Stereographic projection

Section 1.3 Stereographic projection

Subsection 1.3.1 Stereographic projection \(S^1\to \extR\)

Let \(S^1= \{(x,y)\in \R^2 \colon x^2+y^2=1\}\) denote the unit circle in the \(x,y\)-plane. Let \(N=(0,1)\) denote the “north pole” (that is, the point at the “top” of the unit circle). Given a point \(P=(x,y)\neq N\) on the unit circle, let \(s(P)\) denote the intersection of the line \(\overline{NP}\) with the \(x\)-axis. See Figure 1.3.1. The map \(s\colon S^1\setminus \{N\}\to \R\) given by this rule is called stereographic projection. Using similar triangles, it is easy to see that \(s(x,y)=\frac{x}{1-y}\text{.}\)

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Figure 1.3.1. Stereographic projection
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Checkpoint 1.3.2.

Draw the relevant similar triangles and verify the formula \(s(x,y) = \frac{x}{1-y}\text{.}\)

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We extend stereographic projection to the entire unit circle as follows. We call the set

\begin{equation} \extR=\R\cup \{\infty\}\tag{1.3.1} \end{equation}

the extended real numbers, where “\(\infty\)” is an element that is not a real number. Now we define stereographic projection \(s\colon S^1 \to \extR\) by

\begin{equation} s(x,y) = \begin{cases} \frac{x}{1-y} \amp y\neq 1\\ \infty \amp y=1. \end{cases}\tag{1.3.2} \end{equation}

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Subsection 1.3.2 Stereographic projection \(S^2\to \extC\)

The definitions in the previous subsection extend naturally to higher dimensions. Here are the details for the main case of interest.

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Let \(S^2 = \{(a,b,c)\in \R^3\colon a^2+b^2+c^2=1\}\) denote the unit sphere in \(\R^3\text{.}\) Let \(N=(0,0,1)\) denote the “north pole” (that is, the point at the “top” of the sphere, where the positive \(z\)-axis is “up”). Given a point \(P=(a,b,c)\neq N\) on the unit sphere, let \(s(P)\) denote the intersection of the line \(\overline{NP}\) with the \(x,y\)-plane, which we identify with the complex plane \(\C\text{.}\) See Figure 1.3.3. The map \(s\colon S^2\setminus \{N\}\to \C\) given by this rule is called stereographic projection. Using similar triangles, it is easy to see that \(s(a,b,c)=\frac{a+ib}{1-c}\text{.}\)

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Figure 1.3.3. Stereographic projection
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We extend stereographic projection to the entire unit sphere as follows. We call the set

\begin{equation} \extC=\C\cup \{\infty\}\tag{1.3.3} \end{equation}

the extended complex numbers, where “\(\infty\)” is an element that is not a complex number. Now we define stereographic projection \(s\colon S^2 \to \extC\) by

\begin{equation} s(a,b,c) = \begin{cases} \frac{a+ib}{1-c} \amp c\neq 1\\ \infty \amp c=1. \end{cases}\tag{1.3.4} \end{equation}

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Subsection 1.3.3 Conjugate Transformations

Let \(\mu \colon X\to Y\) be a bijective map. We say that maps and \(f\colon X\to X\) and \(g\colon Y\to Y\) are conjugate transformations (with respect to the bijection \(\mu\)) if \(f = \mu^{-1}\circ g\circ \mu\text{.}\) Exercise Group 1.3.4.3–6 demonstrate examples of this definition for which \(\mu\) is stereographic projection.

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Checkpoint 1.3.4.

Show that \(f=\mu^{-1}\circ g \circ \mu\) if and only if \(\mu\circ f = g\circ \mu\text{.}\)

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Exercises 1.3.4 Exercises

Formulas for inverse stereographic projection.

It is intuitively clear that stereographic projection is a bijection. Make this rigorous by finding a formula for the inverse.

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1.

