Chapter Questions
Show that the area of an (infinitesimal) triangle with vertices $(x, y) \cdot(x+d x, y+d y) \cdot(x+\delta x, y+\delta y)$ is equal to $\frac{1}{2}(d x \delta y-d y \delta x)$.
Show that if a surface is given in the form $z=z(x, y)$, then the measure of curvature $k$ can be expressed as$$k=\frac{z_{u s} z_{y y}-z_{x p}^2}{\left(1+z_z^2+z_y^2\right)^2} .$$
Calculate the curvature function $k$ of the paraboloid $z=$ $x^2+y^2$
If $x=x(u, v), y=y(u, v), z=z(u, v)$ are the parametric equations of a surface and if $E=x_{i n}^2+y_x^2+z_{i n}^2, F=$ $x_u x_{\mathrm{r}}+y_{\mathrm{u}} y_{\mathrm{r}}+z_{\mathrm{u}} \varepsilon_{\mathrm{r}}$. and $G=x_r^2+y_r^2+z_r^2$. show that$$d x^2+d y^2+d z^2=E d u^2+2 F d u d v+G d v^2 .$$
Calculate E, F, G on the unit sphere parametrized by $x=\cos u \cos v, y=\cos u \sin v, z=\sin u$, and show that $d s^2=d u^2+\cos ^2 u d v^2$.
Derive the formula $\cos a=\cos b \cos c+\sin b \sin c \cos A$ for an arbitrary spherical triangle with sides $a, b, c$ and opposite angles $A, B, C$ on a sphere of radius 1 by dividing the triangle into two right triangles and applying the formulas of Chapter 4.
Show that the formula in exercise 6 changes to$$\cos \frac{a}{K}=\cos \frac{b}{K} \cos \frac{c}{K}+\sin \frac{b}{K} \sin _K^c \cos A$$if the sphere has radius $K$.
By using power series, show that Taurinus's "log-spherical" formula$$\cosh \frac{a}{K}=\cosh \frac{b}{K} \cosh \frac{c}{K}-\sinh \frac{b}{K} \sinh \frac{c}{K} \cos A$$reduces to the law of cosines as $K \rightarrow \infty$.
Show that Taurinus's formula for an asymptotic right triangle on a sphere of imaginary radius $i$, namely $\sin B=$ $1 / \cosh x$, is equivalent to Lobachevsky's formula for the angle of parallelism, tan $\frac{\beta}{2}=e^{-x}$.
Show that the circumference of a circle of radius $r$ on the sphere of imaginary radius $i K$ is $2 \pi K \sinh \frac{r}{\kappa}$. Show that this value approaches $2 \pi r$ as $K \rightarrow \infty$.
Given that $\tan \frac{1}{2} \Pi(x)=e^{-x}$ where $\Pi(x)$ is Lobachevsky's angle of parallelism, derive the formulas$$\sin \Pi(x)=\frac{1}{\cosh x} \quad \text { and } \quad \cos \Pi(x)=\tanh x$$and show that their power series expansions up to degree 2 are $\sin \Pi(x)=1-\frac{1}{2} x^2$ and $\cos \Pi(x)=x$. respectively.
Substitute the results of exercise 11 into Lobachersky's formulas$$\begin{aligned}& \sin A \tan \Pi(a)=\sin B \tan \Pi(b) \\& \cos A \cos \Pi(b) \cos \Pi(c)+\frac{\sin \Pi(b) \sin \Pi(c)}{\sin \Pi(a)}=1\end{aligned}$$to derive the laws of sines and cosines when the sides $a, b$. $c$ of the non-Euclidean triangle are "small."
Show that if $A B C$ is an arbitrary triangle with sides $a, b, c$, then the formulas $a \sin (A+C)=b \sin A$ and $\cos A+\cos (B+C)=0$, along with the law of sines, imply that $A+B+C=\pi$.
Show that Lobachersky's basic triangle formulas (Eqs. 17.3-17.6) transform into standard formulas of spherical trigonometry if one replaces the sides $a, b, c$ of the triangle by $i a, i b, i c$, respectively. (For simplicity, assume that angle $C$ is a right angle.)
Describe geometrically Beltrami's parametrization of the sphere of radius $k$ given by$$\begin{aligned}& x=\frac{u k}{\sqrt{a^2+u^2+\mathrm{v}^2}} \quad y=\frac{v k}{\sqrt{a^2+u^2+v^2}} \\& z=\frac{a k}{\sqrt{a^2+u^2+v^2}} .\end{aligned}$$
Show that replacing $u, v$ by $i u, i$, respectively, transforms the sphere of exercise 15 with curvature $1 / k^2$ to a pseudosphere with curvature $-1 / k^2$.
