Compute the Gauss map and its derivative for the cone parameterized as follows

X(u,v) =(vcosu, vsinu, v)

What does the image of the Gauss map on the sphere look like?

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- Apr 10th 2013, 10:21 AMarisederholmGauss map and its derivative
Compute the Gauss map and its derivative for the cone parameterized as follows

X(u,v) =(vcosu, vsinu, v)

What does the image of the Gauss map on the sphere look like? - Apr 10th 2013, 01:54 PMxxp9Re: Gauss map and its derivative
Xu = ( -vsinu, vcosu, 0 )

Xv = (cosu, sinu, 1 )

So the normal vector N = Xu x Xv = v(cosu, sinu, -1), the unit normal is n=(cosu, sinu, -1)/\sqrt{2}, and it is the Gauss map G. Its image is a circle on the unit sphere.

The derivative at p=X(u,v) is a linear map dG from tangent space Xp to the tangent space of the unit sphere S_G(p) such that:

dG(Xu) = (-sinu, cosu, 0)/\sqrt{2}

dG(Xv) = 0