For \(s\colon S^1\to \extR\text{,}\) find a formula for \(s^{-1}\colon \extR\to S^1\text{.}\) Find \(s^{-1}(3)\text{.}\)

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Answer.

\begin{equation*} s^{-1}(r) = \begin{cases} \left(\frac{2r}{r^2+1},\frac{r^2-1}{r^2+1}\right) \amp \text{ if } r\neq \infty\\ (0,1)\amp \text{ if } r=\infty \end{cases} \end{equation*}

\begin{equation*} s^{-1}(3) = (3/5,4/5) \end{equation*}

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2.

For \(s\colon S^2\to \extC\text{,}\) find a formula for \(s^{-1}\colon \extC\to S^2\text{.}\) Find \(s^{-1}(3+i)\text{.}\)

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Answer.

\begin{equation*} s^{-1}(z) = \begin{cases} \left(\frac{2\re(z)}{|z|^2+1},\frac{2\im(z)}{|z|^2+1}, \frac{|z|^2-1}{|z|^2+1}\right)\amp \text{ if } z\neq \infty\\ (0,0,1) \amp \text{ if } z=\infty \end{cases} \end{equation*}

\begin{equation*} s^{-1}(3+i) = (6/11,2/11,9/11) \end{equation*}

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Transformations that are conjugate via stereographic projection.

Suggestion: Use Checkpoint 1.3.4 for the exercises that follow.

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3.

Let \(f\colon S^1\to S^1\) and \(g\colon \extR\to \extR\) be given by

\begin{align*} f(x,y)\amp =(x,-y)\\ g(x)\amp =\begin{cases}1/x \amp x\neq 0,\infty\\ \infty \amp x=0\\ 0 \amp x=\infty.\end{cases} \end{align*}

Show that \(f,g\) are conjugate transformations with respect to stereographic projection.

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4.

Let \(R_{Z,\theta}\colon S^2\to S^2\) and \(T_{Z,\theta}\colon \extC\to \extC\) be given by

\begin{align*} R_{Z,\theta}(a,b,c)\amp = (a\cos\theta-b\sin\theta,a\sin\theta+b\cos\theta,c)\\ T_{Z,\theta}(z)\amp = \begin{cases} e^{i\theta}z \amp z\neq \infty\\ \infty\amp z=\infty.\end{cases} \end{align*}

Show that \(R_{Z,\theta},T_{Z,\theta}\) are conjugate transformations with respect to stereographic projection.

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5.

Let \(R_{X,\pi}\colon S^2\to S^2\) and \(T_{X,\pi}\colon \extC\to \extC\) be given by

\begin{align*} R_{X,\pi}(a,b,c)\amp =(a,-b,-c)\\ T_{X,\pi}(z)\amp =\begin{cases}1/z \amp z\neq 0,\infty\\ \infty \amp z=0\\ 0 \amp z=\infty.\end{cases} \end{align*}

Show that \(R_{X,\pi},T_{X,\pi}\) are conjugate transformations with respect to stereographic projection.

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6.

Let \(R_{X,\pi/2}\colon S^2\to S^2\) and \(T_{X,\pi/2}\colon \extC\to \extC\) be given by

\begin{align*} R_{X,\pi/2}(a,b,c)\amp =(a,-c,b)\\ T_{X,\pi/2}(z)\amp =\begin{cases} \frac{z+i}{iz+1}\amp z\neq i,\infty\\ \infty \amp z=i\\ -i \amp z=\infty.\end{cases} \end{align*}

Show that \(R_{X,\pi/2},T_{X,\pi/2}\) are conjugate transformations with respect to stereographic projection.

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7. Projections of endpoints of diameters.

Show that \(s(a,b,c)(s(-a,-b,-c))^\ast=-1\) for any point \((a,b,c)\) in \(S^2\) with \(|c|\neq 1\text{.}\) Conversely, suppose that \(zw^\ast=-1\) for some \(z,w\in \C\text{.}\) Show that \(s^{-1}(z)=-s^{-1}(w)\text{.}\)

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