16.5: Central projection
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- Sep 5, 2021
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\triangle A'B'C'\).
