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 $\triangle A'B'C'$.