# 7.2: Elliptic Geometry with Curvature k > 0

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One may model elliptic geometry on spheres of varying radii, and a change in radius will cause a change in the curvature of the space as well as a change in the relationship between the area of a triangle and its angle sum.

For any real number $$k \gt 0\text{,}$$ we may construct a sphere with constant curvature $$k\text{.}$$ According to Example $$7.1.1$$, the sphere centered at the origin with radius $$\dfrac{1}{\sqrt{k}}$$ works. Geometry on this sphere can be modeled down in the extended plane via stereographic projection. This geometry will be called elliptic geometry with curvature $$k > 0\text{.}$$

Consider the sphere $$\mathbb{S}^2_k$$ centered at the origin of $$\mathbb{R}^3$$ with radius $$\dfrac{1}{\sqrt{k}}\text{.}$$ Define stereographic projection $$\phi_k: \mathbb{S}^2_k \to \mathbb{C}^+\text{,}$$ just as we did in Section 3.3, to obtain the formula

$\phi_k(a,b,c) = \begin{cases} \dfrac{a}{1-c\sqrt{k}}+\dfrac{b}{1-c\sqrt{k}}i & \text{ if $$c \neq \dfrac{1}{\sqrt{k}}$$; } \\ \infty & \text{ if $$c=\dfrac{1}{\sqrt{k}}$$ } \end{cases}\text{.}$

Diametrically opposed points on $$\mathbb{S}^2_k$$ get mapped via $$\phi_k$$ to points $$z$$ and $$z_a$$ that satisfy the equation $$z_a =\dfrac{-1}{k\overline{z}}\text{,}$$ by analogy with Lemma $$6.1.1$$. We call two such points in $$\mathbb{C}^+$$ antipodal with respect to $$\mathbb{S}^2_k\text{,}$$ or just antipodal points if the value of $$k$$ is understood.

Our model for elliptic geometry with curvature $$k$$ has space $$\mathbb{P}^2_k$$ equal to the closed disk in $$\mathbb{C}$$ of radius $$\dfrac{1}{\sqrt{k}}\text{,}$$ with antipodal points of the boundary identified. This space is simply a scaled version of the projective plane from Chapter 6.

The group of transformations, denoted $${\cal S}_k\text{,}$$ consists of those Möbius transformations that preserve antipodal points with respect to $$\mathbb{S}^2_k\text{.}$$ That is, $$T \in {\cal S}_k$$ if and only if the following holds:

$\text{if }~ z_a = -\dfrac{1}{k\overline{z}}~~ \text{then }~ T(z_a) = -\dfrac{1}{k\overline{T(z)}}\text{.}$

The geometry $$(\mathbb{P}^2_k,{\cal S}_k)$$ with $$k \gt 0$$ is called elliptic geometry with curvature $$k$$. Note that $$(\mathbb{P}^2_1,{\cal S}_1)$$ is precisely the geometry we studied in Chapter 6.

The transformations of $$\mathbb{C}^+$$ in the group $${\cal S}_k$$ correspond precisely with rotations of the sphere $$\mathbb{S}^2_k\text{.}$$ One can show that transformations in $${\cal S}_k$$ have the form

$T(z) = e^{i\theta}\dfrac{z-z_0}{1+k\overline{z_0}z}\text{.}$

We define lines in elliptic geometry with curvature $$k$$ to be clines with the property that if they go through $$z$$ then they go through $$z_a = -\dfrac{1}{k\overline{z}}\text{.}$$ These lines correspond precisely to great circles on the sphere $$\mathbb{S}^2_k\text{.}$$

The arc-length and area formulas also slip gently over from Chapter 6 to this more general setting.

