13.3 Arc Length, Slides of Pre-Calculus

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Arc Length

Review:

curve in space: r (^)   t (^)  f (^)   t (^) i  g t   (^) j  h t   k Tangent vector: r '( t (^) 0 )  f ' (^)   t (^) i  g ' (^)   t (^) j  h '  t k

Tangent line at t  t 0 (^) : s  r( t (^) 0 (^) )  s r '( t 0 )

Projectile motion: ( ) (^120 ) r t   2 gt j  v t  r

r ( ) t   (^)  v 0 (^) cos  t , h  (^)  v 0 sin  t  12 gt^2 

initial velocity v (^) 0 , initial position r 0

Initial speed | v 0 | , initial height h , launching angle  :

g  32 ftsec 2 gravitational constant

Velocity: v ( ) t  r '( ) t speed = | ( ) | v t

Acceleration a ( ) t  v '( ) t  r ''( ) t

Example 1:

A line from a to b : 1 0  b^ ^ a^ dt  b^  a = distance from^ a^ to b

Example 2:

Compute the length (circumference) of circle: ( x  x 0 (^) )^2  ( y  y 0 )^2  r^2 parametric description: x  x 0 (^)  r cos( ), t y  y 0  r sin( ) 0 t  t  2 

 

2 0

length = t dt

  r^ ^ 

(^22 2 2 ) 0

r sin ( ) t r cos ( ) t dt

 ^ ^ 

2 0

rdt 2 r

   

1 0

length  (^)  r  t dt 

r ( ) t  a  t ( b  a ) ,^0 ^ t ^1

A drunken bee travels along the path ( ) cos(2 ),sin(2 t), t

for 10 seconds. It then travels at constant speed in a straight line for 10

more seconds. How far did the bee travel?

Problem : r t   t 

velocity: v ( ) t  r '( ) t  2sin(2 ), 2 cos(2 t),1 t  Speed = |^ v ( ) | t^^ ^9 ^3

At time t (^) 0 10, it travels along the tangent line

tangent line: u ( ) s  r( t 0 )  s v ( t 0 )

distance traveled along the helix:

20 10

length = (^)  | u '(s) | ds 

10 0  | r^ '( ) | t^^ dt^ ^30

Total distance traveled: 60 ft

20 10 ^ | v^ (^ t^^0 ) | ds^  3 (20^ ^ 10)^ ^30

Example:

Parametrize the following curve by arc length: r (^)   t (^)  3sin(t )^2 i 3cos( t^2 ) k r ' (^)   t (^)  6 cos( t t^2 ) i 6 sin( t t^2 ) k

| ' r (^)   t (^) |  36 t^2 cos (^2 t^2 )  36 t^2 sin( t^2 ) ^36 t^^^2 ^6 t

0

( ) 6

t s  s t  (^)  udu Solve for t : t= 3^ s

Plug into r ( ) : t r ( ( )) t s  r ( s/ 3)^ ^ 3sin(^3^ s^ ) i^ 3cos(^3 s ) k

Arc length parametrization: ( ) 3sin( ) 3cos( ) 3 3

r s ^ s^ i  s k

Notice: | '( ) | r s  1 length from to :   1

b b a a

s  a s  b (^)  r  s ds  (^)  ds  b  a

t is an arc length parameter if | '( ) | r t 1!

 3 u^2 | uu^  0 t  3 t^2

Curvature

Examples:

a ) The curvature of a straight line:

r ( ) t  r + t v 0 r' ( ) t  v what is the arc length parametrization of the line:

 ^ dds^^ T^  r   s  0 ( if not arc lenght:^ r^ ( ) t^^ ^ r + 0^ t^^2 v , then^^ r^ ''( ) t^^ ^2 t v )

b) The curvature of a circle of radius r : r ( ) t   r cos( ), t r sin( ) t  r '( ) t   r sin( ), t r cos( ) t  r ' (^)   t  r arc length parametrization: r (s)   r cos( (^) r^ s ), r sin( sr ) T^^ ^ r '(s)^  ^ sin(^^ sr^ ), cos(^ rs ) '^ d^^1 cos( s^ ), 1 sin( s )  (^) ds   (^) r r  (^) r r  T^ T 2

d 1 1   (^) ds  (^) r  r T

small radius means large curvature!

hence (^0)   or /

t

s   r  u du  rt t  r s

(s)  (^0) | | r r + s v v since then |^ (s) |^ ^ |^ | ^1 r' v v

d

  ds

T

d dt ds dt

T

   

t t

T

r

   

t  (^) t

T

r

Recall ( )  

t a

s t  (^)  r  u du hence:^ ds^ | '( ) | t dt ^ r

It is usually too difficult to parametrize a curve by arc length.

