scorpion's blog

By scorpion, 10 years ago, In English

Hi, everybody.

I'm interested in how to solve the next problem 18.

It's obvious, that if curve is plane.

Transform torsion at point ρ(s) as , where — is geodesic torsion at the point, and φ is angle between osculating plane and tangent plane. , where k1 and k2 are principal curvatures.

Also, if we consider the case, when curve doesn't lay in plane, and surface is a sphere. At any point on sphere k1(s) = k2(s), and we have to prove that .

In other cases we are able to find a point, where k1 ≠ k2. And I guess we should constract such curve in a neighbourhood of that point .

Any ideas how to complete this solution? Or any others solutions are welcome.

UPD1 Solution for sphere.

Let . If the curve lays on the sphere then the equality holds. By means of a little transformation of this equality, we can obtain . , because the radius of curvature is constannt in any point on the sphere.

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You should've start with:

"I solved first 17 problems and got 2147 rating, but I can't solve 18'th, help me, I want to be red"

that's the only way to make everyone willing to help