Arc Length and Curvature

Chapter 14.3

We develop …

• A natural extension of arc length via parameterization

• Introduce the concept of curvature

example…

• A surgeon studies the x-ray of the spine of an adolescent male

• There is a clear indication of scoliosis – but how do you measure this?

• By taking a series of x-rays from different positions a spacecurve can be generated that represents the spine

Curvature and Arclength

• We now that something is curving because its tangent vector is changing direction! The more it changes in a given distance the greater the curvature. We can define curvature as:

curvature = rate of change of unit tangent vector wrt length, or

= |dT/ds|

Arc Length

• This has a very simple “intuitive” idea – set a bunch of meter sticks along the trace of the curve!

Different ways to define Arclength…

2 2 2[ '( )] [ '( )] [ '( )]b

a

L f t g t h t dt

2 2 2( ) ( ) ( )b

a

dx dy dzL dt

dt dt dt

'( )b

a

L r t dt2 2 2( ) ( ) ( ) ( )

t

a

dx dy dzs t du

du du du

'( )ds

r tdt

• example 2 pg 900:

– Parameterize wrt arc length

• try 14.3#10

Curvature

dTk

ds

• There are several different ways to determine the curvature:

3

'( ) "( )

'( )

r t r tk

r t

''( )

T tk

r t

2 3/ 2

"( )

[1 ( '( )) ]

f xk

f x

• Examples:– Pg 900 #3– Pg 902 #4– Pg 902 #5

Tangents, Normals and Binormals

• Tangents T

• Normals N

• Binormals B

'( )( )

'( )

r tT t

r t

'( )( )

'( )

T tN t

T t

'( )( )

'( )

r tT t

r t

( ) ( ) ( )B t T t N t

Curvature and Torsion

• Curvature and torsion are ways of describing how a curve can “bend”

dTkN

ds dB

Nds

Example pg 907 #55 or …How long are YOUR genes?

Can you model this with a parametric equation?

Case I: The Snowbirds fly in a circular path given as

What do the path and velocity and acceleration vectors look like?

The Snowbirds!

2cos( ),2.5,2sin( )t t

Case II: The Snowbirds fly in tightening spiral path beginning 2.5 km overhead and descending to 500 m and described by:

What do the path and velocity and acceleration vectors look like?

2(2 0.15 )cos( ),2.5 .2 ,(2 0.15 )sin( )t t t t t