Show that Beltrami's formulas for the lengths $p, s, t$ of the sides of a right triangle on his pseudosphere transform into$$\begin{aligned}& \frac{r}{a}=\tanh \frac{\rho}{k} \quad \frac{r}{a} \cos \theta=\tanh \frac{s}{h} \\& \frac{v}{\sqrt{a^2-u^2}}=\tanh \frac{t}{k}\end{aligned}$$and then show that$$\cosh \frac{s}{k} \cosh \frac{l}{k}=\cosh \frac{p}{h} .$$
Demonstrate how a central projection can transform parallel lines into intersecting lines.
Demonstrate the following relationships for the cross ratio:$$(A B, C D)=1-(A C, B D) \quad(A B, C D)=\frac{1}{(A B, D C)}$$
Denote the cross ratios $(A B, C D),(A C, D B),(A D, B C)$ by $\lambda, \mu, \nu$, respectively, Show that$$\lambda+\frac{1}{\mu}=\mu+\frac{1}{\nu}=v+\frac{1}{\lambda}=-\lambda \mu \nu=1 .$$
Show that if the point $p^{\prime}$ lies on the polar $\pi$ of a point $p$ with respect to a conic $C$, then $\pi^{\prime}$, the polar of $p^{\prime}$, goes through p.
Determine homogeneous coordinates of the points (3,4) and $(-1.7)$.
Write the homogeneous coordinates of the point at infinity on the line $2 x-y=0$.
Determine rectangular coordinates of the points (3,1,1) and (4, $-2,2$ ) given in homogeneous coordinates.
Determine the equation (in rectangular coordinates) of a line that passes through the point at infinity ( $2,1,0$ ) and the point ( $6,2,2$ ).
Show that every circle in the plane passes through the two points at infinity $(1, i, 0)$ and $(1,-i, 0)$.
Given three collinear points $A, B, P$, show that the point $Q$ determined by the construction in Fig. 17.17 makes $A$. $B$. $P, Q$ into a harmonic tetrad, that is, makes the cross ratio $(A B, P Q)$ equal to -1 .(FIGURE CAN'T COPY)
Using Klein's definition of distance $d$ in the interior of the circle representing the Lobachevskian plane, show that if $P$. $Q \cdot Q^{\prime}$ are three points on a line, then $d(P, Q)+d\left(Q \cdot Q^{\prime}\right)=$ $d(P, Q)$.
Letting $i, j, k$ be first-order units in three-dimensional space, determine the combinatory product of $2 i+3 j-4 k$. $3 i-j+k . i+2 j-k$.
Show that in Grassmann's combinatory multiplication.$$\begin{gathered}\left(\sum \alpha_{l i} \epsilon_i\right)\left(\sum \alpha_{2, \epsilon_i}\right) \cdots\left(\sum \alpha_{m i} \epsilon_{\mathrm{l}}\right) \\=\operatorname{det}\left(\alpha_{i j}\right)\left[\epsilon_1 \epsilon_2 \cdots \epsilon_n\right] .\end{gathered}$$where each linear combination is of a given set of $n$ firstorder units and where $\left\{\epsilon_1 \epsilon_2 \cdots \epsilon_n \mid\right.$ is the single unit of $n$th order.
If $\omega$ is the differential one-form in three dimensions given by $\omega=A d x+B d y+C d z$, show that$$\begin{aligned}d \omega & =\left(\frac{\partial C}{\partial y}-\frac{\partial B}{\partial z}\right) d y d z+\left(\frac{\partial A}{\partial z}-\frac{\partial C}{\partial x}\right) d z d x \\& +\left(\frac{\partial B}{\partial x}-\frac{\partial A}{\partial y}\right) d x d y\end{aligned}$$
Show that the exterior derivative of $\omega=A d y d z+$ $B d z d x+C d x d y$ is the three-form$$\left(\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}\right) d x d y d z$$
Show that $d(d \omega)=0$ for $\omega$ a differential one-form or twoform in three-dimensional space.
Let $\omega$ be the two-form in $R^3-\{0\}$ given by$$\omega=\frac{x d y d z+y d z d x+z d x d y}{\left(x^2+y^2+z^2\right)^{3 / 2}} .$$Show that $d \omega=0$, but that there is no one-form $\eta$ such that $d \eta=\omega$.
Study several new high school geometry texts. Do they follow Euclid's axioms or Hilbert's axioms or some combination? Comment on the usefulness of using Hilbert's reformulation in teaching a high school geometry class.
Is the analytic form of non-Euclidean geometry as presented by Taurinus, Lobachevsky, and Beltrami a better way of presenting the subject than the synthetic form? How can one make sense of a sphere of imaginary radius?
Why were both of the mathematicians who first published accounts of non-Euclidean geometry from countries not in the mainstream of nineicenth-century mathematics? Is this by chance or are there substantive reasons?
Read a complete version of Riemann's lecture "On the Hypotheses which lie at the Foundation of Geometry." Describe Riemann's major new ideas and comment on how they have been followed up in the twentieth century. In particular, comment on the oft-repeated statement that Riemann's work was a precursor of Einstein's general theory of relativity.