Parameterizations and Arc Length Function

In this video, I show that a curve can be represented by multiple different vector functions, called parameterizations of the curve, and that it is often beneficial to reparameterize a curve as a function of its arc length. This is because the arc length arises naturally from the shape of the curve and does not depend on any particular coordinate system.

To do this, I first define the arc length function, which is the arc length up to the parameter t. From the Fundamental Theorem of Calculus, the derivative of the arc length function is just the integrand and is equal to the length of the second derivative of the vector function. From here, we can obtain the parameter t in terms of the arc length s. Plugging t(s) back into the vector function for the curve reparameterizes it in terms of the arc length function. I illustrate with an example with the earlier circular helix.

#math #vectors #calculus #arclength #education

2 Parameterization Arc Length Function.png

Timestamps

Single curve can be represented by multiple vector functions such as the twisted cubic – 0:00
Each vector function equation is called a parametrization of the curve – 2:35
Definition of arc length function – 3:56
Graph of the arc length function – 5:41
Differentiating arc length function using Part 1 of the Fundamental Theorem of Calculus (FTOC) – 7:37
Recap on the FTOC – 8:35
Parametrize a curve in terms of arc length since arc length is independent of the coordinate system – 10:20
Graphing the parameterization in terms of arc length – 11:55
Example 2: Arc Length Function of a Helix – 13:07
- Integrating to obtain t as a function of arc length – 15:37
- Reparameterization of the circular helix as a function of arc length – 17:53