# 7.4: The Family of Geometries (Xₖ, Gₖ)

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A strong connection exists between the family $$(\mathbb{P}^2_k,{\cal S}_k)$$ of elliptic geometries with curvature $$k\gt 0$$ and the family $$(\mathbb{D}_k,{\cal H}_k)$$ of hyperbolic geometries with curvature $$k\lt 0\text{.}$$ The two families sport identical descriptions of the transformation group, identical descriptions of straight lines, identical arc-length and area formulas, as well as identical formulas for the area of a triangle.

We may symbolize this connection with the following description of an infinite family of geometries, one for each real number $$k\text{.}$$ This general description will allow us to elegantly express some important features common to hyperbolic, Euclidean, and elliptic geometry.

##### Definition: Geometry $$(X_k,G_k)$$

For each real number $$k$$ the geometry $$\boldsymbol{(X_k,G_k)}$$ has space

$X_k = \begin{cases} \mathbb{D}_k & \text{ if $$k \lt 0$$; } \\ \mathbb{C} & \text{ if $$k = 0$$; } \\ \mathbb{P}^2_k & \text{ if $$k \gt 0$$, } \end{cases}$

and group of transformations $$G_k$$ consisting of all Möbius transformations of the form

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

where $$\theta \in \mathbb{R}$$ and $$z_0$$ is a point in $$X_k\text{.}$$ Moreover, the unique line through two points $$p$$ and $$q$$ in $$(X_k,G_k)$$ is the unique cline through the points $$p, q\text{,}$$ and $$-\dfrac{1}{(k\overline{p})}\text{.}$$ A smooth curve $$\boldsymbol{r}:[a,b]\to X_k$$ has arc-length given by

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

The area of a polar region $$R$$ in $$(X_k,G_k)$$ is given by

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

As we have seen, these geometries manifest themselves in strikingly different ways. If $$k \gt 0\text{,}$$ the sum of the angles of a triangle must be greater than $$\pi\text{;}$$ and if $$k \lt 0$$ the sum of the angles of a triangle must be less than $$\pi\text{.}$$ If $$k \gt 0$$ the space $$X_k = \mathbb{P}^2_k$$ has finite area; if $$k \lt 0\text{,}$$ $$X_k = \mathbb{D}_k$$ has infinite area. If $$k \gt 0$$ the circumference of a circle with radius $$r$$ is less than $$2\pi r\text{;}$$ if $$k \lt 0$$ the circumference is greater than $$2\pi r\text{.}$$

What about when $$k = 0\text{?}$$ We may check that $$(X_0,G_0)$$ corresponds to Euclidean geometry, though with a scaled metric. In particular, lines in $$(X_0,G_0)$$ correspond to Euclidean lines (since $$k = 0\text{,}$$ $$-\dfrac{1}{(k\overline{p})} = \infty\text{,}$$ so the unique line through $$p$$ and $$q$$ is the Euclidean line), and when $$k = 0$$ the arc-length is simply twice the usual Euclidean arc-length. So while we are scaling distances in $$(X_0,G_0)\text{,}$$ Euclidean geometry applies: triangles are Euclidean triangles and have angle sum equal to $$180^{\circ}\text{.}$$ Triangles with a right angle are Euclidean right triangles and satisfy the Pythagorean theorem.

Thus, we treat $$(X_k,G_k)$$ as one big family of geometries. The sign of $$k$$ dictates the type of geometry we have, and the magnitude of $$k$$ dictates the radius of the disk in which we model the geometry (unless $$k = 0$$ in which case the space is $$\mathbb{C}$$). Morevoer, Euclidean geometry $$(X_0,G_0)$$ marks the edge of the knife from which we move into a hyperbolic world is $$k$$ drops below $$0$$, and into an elliptic world if $$k$$ rises above $$0$$.

