16.5: Central projection
The central projection is analogous to the projective model of hyperbolic plane which is discussed in Chapter 17 .
Let \(\Sigma\) be the unit sphere centered at the origin which will be denoted by \(O\) . Suppose that \(\Pi^+\) denotes the plane defined by the equation \(z=1\) . This plane is parallel to the \(xy\) -plane and it passes thru the north pole \(N =(0,0,1)\) of \(\Sigma\) .
Recall that the northern hemisphere of \(\Sigma\) , is the subset of points \((x,y,z)\in \Sigma\) such that \(z>0\) . The northern hemisphere will be denoted by \(\Sigma^+\) .
Given a point \(P\in \Sigma^+\) , consider the half-line \([OP)\) . Suppose that \(P'\) denotes the intersection of \([OP)\) and \(\Pi^+\) . Note that if \(P=(x,y,z)\) , then \(P'=(\dfrac{x}{z},\dfrac{y}{z},1)\) . It follows that \(P\leftrightarrow P'\) is a bijection between \(\Sigma^+\) and \(\Pi^+\) .
The described bijection \(\Sigma^+ \leftrightarrow \Pi^+\) is called the central projection of the hemisphere \(\Sigma^+\) .
Note that the central projection sends the intersections of the great circles with \(\Sigma^+\) to the lines in \(\Pi^+\) . The latter follows since the great circles are intersections of \(\Sigma\) with planes passing thru the origin as well as the lines in \(\Pi^+\) are the intersection of \(\Pi^+\) with these planes.
The following exercise is analogous to Exercise 17.2.1 in hyperbolic geometry.
Let \(\triangle_sABC\) be a nondegenerate spherical triangle. Assume that the plane \(\Pi^+\) is parallel to the plane passing thru \(A\) , \(B\) , and \(C\) . Let \(A'\) , \(B'\) , and \(C'\) denote the central projections of \(A\) , \(B\) and \(C\) .
- Show that the midpoints of \([A'B']\) , \([B'C']\) , and \([C'A']\) are central projections of the midpoints of \([AB]_s\) , \([BC]_s\) , and \([CA]_s\) respectively.
- Use part (a) to show that the medians of a spherical triangle intersect at one point.
- Hint
-
(a). Observe and use that \(OA' = OB' = OC'\).
(b). Note that the medians of spherical triangle ABC map to the medians of Euclidean a triangle \(A'B'C'\). It remains to apply Theorem 8.3.1 for \(\triangle A'B'C'\).