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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.
Plane Euclidean Geometry | Spherical Geometry |
---|---|
Line | Great Circle |
Line Segment | Arc of a great circle |
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 ℓ (like the x-axis). We will mark the north pole of this great circle.