of the disk). An ideal triangle is one with all three “vertices” (the vertices do not exist in hyperbolic space) on the boundary. Using a conformal map of the disk (which is an isometry of hyperbolic space), we can move any three points on the boundary to any other three points, so up to isometry, there is only one ideal triangle. We have fixed our metric, so we can find the area of this triangle. The logically consistent way to find this is with an integral since we will use this fact in our proof sketch of Gauss-Bonnet, but as a remark, suppose we know Gauss-Bonnet. Imagine a triangle very close to ideal. The curvature is {-1}, and the euler characteristic is {1}. The sum of the exterior angles is just slightly under {3\pi}, so using Gauss-Bonnet, the area is very close to {\pi}, and goes to {\pi} as we push the vertices off to infinity.

One note is that suppose we know what the geodesics are, and we know what the area of an ideal triangle is (suppose we just defined it to be {\pi} without knowing the curvature). Then by pasting together ideal triangles, as we will see, we could find the area of any triangle. That is, really the key to understanding area is knowing the area of an ideal triangle.

As mentioned above, there is a single triangle, up to isometry, with given angles, so denote the triangle with angles {\alpha, \beta, \gamma} by {\Delta(\alpha, \beta, \gamma)}.

2.1. Area

Knowing the area of an ideal triangle allows us to calculate the area of any triangle. In fact:

Theorem 1 (Gauss) {\mathrm{area}(\Delta(\alpha, \beta, \gamma)) = \pi - (\alpha + \beta + \gamma)}

This geometric proof relies on the fact that the angles in the Poincare model are the euclidean angles in the model. Consider the generic picture:

We have extended the sides of {\Delta(\alpha, \beta, \gamma)} and drawn the ideal triangle containing these geodesics. Since the angles are what they look like, we know that the area of {\Delta(\alpha,\beta,\gamma)} is the area of the ideal triangle ({\pi}), minus the sum of the areas of the smaller triangles with two points at infinity:

\displaystyle \mathrm{area}(\Delta(\alpha, \beta, \gamma)) = \pi - \mathrm{area}(\Delta(\pi-\alpha, 0,0)) - \mathrm{area}(\Delta(\pi-\beta, 0, 0)) - \mathrm{area}(\Delta(\pi-\gamma, 0, 0))

Thus it suffices to show that {\mathrm{area}(\Delta(\pi - \alpha, 0, 0)) = \alpha}.

For this fact, we need another picture:

Define {f(\alpha) = \mathrm{area}(\Delta(\pi-\alpha, 0, 0))}. The picture shows that the area of the left triangle (with two vertices at infinity and one near the origin) plus the area of the right triangle is the area of the top triangle plus the area of the (ideal) bottom triangle:

\displaystyle f(\alpha) + f(\beta) = f(\alpha+\beta-\pi) + \pi

We also know some boundary conditions on {f}: we know {f(0) = 0} (this is a degenerate triangle) and {f(\pi) = \pi} (this is an ideal triangle). We therefore conclude that

\displaystyle f(\frac{\pi}{2}) + f(\frac{\pi}{2}) = f(0) + \pi \qquad \Rightarrow \qquad f(\frac{\pi}{2}) = \frac{\pi}{2}

Similarly,

\displaystyle 2f(\frac{3\pi}{4}) = f(\frac{\pi}{2}) + \pi \qquad \Rightarrow \qquad f(\frac{3\pi}{4}) = \frac{3\pi}{4}

And we can find {f(\pi/4) = \pi/4} by observing that

\displaystyle f(\frac{3\pi}{4}) + f(\frac{\pi}{2}) = f(\frac{\pi}{4}) + \pi

Similarly, if we know {f(\frac{k\pi}{2^n}) = \frac{k\pi}{2^n}}, then

\displaystyle f(\frac{(2^{n+1}-1)\pi}{2^{n+1}}) = \frac{(2^{n+1}-1)\pi}{2^{n+1}}

And by subtracting {\pi/2^n}, we find that {f(\frac{k\pi}{2^{n+1}}) = \frac{k\pi}{2^{n+1}}}. By induction, then, {f(\alpha) =\alpha} if {\alpha} is a dyadic rational times {\pi}. This is a dense set, so we know {f(\alpha) = \alpha} for all {\alpha \in [0,\pi]} by continuity. This proves the theorem.

3. Triangles On Spheres

We can find a similar formula for triangles on spheres. A lune is a wedge of a sphere:

A lune.

