
# 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.

Exercise $$\PageIndex{1}$$

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 $$C$$.

1. 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.

2. 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 $$\triangle A'B'C'$$.