The arc-length of a smooth curve $$\boldsymbol{r}$$ in $$\mathbb{P}^2_k$$ is

${\cal L}(\boldsymbol{r}) = \int_a^b \dfrac{2|\boldsymbol{r}^\prime(t)|}{1 + k|\boldsymbol{r}(t)|^2}~dt\text{.}$

As before, arc-length is an invariant, and the shortest path between two points is along the elliptic line through them. In the exercises we derive a formula for the distance between points in this geometry. The greatest possible distance between two points in $$(\mathbb{P}^2_k,{\cal S}_k)$$ turns out to be $$\dfrac{\pi}{(2\sqrt{k})}\text{.}$$

The area of a region $$R$$ given in polar form is computed by the formula

$A(R) = \iint_R \dfrac{4r}{(1+kr^2)^2}dr d\theta\text{.}$

To compute the area of a triangle, proceed as in Chapter 6. First, tackle the area of a lune, a $$2$$-gon whose sides are elliptic lines in $$(\mathbb{P}^2_k,{\cal S}_k)\text{.}$$

##### Lemma $$\PageIndex{1}$$

Assume $$k \gt 0\text{.}$$ A lune in $$(\mathbb{P}^2_k,{\cal S}_k)$$ with interior angle $$\alpha$$ has area $$\dfrac{2\alpha}{k}\text{.}$$

Proof

Without loss of generality, we may consider the vertex of our lune to be the origin. As before, elliptic lines through the origin must also pass through $$\infty\text{,}$$ so our two lines forming the lune are Euclidean lines. After a convenient rotation, we may further assume one of these lines is the real axis, so that the lune resembles the one in Figure $$6.3.4$$. To compute the area of the lune, compute the integral

$A = 2 \int_0^\alpha \int_0^{1/\sqrt{k}}\dfrac{4r}{(1+kr^2)^2}drd\theta\text{.}$

Letting $$u = 1 + kr^2$$ so that $$du = 2krdr\text{,}$$ the bounds of integration change from $$[0,\dfrac{1}{\sqrt{k}}]$$ to $$[1,2]\text{.}$$ Then,

$A = 2 \int^\alpha_0 \dfrac{2}{k}\int_1^2 \dfrac{du}{u^2}d\theta = \dfrac{4}{k}\int_0^\alpha \dfrac{1}{2} d\theta =\dfrac{2\alpha}{k}\text{.}$

Thus, the angle of a lune with interior angle $$\alpha$$ is $$\dfrac{2\alpha}{k}\text{.}$$

We remark that the lune with angle $$\pi$$ actually covers the entire disk of radius $$\dfrac{1}{\sqrt{k}}\text{.}$$ Thus, the area of the entire space $$\mathbb{P}^2_k$$ is $$\dfrac{2\pi}{k}\text{,}$$ which matches half the surface area of a sphere of radius $$\dfrac{1}{\sqrt{k}}\text{.}$$ We often call $$s = \dfrac{1}{\sqrt{k}}$$ the radius of curvature for the geometry; it is the radius of the disk on which we model the geometry.

Also, the integral computation in the proof of Lemma $$7.2.1$$ reveals the following useful antiderivative:

$\int \dfrac{4r}{(1+kr^2)^2}dr = \dfrac{-2}{k(1+kr^2)} + C\text{.}$

This fact may speed up future integral computations.

##### Lemma $$\PageIndex{2}$$

In elliptic geometry with curvature $$k\text{,}$$ the area of a triangle with angles $$\alpha, \beta,$$ and $$\gamma$$ is

$A = \dfrac{1}{k}(\alpha+\beta+\gamma -\pi)\text{.}$

Proof

As in the case $$k = 1\text{,}$$ the area of any triangle may be determined from the area of three lunes and the total area of $$\mathbb{P}^2_k\text{,}$$ as depicted in Example $$6.3.1$$.

Example 7.2.3: Triangles on the Earth.