Express curvature in terms of a general parameter t :

Example: Compute the curvature of an ellipse:

2 2 a^ x^^2 ^ by^2 ^1 r ( ) t   a cos( ), b sin( ) t t  r '( ) t   a sin( ), bcos( ) t t  | r^ '( ) | t^^ ^ a^^2 sin ( )^2 t^^  b^^2 cos ( )^2 t

2 2 2 2 T( ) sin( ), bcos( ) sin ( ) cos ( )

t a^ t^ t a t b t

 ^  

2 2 2 2 2 2 3/ T'( ) cos( ),^ ba sin( ) ( sin ( ) cos ( )) t ab^ t^ t a t b t  ^ ^  

| T'( ) | (^2) sin ( ) (^2 2) cos ( ) 2 t ab  (^) a t  b t     (^2 sin ( )^2 2 cos ( ))^2 3/

t (^) ab

 t a t b t

^ T   

r

dt d

ds dt

 T

(a lengthy computation…)

r ( ) t   a cos( ), b sin( ) t t 

r '( ) t   a sin( ), bcos( ), 0 t t  r ''( ) t   a cos( ), t  b sin( ), 0 t 

or r ( ) t   a cos( ), b sin( ), 0 t t 

Example: Ellipse revisited:

    sin( )^ cos( )^0 cos( ) sin( ) 0

t t a t b t a t b t

 (^)       

i j k r r ( ab sin ( )^2 t  ab cos ( ))^2 t k = ab k

𝜅 = |𝐫

′ (^) 𝑡 × 𝐫″ (^) 𝑡 | |𝐫′^ 𝑡 |^3 =^

𝑎𝑏 𝑎^2 sin^2 (𝑡) + 𝑏^2 𝑐𝑜𝑠^2 (𝑡) 3 2

Example: Curvature of the (elliptical) helix:

r ( ) t   a cos( ), bsin(t), t c t r '( ) t   a sin( ), bcos( ), c t t  r ''( ) t   a cos( ), t  bsin(t), 0

    sin( )^ cos( ) cos( ) sin( ) 0

t t a t b t c a t b t

       

i j k r r (0  bc sin( )) t i  ( ac cos( ) t  0) j  ( ab sin ( )^2 t  ab cos ( ))^2 t k

r ^   t  r   t  b c^2 2^ sin ( )^2 t  a c^2^2 cos ( )^2 t  a b^2

 bc sin( ) t i  ac cos( ) t j  ab k

| r '( ) | t  a^2 sin ( )^2 t  b cos ( )^2 2 t c^2

𝜅 = |𝐫

′ (^) 𝑡 × 𝐫″ (^) 𝑡 | |𝐫′^ 𝑡 |^3  

2 2 2 2 2 2 2 2 ^ b ca 2^ (^) sin ( )^ sin ( ) (^2) tt ^  b cos ( )^ a c 2^ cos ( ) (^2) t  t^ c 2  a b 3/

Circular helix: b  a , 2 2

c

a  ^  ^ a  constant!

2 2 2 2  ^ a ca (^2) ^ c 2 a a 3/2   

2 2 aa (^) 2 c  (^) c  2 a 3/

Summary:

Length of a curve | '( ) |

b a

 (^)  r t dt

Arc length function ( )  

t a

s t  (^)  r  u du^ ds^ | '( ) | t dt ^ r Arc length parametrization r ( ) with s | '( ) | r s  1

Unit tangent vector '( ) '(s) | '( ) |

t t

T ^ r  r r

Curvature: (^)   d (^) s ds

 ^ T^  r  

   

t

t

T

r

      3

t t

t

r r

r