Wikipedia:Reference desk/Archives/Mathematics/2016 March 21

= March 21 =

Mapping the sphere to the plane
Suppose I have a unit sphere, described using spherical polar coordinates. A surface area element has area $$\mathrm{d}A=\sin\theta\mathrm{d}\theta\mathrm{d}\phi$$.

Next, suppose I want to project this to a plane in such a way that distances from the point with $$\theta=0$$ are correct. This requires that my projection, if described in plane polar coordinates, has radius $$r=\theta$$.

Further, I want areas on the sphere to correspond to areas on the plane. Since an area element on the plane is given by $$\mathrm{d}A=r\mathrm{d}r\mathrm{d}\Phi$$, by comparing area elements it seems irresistible to conclude that one way of achieving this is by $$\Phi=\frac{\sin\theta}{\theta}\phi$$, or perhaps $$\Phi=\frac{\sin\theta}{\theta}(\phi-\pi/2)+\pi/2$$. Obviously, this won't fill the whole plane, but this is not my intention.

Does this work, or have I gone wrong somewhere?--Leon (talk) 21:10, 21 March 2016 (UTC)


 * Various possibilities are discussed at Map_projection.  S ławomir  Biały  21:17, 21 March 2016 (UTC)


 * I think that's the Werner projection (which does have the properties you want). -- BenRG (talk) 07:02, 22 March 2016 (UTC)