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Parameterizing a curve by arc length

To parameterize a curve by arc length, the procedure is

- Find the arc length
. If the curve is regular then
is a
monotonically increasing function.
- Calculate the inverse of the arc length
. This is another
monotonically increasing function.
- Set the new parameter
. Now
has arc length
parameterization.

Intuitively, if the curve if ``moving'' quickly in some region, then its arc
length will increase rapidly and the inverse
will increase
proportionally slowly. In the parameterized curve
these two
factors will cancel out and the curve will increase in length uniformly. What
is the velocity of
?
We have

where we have used the fact that
when
and
are inverses of each other, and the definition of arc length in
equation (2.4). Hence we have found that the velocity
is one for an arc length parameterized curve.

Copyright © 2005 Adrian Secord. All rights reserved.