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I have fun with this website. It is very helpful and great. I have a question in geometry. I try to solve it but it needs imagination to figure out the point on the surface. This question makes me confused about other exercises. I hope someone solves this question. (I see this question needs a real expert in geometry.)

(Lambert's cylindrical projection) Consider the projection of $S^2−(0,0,\pm 1)$ onto a unit cylinder by radial projection; that is

$$\Pi: (x,y,z) \to( \frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}, z)$$

Check that this map preserves area that it is not a conformal map. Thank you so much for helping. Good luck.

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Oh, dear: please do use LaTex when writing mathematics in this site! Go to the FAQ section for diretions. –  DonAntonio Feb 25 '13 at 12:24
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Hint: What is the surface element on the sphere, in spherical coordinates $(\theta,\phi)$? And what is the surface element on the cylinder, in $(\theta,z)$ coordinates? Note that $z=\cos\phi$ and try to connect the two. As for non-conformality, check how infinitimal arclengths along the sphere map to the cylinder, in the latitudinal and longitudinal direction respectively. –  Harald Hanche-Olsen Feb 25 '13 at 12:52

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