It is said that we shall naturally find the Hyperbolic Non-Euclidean world in projective geometry. And we

Let us go back to the Euclidean world, and forget about the Hyperbolic Non-Euclidean world for the time being. Then we will try to see infinity with pictures as usual.

Look at Fig. 1. The camera is the projection center and takes the picture of triangle

Fig. 2 is a bird's-eye view of the situation in Fig. 1. It shows the correct shape of triangle

Fig. 3 is the side view of the situation in Fig. 1. The black dotted line indicates the direction of horizon that we can see through the viewfinder of camera.

Fig. 4 is the front view of Panel-screen with triangle

Fig. 5 explains why a straight line is unchanged by projective transformation. It is self-evident, isn't it?

Fig. 6 is a projection of parallel lines. For easy to see, the camera etc. are omitted, and the ground color of Panel-screen is in yellow. The parallel lines on Panel-screen look to have end points and meet each other at horizon .

Look at Fig. 7. We drew two pairs of parallel lines on the ground and projected them to Panel-screen. Projective rays are omitted. The blue prallel lines also meet each other at horizon.

Fig. 8 shows the two pairs of parallel lines on Panel-screen. The lower right is the parallelogram formed by them. Mathematicians call the horizon on Panel-screen a

Fig. 9 shows 180 camera shoot with many rectangular Panel-screens. The chained Panel-screens form a half cylinder. It helps us to take a full picture of the entire range of parallel lines. Parallel lines on the ground look like a railroad, but not on chained Panel-screens.

Fig. 10 shows unrolled Panel-screens of Fig. 9. Points

Fig. 11 shows two pairs of parallel lines on a sphere. They are projected from the ground to a hemisphere instead of the chained Panel-screens. The center of hemisphere is the projection center. The hemisphere can be considered as that it consists of infinite number of micro Panel-screens (tangent planes). The edge (circumference of the equator) of hemisphere is now a line at infinity.

Fig. 11' shows the longitudinal section with rays through the center of hemisphere in Fig. 11. The rays are from points at regular intervals on the ground. The left area and rays gettig horizontally are omitted.

Now let us regard the hemisphere as a plane and think of a geometry on it. The great circle is naturally considered as a straight line. However, we have a problem. In Euclidean geometry two points defines only one straight line. Two red straight lines reach the two points

OK. Let us accept it anyway. But, what plane is it? How can we join point

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