Exercise 7 Page 891

We are told that, in plane Euclidean geometry, the points on any line or line segment can be put into one-to-one correspondence with the real numbers. Thus, let's try to transform a great circle into a line or line segment.

To do that, we start by marking some angles around the circle.

Next, we will unfold the circle so that we get a line segment. We will assign each point on the circle to its corresponding angle. We will show just a few points.

Finally, the line segment at the left can be put into one-to-one correspondence with the real numbers. That way, we've put all the points on any circle in correspondence with the real numbers. Before concluding, let's recall some correspondences between plane geometry and spherical geometry.

Alternative Solution

Alternative Solution

We know that the points of a line can be put into one-to-one correspondence with the real numbers. Below we show an example.

Next, let's draw a great circle of radius $3,$ for example, and one numbered line $\ell$ (like the $x\text{-}$axis). We will mark the north pole of this great circle.

Then, for any point $P$ on the circle, we will draw the line passing through it and the north pole $N.$ Let $Q$ be the intersection of this line and line $\ell .$

One way to put the points on a great circle into one-to-one correspondence with real numbers is to assign each point $P$ with the corresponding point $Q.$ That way, any point on the great circle is assigned to one and only one real number.

$$\begin{array}{c}P\stackrel{f}{⟶}Q\end{array}$$

To write the corresponding statement, recall that in spherical geometric a line is a $\text{great circle}$ and a line segment is an $\text{arc of a great circle}.$

$$\begin{array}{c}\text{The points on any great circle or arc of}\\ \text{a great circle can be put into one-to-one}\\ \text{correspondence with real numbers}\end{array}$$

The function $f$ we presented above is called the stereographic projection.