Reparametrization - Lecture notes 8 PDF

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Math 11 Santa Monica College...

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Arc Length Parametrization and Unit Speed Curves

Math 11 S. Soleymani

The vector-valued function that determines the position of a particle in space can be parametrized (described) as a function of time t or as a function of s with the following interpretations: r (t ) gives the position of the particle after time t has elapsed r( s ) gives the position of the particle after having traveled s units (arc length)

We can change the parameter of a vector valued function r (t ) from any parameter t to arc length s.

Example: The circle of radius R centered at the origin is determined by r( t)  Rcos t, Rsin t

for 0  t  2 . We

are going to reparametrize it using arc length from t  0 in the direction of increasing t. Here are the steps: i.

Find an expression in terms of t that describes arc length s from the time the particle begins its motion u  0 to any time arbitrary time u  t. We use the arc length formula: s  s (t ) 



t

0

v (u ) du 



t

0

r (u ) du 



t

0

( R sin u )2  (R sin u )2 du 



t

0

R du  Rt .

This value should make sense if you chose 2 for t to get s  2 R which is the circumference of the circle. ii.

iii.

Solve the equation obtained in the first step for t as a function of s. s s  Rt  t  . R Replace the parameter t in the vector-valued function r( t)  Rcos t, Rsin t for 0  t  2 becomes

 

 

r(s)  R cos s , R sin s R R which is another parametrization of a circle of radius R.

for 0  s  2 R

Exercise 1: Reparametrize the line r (t )  1  2t , 3t , 2  t using arc length measured from t  0 in the direction of increasing t.

Exercise 2: Show that r ( s)  v ( s)  1 for both the Example and Exercise 1. Explain why v ( s )  1 for all vectorvalued functions given that s is the arc length parameter.

Theorem: If r (t ) is a vector-valued function such that r(t )  1 then t is arc length parameter. Definition: The curve C determined by r (t ) with r(t )  1 is called a unit speed curve. Note: r  ( t)  v (t )  v (t ) means speed....