The surface of the Earth is approximately spherical with radius about $$6375$$ km. Therefore, the geometry on the surface of the Earth can be reasonably modeled by $$(\mathbb{P}^2_k, {\cal S}_k)$$ where $$k = \dfrac{1}{6375^2}~ \text{km}^{-2}\text{.}$$ The area of a triangle on the Earth's surface having angles $$\alpha, \beta,$$ and $$\gamma$$ is

$A = \dfrac{1}{k}(\alpha + \beta + \gamma - \pi)\text{.}$

Can you find the area of the triangle formed by Paris, New York, and Rio? Use a globe, a protractor, and some string. The string follows a geodesic between two points when it is pulled taut.

## Exercises

##### Exercise $$\PageIndex{1}$$

Prove that for $$k > 0\text{,}$$ any transformation in $${\cal S}_k$$ has the form

$T(z) = e^{i\theta}\dfrac{z-z_0}{1+k\overline{z_0}z}\text{,}$

where $$\theta$$ is any real number and $$z_0$$ is a point in $$\mathbb{P}^2_k\text{.}$$

Hint

Follow the derivation of the transformations in $${\cal S}$$ found in Chapter 6.

##### Exercise $$\PageIndex{2}$$

Verify the formula for the stereographic projection map $$\phi_k\text{.}$$

##### Exercise $$\PageIndex{3}$$

Assume $$k \gt 0$$ and let $$s = \dfrac{1}{\sqrt{k}}\text{.}$$ Derive the following measurement formulas in $$(\mathbb{P}^2_k,{\cal S}_k)\text{.}$$

1. The length of a line segment from $$0$$ to $$x\text{,}$$ where $$0 \lt x \leq s$$ is $d_{k}(0,x) = 2s\arctan(\dfrac{x}{s})\text{.}$
2. The circumference of the circle centered at the origin with elliptic radius $$r \lt \dfrac{\pi}{(2\sqrt{k})}$$ is $$C=2\pi s\sin(\dfrac{r}{s})\text{.}$$
3. The area of the circle centered at the origin with elliptic radius $$r \lt \dfrac{\pi}{(2\sqrt{k})}$$ is $$\displaystyle A = 4\pi s^2 \sin^2\left(\dfrac{r}{2s}\right)\text{.}$$
##### Exercise $$\PageIndex{4}$$

In this exercise we investigate the idea that the elliptic formulas in Exercise $$7.2.3$$ for distance, circumference, and area approach Euclidean formulas when $$k \to 0^+\text{.}$$

1. Show that the elliptic distance $$d_{k}(0,x)$$ from $$0$$ to $$x\text{,}$$ where $$0 \lt x \leq s,$$ approaches $$2x$$ as $$k \to 0^+$$ (twice the usual notion of Euclidean distance).
2. Show that the elliptic circumference of a circle with elliptic radius $$r$$ approaches $$2\pi r$$ as $$k \to 0^+\text{.}$$
3. Show that the elliptic area of this circle approaches $$\pi r^2$$ as $$k \to 0^+\text{.}$$
##### Exercise $$\PageIndex{5}$$

Triangle trigonometry in $$(\mathbb{P}^2_k,{\cal S}_k)\text{.}$$

Suppose we have a triangle in $$(\mathbb{P}^2_k,{\cal S}_k)$$ with side lengths $$a,b,c$$ and angles $$\alpha, \beta, \gamma$$ as pictured in Figure $$6.3.5$$.

1. Prove the elliptic law of cosines in $$(\mathbb{P}^2_k,{\cal S}_k)\text{:}$$ $\cos(\sqrt{k}c)=\cos(\sqrt{k}a)\cos(\sqrt{k}b)+\sin(\sqrt{k}a)\sin(\sqrt{k}b)\cos(\gamma)\text{.}$
2. Prove the elliptic law of sines in $$(\mathbb{P}^2_k,{\cal S}_k)\text{:}$$ $\dfrac{\sin(\sqrt{k}a)}{\sin(\alpha)}=\dfrac{\sin(\sqrt{k}b)}{\sin(\beta)}=\dfrac{\sin(\sqrt{k}c)}{\sin(\gamma)}\text{.}$

This page titled 7.2: Elliptic Geometry with Curvature k > 0 is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael P. Hitchman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.