Post on 30-Dec-2015
description
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
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
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