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Unformatted text preview: Math 4441 Aug 28, 2007 1 Differential Geometry Fall 2007, Georgia Tech Lecture Notes 1 1 Curves 1.1 Definition and Examples A (parametrized) curve (in Euclidean space) is a mapping α : I → R n , where I is an interval in the real line. We also use the notation I 3 t α 7-→ α ( t ) ∈ R n , which emphasizes that α sends each element of the interval I to a certain point in R n . We say that α is (of the class of) C k provided that it is k times continuously differentiable. We shall always assume that α is continuous ( C ), and whenever we need to differentiate it we will assume that α is differentiable up to however many orders that we may need. Some standard examples of curves are a line which passes through a point p ∈ R n , is parallel to the vector v ∈ R n , and has constant speed k v k [0 , 2 π ] 3 t α 7-→ p + tv ∈ R n ; a circle of radius R in the plane, which is oriented counterclockwise, [0 , 2 π ] 3 t α 7-→ ( r cos( t ) , r sin( t ) ) ∈ R 2 ; and the right handed helix (or corkscrew) given by R 3 t α 7-→ ( r cos( t ) , r sin( t ) , t ) ∈ R 3 . Other famous examples include the figure-eight curve [0 , 2 π ] 3 t α 7-→ ( sin( t ) , sin(2 t ) ) ∈ R 2 , 1 Last revised: August 28, 2007 1 the parabola R 3 t α 7-→ ( t, t 2 ) ∈ R 2 , and the cubic curve R 3 t α 7-→ ( t, t 2 , t 3 ) ∈ R 3 . Exercise 1. Sketch the cubic curve ( Hint: First draw each of the projections into the xy , yz , and zx planes). Exercise 2. Find a formula for the curve which is traced by the motion of a fixed point on a wheel of radius r rolling with constant speed on a flat surface ( Hint: Add the formula for a circle to the formula for a line generated by the motion of the center of the wheel. You only need to make sure that the speed of the line correctly matches the speed of the circle). Exercise 3. Let α : I → R n , and β : J → R n be a pair of differentiable curves . Show that α ( t ) , β ( t ) = α ( t ) , β ( t ) + α ( t ) , β ( t ) and k α ( t ) k = α ( t ) , α ( t ) k α ( t ) k . ( Hint: The first identity follows immediately from the definition of the inner- product, together with the ordinary product rule for derivatives. The second identity follows from the first once we recall that k · k := h· , ·i 1 / 2 ). Exercise 4. Show that if α has unit speed, i.e., k α ( t ) k = 1, then its velocity and acceleration are orthogonal, i.e., h α ( t ) , α 00 ( t ) i = 0. Exercise 5. Show that if the position vector and velocity of a planar curve α : I → R 2 are always perpendicular, i.e., h α ( t ) , α ( t ) i = 0, for all t ∈ I , then α ( I ) lies on a circle centered at the origin of R 2 ....
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