Transition Between Stereographic Projections

Level 1

Let $f_{+}: S^2 - \{N\} \to \mathbb{R}^2$ denote stereographic projection from the north pole of $S^2$ , as defined in the wiki Homeomorphism . Similarly, let $f_{-} : S^2 - \{S\} \to \mathbb{R}^2$ denote stereographic projection from the south pole of $S^2$ , i.e. from the point $S = (0,0,-1)$ .

On $\mathbb{R}^2$ , the function $g:= f_{+} \circ f_{-}^{-1}$ is well-defined. If $g(3,4) = (a,b)$ , what is $a+b$ ?

\frac{6}{25}

\frac{7}{25}

\frac{8}{25}

\frac{9}{25}

1 solution

Sameer Kailasa
Mar 7, 2016

Note that $g$ is performing inversion with respect to the circle $x^2 + y^2 = 1$ . Thus, for a point $p\in \mathbb{R}^2$ , we have $g(p) = p/\|p\|^2$ . It follows $g(3,4) = (3,4)/25 = (3/25, 4/25)$ , so $a+b = \mathbf{7/25}$ .

I don't understand why $g$ is performing inversion with respect to the circle, it does not seem obvious at all. Did you prove it algebraically? I could not find the reciprocal for $f_-$ .

Alexis D - 3 months, 3 weeks ago

Transition Between Stereographic Projections

1 solution

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