Let x = x(u, v) be a regular parametrized surface. A parallel surface to x is a parametrized surface y(u, v) = x(4, v) + aN(u, v), where a is a constant. a. Prove that y, A Y, = (1 – 2Ha + Ka²)(xu ^ X,), where K and H are the Gaussian and mean curvatures of x, respectively. b. Prove that at the regular points, the Gaussian curvature of y is K 1– 2Ha + Ka²

Let x = x(u, v) be a regular parametrized surface. A parallel surface to x is a parametrized surface y(u, v) = x(4, v) + aN(u, v), where a is a constant. a. Prove that y, A Y, = (1 – 2Ha + Ka²)(xu ^ X,), where K and H are the Gaussian and mean curvatures of x, respectively. b. Prove that at the regular points, the Gaussian curvature of y is K 1– 2Ha + Ka²

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let x = x(u, v) be a regular parametrized surface. A parallel surface to x is a
parametrized surface
y(u, v) = x(u, v) + aN(u, v),
where a is a constant.
a. Prove that y, A y, = (1 – 2Ha + Ka²)(xu ^ x,), where K and H are the
Gaussian and mean curvatures of x, respectively.
b. Prove that at the regular points, the Gaussian curvature of y is
K
1- 2На + Ка2
and the mean curvature of y is
Н - Ка
1- 2На + Ка?

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