Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 8.

1. Problem 1.

a) Consider a curve α(u) = (ρ(u), z(u)) in the (ρ, z) plane, ρ = ρ(u) > 0. If this curve is rotated about the z-axis, we obtain a surface of revolution. We may parametrize this surface as follows

r(u, v) = (ρ(u) cos v, ρ(u) sin v, z(u)), −π < v < π.

Prove that r is a simple surface if the original curve α was regular and one-to-one by computing r_{u}, r_{v}, and n.

b) Consider the Monge patch r(u, v) = (u, v, uv). Find the equation of the tangent plane at the point r(1, 2). Prove that this tangent plane intersects the surface in two lines:

α(t) = r(t, 2) = (t, 2, 2t) and β(t) = r(1, t) = (1, t, t).

Hint: Prove that α and β do actually lie in the tangent plane.

c) Write out the parametrization r(u, v) of the Möbius band. Compute n(u, 0) and show that

lim_{u→−π} n(u, 0) = − lim_{u→π} n(u, 0)

while

lim_{u→−π} r(u, 0) = lim_{u→π} r(u, 0).

2. Problem 2.

a) Let S^{2} = {(u, v, w) ∈ ℝ^{3} : u^{2} + v^{2} + w^{2} = 1} and ℝ^{2} = {(u, v, w) ∈ ℝ^{3} : w = 0}. If (u, v, 0) belongs to ℝ^{2}, the line determined by (u, v, 0) and (0, 0, 1) intersects S^{2} in a point other than (0, 0, 1). Denote this point by r(u, v). Compute the actual form of r(u, v) and show that r : ℝ^{2} → ℝ^{3} is a simple surface. The inverse mapping to r is called the stereographic projection.

b) Let α : (a, b) → ℝ^{3} be a unit speed curve with k ≠ 0 and let

U = {(u, v) ∈ ℝ^{2} : a < u < b, v ≠ 0}.

Define r : U → ℝ^{3} by r(u, v) = α(u) + vα′(u). Prove that r is a simple surface, provided r one-to-one. It is called the tangent developable surface of α.

c) Let

r(u, v) = (sin u cos v, 2 sin u sin v, 3 cos u), −1 < u < 1, 0 < v < π.

Show that r is a simple surface. What is it?

3. Problem 3.

Let P be a convex polygon in ℝ^{2} with faces (that are actually edges) F_{j} of length c_{j} = l(F_{j}) and unit normals (to faces) n_{j}.

a) Prove that ∑_{j} c_{j}n_{j} = 0.

b) Let (ξ, η) ∈ ℝ^{2} be such that ξ^{2} + η^{2} = 1. Prove the Cauchy projection formula

length(P |(ξ, η)^{⊥}) = \frac{1}{2} ∑_{j} c_{j}|(ξ, η) · n_{j}|,

where P |(ξ, η)^{⊥} is the projection of P onto the line

(ξ, η)^{⊥} = {(x, y) ∈ ℝ^{2} : (x, y) · (ξ, η) = 0},

i.e.,

P |(ξ, η)^{⊥} = {(x, y) ∈ ξ^{⊥} : (x, y) + λ(ξ, η) ∈ P for some λ ∈ ℝ}.

c)^{∗} Let c_{j} > 0 and let n_{j} be unit vectors in ℝ^{2}, j = 1, 2, ..., m, m ≥ 3, such that the condition in a) is true. Does there exist a convex polygon P such that c_{j} are lengths of its faces F_{j} and n_{j} are their normals?