Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 3.

1. Problem 1. Let α ∈ ℝ^{3} be a trace of a regular smooth curve α(t), and let P be a point on α. Assume that l is the line passing through P. If Q is another point on α, and h = dist(Q, l), d = dist(P, Q), the line l is called a tangent line provided h/d → 0 as Q → P. Prove that α(t) has a unique tangent line, and the tangent at P corresponding to t has a direction of the vector α′(t).

2. Problem 2. a) For a helix x = cos t, y = sin t, z = t at the point (1, 0, 0) write the equations of the tangent line, osculating plane, normal plane, normal, binormal.

b) What is the angle between the curves xy = c_{1} and x^{2} − y^{2} = c_{2}?

c) Prove that if the trace of the plane regular curve γ has the angle π/2 with the curves of the family φ(x, y) = const, φ_{x}^{2} + φ_{y}^{2} ≠ 0, then dx/φ_{x} = dy/φ_{y}.

d) Find the equation of the circle tangent of the second order to the parabola y = x^{2} at (0, 0).

3. Problem 3.

Write the equation of the osculating plane of the curve given by the equations

φ(x, y, z) = 0, ψ(x, y, z) = 0

at the point (x_{o}, y_{o}, z_{o}).

4. Problem 4. a) Prove that kτ = −T′ · B′.

b) Prove that [α′, α′′, α′′′] = k^{2}τ, provided α(s) is a unit speed curve.

c) Let α(t) be a regular curve. Suppose that there is a point a ∈ ℝ^{3} such that α(t)−a is orthogonal to T (t) for all t. Prove that α(t) lies on the sphere.

d) Let α(t) be a regular curve and let a be a point that belongs to each normal plane of α. Prove that α is a sphere a curve.

5. Problem 5^{∗} . a) Prove that if all tangent lines of a given curve are parallel to a plane, then the curve is plane.

b) Prove that if all osculating planes of a given curve pass through a fixed point, then the curve is plane.