Key to Homework #3
1. Compute the curvature of the following curves (warning: the curves may not be
unit-speed).
!
1
t
3/2 1
3/2
(a) (t) =
(1 + t) , (1 t) , ;
3
3
2
3
3
(b) (t) = cos t, sin t .
Solution.
!
Solution: (a)
0
(t) =
1
1
1
(1 + t)1/2 , (1 t)1/

Homework #4
1. Let (t) = (t +
its Frenet frame.
3 sin t, 2 cos t, 3t sin t). Calculate its curvature, torsion and
2 For the curve defined in problem #1, what can you say about the curve? Is
it a helix? If so, can you reparametrize the curve in terms of th

Key to Homework #2
1. Sketch the following parametrized curves:
(a) (t) = (cos t, sin t), 0 < t < 3/2;
(b) (t) = (et , t2 ), t R.
2. Is (t) = (t2 , t4 ) a parametrization of the parabola y = x2 ?
Solution: No, it is only part of the parabola y = x2 since

CHAPTER THREE: SURFACES IN R3
3.4 THE GAUSS MAP AND SHAPE OPERATOR
The Gauss map: Given a surface M parametrized by x :
U R3, we have defined the normal n(P ) at the point P =
x(u0, v0) M as the unit vector
n(P ) =
xu(u0, v0) xv (u0, v0)
.
k u(u0, v0) v (