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 xray 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 xrays 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
K = |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…
b
L   [ f '(t )]2  [ g '(t )]2  [h '(t )]2 dt
a
b
L (
a
dx 2 dy 2 dz 2
)  ( )  ( ) dt
dt
dt
dt
b
L   r '(t ) dt
a
t
s (t )  
a
ds
 r '(t )
dt
dx 2 dy 2
dz 2
( )  ( )  ( ) du
du
du
du
• example 2 pg 900:
– Parameterize wrt arc length
• try 14.3#10
Curvature
• There are several different ways to determine the curvature:
dT
k
ds
T 't 
k
r '(t )
f "( x)
k
2 3/ 2
[1  ( f '( x)) ]
k
r '(t )  r "(t )
r '(t )
3
• Examples:
– Pg 900 #3
– Pg 902 #4
– Pg 902 #5
Tangents, Normals and Binormals
• Tangents T
r '(t )
T (t ) 
r '(t )
• Normals N
T '(t )
N (t ) 
T '(t )
• Binormals B
B(t )  T (t )  N (t )
T (t ) 
r '(t )
r '(t )
Curvature and Torsion
• Curvature and torsion are ways of describing
how a curve can “bend”
dT
 kN
ds
dB
  N
ds
Example pg 907 #55 or …How long
are YOUR genes?
Can you model this with a
parametric equation?
The Snowbirds!
ase I: The Snowbirds fly in a circular path
ven as
 2cos(t ),2.5,2sin(t ) 
What do the path and velocity and
cceleration vectors look like?
Case II: The Snowbirds fly in
tightening spiral path beginning
2.5 km overhead and descending
to 500 m and described by:
 (2  0.15t )cos(t ),2.5  .2t 2 ,(2  0.15t )sin(
What do the path and velocity and
acceleration vectors look like?