Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 2.

1. Problem 1.

a) Find the arc length of astroid x(t) = a cos^{3} t, y(t) = a sin^{3} t.

b) Find the curvature of the curve

x(t) = t − sin t,  y(t) = 1 − cos t,  z(t) = 4 sin \frac{t}{2}.

c) Write the equation of the tangent line to the curve given by the equation

x^{2} + y^{2} + z^{2} = 1, x^{2} + y^{2} = x,

at the point (0, 0, 1).

d) Find the curvature of the line given implicitly by the equations

x + sinh x = sin y + y, z + e^{z} = x + ln(1 + x) + 1

at the point (0, 0, 0)

e) Find the curvature and the torsion of the curve

x = a cosh t,  y = a sinh t,  z = a t.

f) Find the equation of the parabola of the form y = x^{2} + ax + b that is tangent to the circle x^{2} + y^{2} = 2 at the point (1, 1).

2. Problem 2. One often gives a plane curve in polar coordinates by ρ = ρ(θ), θ ∈ (a, b). a) Show that the arc length is

∫_{a}^{b} \sqrt{ρ^{2} + (ρ′)^{2}}dθ,

where the prime denotes the derivative relative to θ.

b) Show that the curvature is

k(θ) = \frac{2(ρ′)^{2} − ρρ′′ + ρ^{2}}{((ρ′)^{2} + ρ^{2})^{3/2}} .

3. Problem 3.

Show that

α(s) = (\frac{(1 + s)^{3/2}}{3} , \frac{(1 − s)^{3/2}}{3} , \frac{s}{\sqrt{2}})

is a unit speed curve and compute its Frenet-Serret apparatus.

4. Problem 4.

Assume that all normals of a regular curve pass through a fixed point. Prove that the trace of the curve is contained in a circle.

5. Problem 5^{∗} . A unit speed curve α(s) with k ≠ 0 is a helix if and only if there is a constant c such that τ = c k.