Geometry Second Semester 2010/11 Homework 9

1. Let α: I → C be a local parametrisation of the oriented curve C of the form α(t) = (x(t), y(t)).

Show that

(κ ∘ α)(t) = \frac{x′(t)y′′(t) − y′(t)x′′(t)}{(x′(t)^{2} + y′(t)^{2})^{\frac{3}{2}}}.

2. Find global parametrisations of f^{−1}(0) oriented by ∇f/∥∇f∥ in each of the following cases:

(a) f(x_{1}, x_{2}) = ax_{1} + bx_{2} − c with (a, b) ≠ (0, 0);

(b) f(x_{1}, x_{2}) = x_{1}^{2}/a^{2} + x_{2}^{2}/b^{2} − 1 with a, b ≠ 0;

(c) f(x_{1}, x_{2}) = x_{2} − ax_{1}^{2} − 1 with a, b ≠ 0; and

(d) f(x_{1}, x_{2}) = x_{1}^{2} − x_{2}^{2} − 1 defined for x_{1} > 0.

3. Find the curvature of each of the oriented plane curves in Exercise 2.

4. Let C be a plane curve oriented by the unit normal vector field n. Let α: I → C be a unit speed local parametrisation for C. For t ∈ I, let T(t) = α̇(t). Show that

\begin{cases}Ṫ = (κ ∘ α)(n ∘ α), & and \\ (n ∘ α)˙ = −(κ ∘ α)T. \end{cases}

These formulae are called the Frenet formulae for a plane curve.

5. Let S = f^{−1}(c) be an n-surface in R^{n+1} oriented by ∇f/∥∇f∥. For v = (p, v_{1}, v_{2}, …, v_{n+1}) ∈ S_{p}, with p ∈ S, we define the second fundamental form of S at p on v to be 𝒮_{p}(v) := L_{p}(v)·v. Show that

𝒮_{p}(v) = \frac{−1}{∥∇f∥} ∑_{i,j=1}^{n+1} ∂_{i,j}^{2}f(p)v_{i}v_{j}.

(When ∥v∥ = 1, then k(v) = 𝒮_{p}(v), so this formula provides a useful way to compute normal curvatures.)

6. Find the normal curvature k(v) for each direction v, the principal curvatures and principal curvature directions at the point p for the 3-surface f^{−1}(0) oriented by ∇f/∥∇f∥ in each of the following cases.

(a) f(x_{1}, x_{2}, x_{3}) = x_{1}^{2}/a^{2} + x_{2}^{2}/b^{2} + x_{3}^{2}/c^{2} − 1, p = (a, 0, 0)

(b) f(x_{1}, x_{2}, x_{3}) = x_{1}^{2}/a^{2} + x_{2}^{2}/b^{2} − x_{3}^{2}/c^{2} − 1, p = (a, 0, 0)

(c) f(x_{1}, x_{2}, x_{3}) = x_{1}^{2} + (\sqrt{x_{2}^{2} + x_{3}^{2}} − 2)^{2} − 1,

i. p = (0, 3, 0) ii. p = (0, 1, 0)