We now summarize some results established in the previous sections and emphasize key features common to all $$(X_k,G_k)\text{.}$$

First and foremost, we note that arc-length is an invariant function of $$(X_k,G_k)$$ and that the arc-length ensures that the shortest path from $$p$$ to $$q$$ in $$(X_k,G_k)$$ is along the line between them. We have discussed these facts in the cases $$k = -1,0,1\text{,}$$ and the result holds for arbitrary $$k\text{.}$$ So, the arc-length formula provides a metric on $$(X_k,G_k)\text{:}$$ Given $$p,q \in X_k\text{,}$$ we define $$d_k(p,q)$$ to be the length of the shortest path from $$p$$ to $$q\text{.}$$ The circle in $$(X_k,G_k)$$ centered at $$p$$ through $$q$$ consists of all points in $$X_k$$ whose distance from $$p$$ equals $$d_k(p,q)\text{.}$$

##### Theorem $$\PageIndex{1}$$

For all real numbers $$k\text{,}$$ $$(X_k,G_k)$$ is homogeneous and isotropic.

Proof

Given any point $$p$$ in $$X_k\text{,}$$ the transformation $$T(z) = \dfrac{z-p}{1+k\overline{p}z}$$ in $$G_k$$ maps $$p$$ to the origin. So all points in $$X_k$$ are congruent to $$0\text{.}$$ By the group structure of $$G_k$$ it follows that any two points in $$X_k$$ are congruent, so the geometry is homogeneous.

To show $$(X_k,G_k)$$ is isotropic we consider three cases.

If $$k \lt 0$$ then $$(X_k,G_k)$$ models hyperbolic geometry on the open disk with radius $$s = \dfrac{1}{\sqrt{|k|}}\text{.}$$ As such, $$G_k$$ contains the sorts of Möbius transformations discussed in Chapter 5 and pictured in Figure $$5.1.3$$. In particular, for any point $$p \in \mathbb{D}_k\text{,}$$ $$G_k$$ contains all elliptic Möbius transformations that swirl points around type II clines of $$p$$ and $$-\dfrac{1}{(k\overline{p})}\text{,}$$ the point symmetric to $$p$$ with respect to the circle at infinity. These maps are preciley the rotations about the point $$p$$ in this geometry: they rotate points in $$X_k$$ around cicles centered at $$p\text{.}$$

If $$k = 0\text{,}$$ then transformations in $$G_0$$ have the form $$T(z)=e^{i\theta}(z-z_0)\text{.}$$ Now, rotation by angle $$\theta$$ about the point $$p$$ in the Euclidean plane is given by $$T(z)=e^{i\theta}(z-p)+p\text{.}$$ Setting $$z_0 = p-pe^{-i\theta}$$ we see that this rotation indeed lives in $$G_0\text{.}$$

If $$k \gt 0$$ then $$(X_k,G_k)$$ models elliptic geometry on the projective plane with radius $$s = \dfrac{1}{\sqrt{k}}\text{.}$$ As such, for each $$p \in X_k\text{,}$$ $$G_k$$ contains all elliptic Möbius transformations that fix $$p$$ and $$p_a\text{,}$$ the point antipodal to $$p$$ with respect to the circle with radius $$s\text{.}$$ Such a map rotates points around type II clines with respect to $$p$$ and $$p_a\text{.}$$ Since these type II clines correspond to circles in $$X_k$$ centered at the fixed point, it follows that $$G_k$$ contains all rotations.

Thus, for all $$k \in \mathbb{R}\text{,}$$ $$(X_k,G_k)$$ is isotropic.

##### Theorem $$\PageIndex{2}$$

Suppose $$k$$ is any real number, and we have a triangle in $$(X_k,G_k)$$ whose angles are $$\alpha, \beta,$$ and $$\gamma$$ and whose area is $$A\text{.}$$ Then

$kA = (\alpha + \beta + \gamma - \pi)\text{.}$

Proof of this tidy result has already appeared in pieces (see Exercise $$1.2.2$$, Lemma $$7.2.2$$, and Lemma $$7.3.1$$); we emphasize that this triangle area formula reveals the locally Euclidean nature of all the geometries $$(X_k, G_k)\text{:}$$ a small triangle (one with area close to $$0$$) will have an angle sum close to $$180^{\circ}\text{.}$$ Observe also that the closer $$|k|$$ is to $$0$$, the larger a triangle needs to be in order to detect an angle sum different from $$180^{\circ}\text{.}$$ Of course, if $$k = 0$$ the theorem tells us that the angles of a Euclidean triangle sum to $$\pi$$ radians.

