Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment I.

1. Problem 1. a) Find a regular curve whose trace is the circle x^{2} + y^{2} = 1 such that α(t) runs clockwise around the circle with α(0) = (0, 1).

b) Let α(t) be a regular curve which does not pass through the origin. If α(t_{o}) is the point of the trace of α closest to the origin, show that the position vector α(t_{o}) is orthogonal to α′(t_{o}).

c) A regular curve α(t) has the property that its second derivative α′′(t) is identically zero. What can be said about α?

d) Let α : (a, b) → ℝ^{3} be a regular curve and let v ∈ ℝ^{3} be a fixed vector. Assume that α′(t) is orthogonal to v for all t ∈ (a, b) and that α(0) is also orthogonal to v. Prove that α(t) is orthogonal to v for all t ∈ (a, b).

e) Let α : (a, b) → ℝ^{3} be a regular curve. Show that |α(t)| is a nonzero constant if and only if α(t) is orthogonal to α′(t) for all t ∈ (a, b).

f) Assume that α : (a, b) → ℝ^{3} is just a differentiable curve (non necessarily regular). Are the corresponding conclusions of a)-e) remained true?

2. Problem 2. a) Which of the following are regular curves?

(i) α(θ) = (cos θ, 1 − cos θ − sin θ, − sin θ).
(ii) β(θ) = (2 sin^{2} θ, 2 sin^{2} θ tan θ, 0).
(iii) γ(θ) = (cos θ, cos^{2} θ, sin θ).

b) Find the tangent line to each of the above curves at θ = π/4.

c) Let α(t) be given by α(t) = (r cos t, r sin t, ht), where r > 0 and h > 0. Show that the angle between (0, 0, 1) and dα/dt is independent of t.

d) Show that f(−1, 1) → (−∞, ∞) given by f(t) = tan(πt/2) is a reparametrization.

3. Problem 3. a) Show that α(s) is a unit speed curve where

α(s) = \frac{1}{2}(s + \sqrt{1 + s^{2}}, \frac{1}{s + \sqrt{1 + s^{2}}} , \sqrt{2} ln(s + \sqrt{1 + s^{2}})).

b) Reparametrize the circular helix from Problem 2 c) by arc length. c) Let α(t) = (ae^{bt} cos t, ae^{bt} sin t), t ∈ ℝ, a > 0 and b < 0 are constants. Show that α(t) is regular. Show also that as t → ∞, α(t) approaches the origin 0, spiraling around it (because of this, the trace is called the logarithmic spiral; draw the picture).

d) Let α(t) be as in the previous exercise. Show that α′(t) → (0, 0) as t → +∞ and that

lim_{t→+∞} ∫_{t_{o}}^{t} |α′(t)| dt

is finite; that is, α has finite arc length in [t_{o}, +∞).

4. Problem 4.

A circular disc of radius 1 in the plane xy rolls without slipping along the x axis. The figure described by a point of the circumference of the disc is called a cycloid (draw a picture or google it!).

a) Obtain a parametrized curve α : ℝ → ℝ^{2} the trace of which is the cycloid, and determine its singular points.

b) Compute the arc length of the cycloid corresponding to a complete rotation of the disc.

5. Problem 5^{∗}. Prove that α(s) (parametrized by an arc length) is a straight line (or a part of it) if and only if there is a point v ∈ ℝ^{3} such that every tangent line to α goes through v.