Since the area of a lune is proportional to the angle at the peak, and the lune with angle {2\pi} has area {4\pi}, the lune {L(\alpha)} with angle {\alpha} has area {2\alpha}. Now consider the following picture:

Notice that each corner of the triangle gives us two lunes (the lunes for {\alpha} are shown) and that there is an identical triangle on the rear of the sphere. If we add up the area of all 6 lunes associated with the corners, we get the total area of the sphere, plus twice the area of both triangles since we have triple-counted them. In other words:

\displaystyle 4\pi + 4\mathrm{area}(\Delta(\alpha, \beta,\gamma)) = 2L(\alpha) + 2L(\beta) + 2L(\gamma) = 4(\alpha + \beta + \gamma)

Solving,

\displaystyle \mathrm{area}(\Delta(\alpha, \beta,\gamma)) = \alpha + \beta + \gamma - \pi

4. Gauss-Bonnet

If we encouter a triangle {\Delta} of constant curvature {K(\Delta)}, then we can scale the problem to one of the two formulas we just computed, so

\displaystyle \mathrm{area}(\Delta) = \frac{\sum \mathrm{angles} - \pi}{K(\Delta)}

This formula allows us to give a slightly handwavy, but accurate, proof of the Gauss-Bonnet theorem, which relates topological information (Euler characteristic) to geometric information (area and curvature). The proof will precede the statement, since this is really a discussion.

Suppose we have any closed Riemannian manifold (surface) {S}. The surface need not have constant curvature. Suppose for the time being it has no boundary. Triangulate it with very small triangles {\Delta_i} such that {\mathrm{area}(\Delta_i) \sim \epsilon^2} and {\mathrm{diameter}(\Delta_i) \sim \epsilon}. Then since the deviation between the curvature and the curvature at the midpoint {K_\mathrm{midpoint}} is {o(\epsilon^2)} times the distance from the midpoint,

\displaystyle \int_{\Delta_i} K d\mathrm{area} = K_\mathrm{midpoint}\cdot \mathrm{area}(\Delta_i) + o(\epsilon^3)

For each triangle {\Delta_i}, we can form a comparison triangle {\Delta^c_i} with the same edge lengths and constant curvature {K_\mathrm{midpoint}}. Using the formula from the beginning of this section, we can rewrite the right hand side of the formula above, so

\displaystyle \int_{\Delta_i} K d\mathrm{area} = \sum_{\Delta_i^c} \mathrm{angles} - \pi + o(\epsilon^3)

Now since the curvature deviates by {o(\epsilon^2)} times the distance from the midpoint, the angles in {\Delta_i} deviate from those in {\Delta_i^c} just slightly:

\displaystyle \sum_{\Delta_i} \mathrm{angles} = \sum_{\Delta_i^c} \mathrm{angles} + o(\epsilon^3)

So we have

\displaystyle \int_{\Delta_i} K d\mathrm{area} = \sum_{\Delta_i} \mathrm{angles} - \pi + o(\epsilon^3)

Therefore, summing over all triangles,

\displaystyle \int_{S} K d\mathrm{area} = \sum_i \left[ \sum_{\Delta_i} \mathrm{angles} - \pi \right] + o(\epsilon)

The right hand side is just the total angle sum. Since the angle sum around each vertex in the triangulation is {2\pi},

\displaystyle \sum_i \left[ \sum_{\Delta_i} \mathrm{angles} - \pi \right] = 2\pi V - \pi T

Where {V} is the number of vertices, and {T} is the number of triangles. The number of edges, {E}, can be calculated from the number of triangles, since there are {3} edges for each triangle, and they are each double counted, so {E = \frac{3}{2} T}. Rewriting the equation,

\displaystyle \int_{S} K d\mathrm{area} = 2\pi (V - \frac{1}{2}T) = 2\pi (V - E + T) = 2\pi\chi(S) + o(\epsilon)

Taking the mesh size {\epsilon} to zero, we get the Gauss-Bonnet theorem {\int_S K d\mathrm{area} = 2\pi\chi(S)}.

4.1. Variants of Gauss-Bonnet

  • If {S} is compact with totally geodesic boundary, then the formula still holds, which can be shown by doubling the surface, applying the theorem to the doubled surface, and finding that euler characteristic also doubles.
  • If {S} has geodesic boundary with corners, then\displaystyle \int_S K d\mathrm{area} + \sum_\mathrm{corners} \mathrm{turning angle} = 2\pi\chi(S) Where the turning angle is the angle you would turn tracing the shape from the outside. That is, it is {\pi - \alpha}, where {\alpha} is the interior angle.

     

  • Most generally, if {S} has smooth boundary with corners, then we can approximate the boundary with totally geodesic segments; taking the length of these segments to zero gives us geodesic curvature ({k_g}):\displaystyle \int_S K d\mathrm{area} + \sum_\mathrm{corners} \mathrm{turning angle} + \int_{\partial S} k_g d\mathrm{length} = 2\pi\chi(S)

4.2. Examples

  • The Euler characteristic of the round disk in the plane is {1}, and the disk has zero curvature, so {\int_{\partial S} k_g d\mathrm{length} = 2\pi}. The geodesic curvature is constant, and the circumference is {2\pi r}, so {2\pi r k_g = 2\pi}, so {k_g = 1/r}.
  • A polygon in the plane has no curvature nor geodesic curvature, so {\sum_\mathrm{corners} \pi - \mathrm{angle} = 2\pi}.

The Gauss-Bonnet theorem constrains the geometry in any space with nonzero curvature. This the “reason” similarities which don’t preserve length and/or area exist in euclidean space; it has curvature zero.