##### Theorem $$\PageIndex{3}$$

Suppose a convex $$n$$-sided polygon $$(n \geq 3)$$ in $$(X_k,G_k)$$ has interior angles $$\alpha_i$$ for $$i = 1, 2, \ldots, n\text{.}$$ The area $$A$$ of the $$n$$-gon is related to its interior angles by

$kA = \bigg(\sum_{i=1}^n\alpha_i\bigg) - (n-2)\pi\text{.}$

Proof

A convex $$n$$-gon can be divided into $$n-2$$ triangles as in Figure $$7.4.1$$. Observe that the area of the $$n$$-gon equals the sum of the areas of these triangles.

By Theorem $$7.4.2$$, the area $$A_i$$ of the $$i$$th triangle $$\Delta_i$$ is related to its angle sum by

$kA_i = (\sum \text{angles in } \Delta_i) - \pi\text{.}$

Thus,

\begin{align*} kA & = \sum_{i=1}^{n-2}kA_i\\ & = \sum_{i=1}^{n-2}(\sum \text{angles in } \Delta_i - \pi)\text{.} \end{align*}

Now, the total angle sum of the $$n-2$$ triangles equals the interior angle sum of the $$n$$-gon, so it follows that

$kA = \bigg(\sum_{i=1}^n \alpha_i \bigg) - (n-2)\pi\text{.}$

This completes the proof.

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

Suppose $$k \in \mathbb{R}\text{,}$$ $$s = \dfrac{1}{\sqrt{|k|}}$$ and $$0 \lt x \lt s$$ is a real number (if $$k = 0\text{,}$$ we just assume $$0 \lt x$$). In $$(X_k,G_k)\text{,}$$ the circle centered at $$0$$ through $$x$$ has area

$\dfrac{4\pi x^2}{1+kx^2}\text{.}$

Proof

Consider the circle centered at the origin that goes through the point $$x$$ on the positive real axis, where $$0 \lt x \lt s\text{.}$$ The circular region matches the polar rectangle $$0 \lt \theta \lt 2\pi$$ and $$0 \lt r \lt x\text{,}$$ so the area is given by

$\int_0^{2\pi} \int_0^x \dfrac{4r}{(1+kr^2)^2}dr d\theta\text{.}$

Evaluating this integral gives the result, and the details are left as an exercise.

##### Theorem $$\PageIndex{4}$$: Unified Pythagorean Theorem

Suppose $$k \in \mathbb{R}\text{,}$$ and we have a geodesic right triangle in $$(X_k,G_k)$$ whose legs have length $$a$$ and $$b$$ and whose hypotenuse has length $$c\text{.}$$ Then

$A(c)=A(a)+A(b)-\dfrac{k}{2\pi}A(a)A(b)\text{,}$

where $$A(r)$$ denotes the area of a circle with radius $$r$$ as measured in $$(X_k,G_k)\text{.}$$

Proof

Suppose $$k \in \mathbb{R}\text{.}$$ If $$k = 0$$ the equation reduces to $$c^2 = a^2 + b^2\text{,}$$ which is true by the Pythagorean Theorem $$1.2.1$$! Otherwise, assume $$k \neq 0$$ and let $$s = \dfrac{1}{\sqrt{|k|}}\text{,}$$ as usual. Without loss of generality we may assume our right triangle is defined by the points $$z = 0\text{,}$$ $$p = x\text{,}$$ and $$q = yi\text{,}$$ where $$0 \lt x,y \lt s\text{.}$$ By construction, the legs $$zp$$ and $$zq$$ are Euclidean segments, and $$\angle pzq$$ is right.

Let $$a = d_k(z,p), b = d_k(z,q),$$ and $$c = d_k(p,q)\text{.}$$ By Lemma $$7.4.1$$,

$A(a) = \dfrac{4\pi x^2}{1+kx^2},~~{\text and}~~A(b) = \dfrac{4\pi y^2}{1+ky^2}\text{.}$

To find $$A(c)\text{,}$$ we first find $$d_k(p,q)\text{.}$$ By invariance of distance,

$d_k(p,q)=d_k(0,|T(q)|)\text{,}$

where $$T(z) = \dfrac{z - p}{1+k\overline{p}z}\text{.}$$ Let $$t = |T(q)| =\dfrac{|yi-x|}{|1+kxyi|}\text{,}$$ which is a real number.

