Gaussian Curvature: Use Gauss Map to Find Sphere/Cylinder Radii

[halvizo1031]

I really need help answering these questions:

use the gauss map to find the gaussian curvature of a sphere of radius r at any point. also, use the gauss map to find the gassian curvature of a cylinder of radius r at any point.

 

[VKint]

Warning: long post ahead!

The Gaussian curvature is defined as the product of the eigenvalues of the Gauss map. Thus, all you need to do is find the matrix of the Gauss map and calculate its determinant.

The way to go about this is to find the first and second fundamental forms. Let the surface be parametrized by r(u,v). The first fundamental form defines the inner product on the tangent space to the surface in the basis {r_u,r_v} (where subscripts indicate partial differentiation). It has matrix
(E F)
(F G) = g,
where E = r_u^.r_v, F = r_u^.r_v, and G = r_v^.r_v (sorry, but Latex seems to be failing to work at the moment). If S = (r_u,r_v) is the matrix with columns equal to the partial derivatives of r, then g = S^TS. The second fundamental form is defined to be
(l m)
(m n) = A,
where l = r_u^.W(r_v) (here, W denotes the Gauss map), m = r_u^.W(r_v), and n = r_v^.W(r_v) (I'm guessing that you know all this already; I'm just making sure we're on the same page). Since W(v) = -DN(v), where N is the unit normal and D denotes the Jacobean, these expressions reduce to
l = r_uu^.N
m = r_uv^.N
n = r_vv^.N
(try verifying this yourself). The first definition of the second fundamental form is equivalent to saying that A = S^TWS, where S is the same as above, and W again denotes the matrix of the Gauss map (again, verify this for yourself). We can rewrite this as
A = S^TSS^-1WS, i.e. (remembering that g = S^TS),
g^-1A = S^-1WS.
Thus, since similar matrices have equal determinants, we have K = det(W) = det(A)det(g^-1) = det(A) / det(g), where K is the Gaussian curvature. This says that all you need to do is calculate the determinants of the first and second fundamental forms to find the Gaussian curvature.

For the sphere, use the usual parametrization x = r sin(p) cos(t), y = r sin(p) sin(t), z = r cos(p) (where p = phi and t = theta). Find a usable expression for N (this should be easy). Then calculate all the necessary partial derivatives, plug into the above formulas for E, F, G, l, m, and n, find the necessary determinants, and you're done; the same process works for the cylinder.

As a final comment, it's interesting to note (perhaps you've already covered this) that the Gaussian curvature can in fact be calculated only using the components of g, the first fundamental form (usually called the Riemannian metric in this context). Specifically, there's a famous formula for K involving only the Christoffel symbols associated to the metric. This is usually not a good way to go about finding K when an explicit parametrization is known, however; the above process is much faster, and less easy to screw up.