Now, $$A(c)$$ is the area of a circle with radius $$c\text{,}$$ and the circle centered at $$0$$ through $$t$$ has this radius, so

$A(c) = \dfrac{4\pi t^2}{1+kt^2}\text{.}$

Using the fact that $$t^2 = \dfrac{x^2+y^2}{1 + k^2x^2y^2}\text{,}$$ one can now check by direct substitution that

$A(c) = A(a) + A(b)-\dfrac{k}{2\pi}A(a)A(b)\text{.}$

While we have proved the theorem, it feels a bit like we have missed the best part - discovery of the relationship. For more on this, we encourage the reader to consult [20].

##### Example $$\PageIndex{1}$$

Suppose a two-dimensional bug in $$(X_k,G_k)$$ walks along a line for $$a \gt 0$$ units, turns left $$90^{\circ}\text{,}$$ and walks on a line for another $$a$$ units, thus creating a right triangle with legs of equal length. Let $$c$$ denote the hypotenuse of this triangle. The diagram below depicts such a route, in each of the three geometry types. For convenience, we assume the journey begins at the origin and proceeds first along the positive real axis.

It turns out the value of $$c$$ as a function of $$a$$ can reveal the curvature $$k$$ of the geometry. If $$k = 0$$ the Pythagorean theorem tells us that $$c^2 = 2a^2\text{.}$$ For $$k \lt 0\text{,}$$ the hyperbolic law of cosines (Exercise $$7.3.4$$) tells us that $$\cosh(\sqrt{|k|} c) = \cosh^2(\sqrt{|k|}a)\text{.}$$ For $$k \gt 0\text{,}$$ the elliptic law of cosines (Exercise $$7.2.5$$) tells us that $$\cos(\sqrt{k} c) = \cos^2(\sqrt{k}a)\text{.}$$ We may solve each of these equations for $$c\text{,}$$ using the fact that $$a$$ and $$c$$ are positive, to give us $$c$$ as a function of $$a$$ and the curvature $$k\text{.}$$ We have plotted these functions for $$k = -1,0,1$$ below. When $$k \lt 0$$ the length of the hypotenuse of such a triangle is slightly longer than that predicted by the Euclidean formula (the Pythagorean theorem); when $$k \gt 0$$ the length is smaller than the length predicted by the Euclidean formula.

We close by summarizing measurement formulas for these geometries. Except for the case $$k=0$$ these formulas were proved in the exercises of the previoius two sections. The case $$k=0$$ is tackled in Exercise $$7.4.1$$.

##### Measurement Formulas in $$(X_k,G_k)$$

Suppose $$k \in \mathbb{R}\text{,}$$ $$s = \dfrac{1}{\sqrt{|k|}}$$ and $$0 \lt x \lt s$$ is a real number (with the convention that if $$k = 0$$ we simply require $$0 \lt x$$).

• In $$(X_k,G_k)\text{,}$$ the distance from 0 to $$x$$ is given by

$d_k(0,x) = \begin{cases}s \ln\bigg(\dfrac{s+x}{s-x}\bigg) & \text{ if $$k \lt 0$$; } \\ 2x & \text{ if $$k=0$$; } \\ 2s \arctan(\dfrac{x}{s}) & \text{if $$k \gt 0$$.} \end{cases}$

• A circle with radius $$r$$ as measured in $$(X_k,G_k)$$ has area $$A(r)$$ given by

$A(r) = \begin{cases} 4\pi s^2 \sinh^2(\dfrac{r}{2s})& \text{ if $$k \lt 0$$; } \\ \pi r^2 & \text{ if $$k=0$$; } \\ 4\pi s^2 \sin^2(\frac{r}{2s}) & \text{if $$k \gt 0$$,} \end{cases}$

• A circle with radius $$r$$ as measured in $$(X_k,G_k)$$ has circumference $$C(r)$$ given by

$C(r) = \begin{cases}2\pi s \sinh(\dfrac{r}{s}) & \text{ if $$k \lt 0$$; } \\ 2\pi r & \text{ if $$k=0$$; } \\ 2\pi s \sin(\dfrac{r}{s}) & \text{if $$k \gt 0$$.} \end{cases}$

## Exercises

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

Check that the measurement formulas in $$(X_k,G_k)$$ are correct when $$k = 0\text{.}$$ In particular, show that $$d_0(0,x)=2x$$ for any $$x \gt 0$$ on the real axis, and that a circle with radius $$r$$ as measured in $$(X_0,G_0)$$ has area $$A(r) = \pi r^2$$ and circumference $$C(r)=2\pi r\text{.}$$

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

Complete the proof of Lemma $$7.4.1$$.

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

Use the definition of the curvature of the space at $$p$$ (Section 7.1) to prove that for all real numbers $$k\text{,}$$ the curvature of $$(X_k, G_k)$$ is indeed equal to $$k\text{.}$$

Hint

Tackle three cases: $$k \lt 0, k = 0\text{,}$$ and $$k \gt 0\text{.}$$

##### Exercise $$\PageIndex{4}$$

Suppose an intrepid team of two-dimensional explorers sets out to determine which $$2$$-dimensional geometry is theirs. Their cosmologists have told them there world is homogeneous, isotropic, and metric, so they believe that the geometry of their universe is modeled by $$(X_k,G_k)$$ for some real number $$k\text{.}$$ They carefully measure the angles and area of a triangle. They find the angles to be $$29.2438^{\circ}\text{,}$$ $$73.4526^{\circ}\text{,}$$ and $$77.2886^{\circ}\text{,}$$ and the area is $$8.81$$ km$$^2\text{.}$$ Which geometry is theirs? What is the curvature of their universe?

##### Exercise $$\PageIndex{5}$$

Prove that in $$(X_k,G_k)$$ the derivative of area with respect to $$r$$ is circumference: $$\dfrac{d}{dr}[A(r)]=C(r)\text{.}$$

##### Exercise $$\PageIndex{6}$$

Suppose a two-dimensional bug in $$(X_k,G_k)$$ traces the right triangle route from 0 to $$p$$ as depicted in Example $$7.4.1$$. Argue that for a given value of $$a\text{,}$$ the hyperbolic hypotenuse length exceeds the Euclidean hypotenuse length, which exceeds the elliptic hypotenuse length.

Hint

One might prove that for all $$k\text{,}$$ when $$a = 0\text{,}$$ $$c = 0$$ and $$\dfrac{dc}{da}\vert_{a=0}=\sqrt{2}\text{;}$$ and then show that for $$a \gt 0\text{,}$$ $$\dfrac{d^2 c}{da^2}$$ is positive for negative values of $$k\text{,}$$ and it is negative for positive values of $$k\text{.}$$

##### Exercise $$\PageIndex{7}$$

Suppose a team of two-dimensional explorers living in $$(X_k,G_k)$$ travels $$8$$ units along a line. Then they turn right ($$90^{\circ}$$) and travel $$8$$ units along a line. At this point they find they are $$12$$ units from their starting point. Which type of geometry applies to their universe? Can they determine the value of $$k$$ from this measurement? If so, what is it?

##### Exercise $$\PageIndex{8}$$

Repeat the previous exercise using the measurements $$a = 8$$ units and $$c = 11.2$$ units.

##### Exercise $$\PageIndex{9}$$

Suppose a team of two-dimensional explorers living in $$(X_k,G_k)$$ finds themselves at point $$p\text{.}$$ They travel $$8$$ units along a line to a point $$z\text{,}$$ turn right ($$90^{\circ}$$) and travel another $$8$$ units along a line to the point $$q\text{.}$$ At this point they measure $$\angle pqz =.789$$ radians. Which type of geometry applies to their universe? Can they determine the value of $$k$$ from this measurement? If so, what is it?

##### Exercise $$\PageIndex{10}$$

Suppose a team of two-dimensional explorers living in $$(X_k,G_k)$$ tethers one of their team to a line $$18$$ scrambles long (a scramble is the standard unit for measuring length in this world - and $$24$$ scrambles equals one tubablast). They swing the volunteer around in a circle, and though he laughs maniacally, he is able to record with confidence that he traveled $$113.4$$ scrambles. Assuming these measurements are correct, which type of geometry applies to their universe? Can they determine the value of $$k$$ from these measurements? If so, what is it?

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