Arc length (and curvature) [9.8]
The Nowitna river (Alaska) - How long is it?
a
- Calculating arclength and curvature
- Arclength is independent of the choice of parameter used to describe a curve.
- Re-parameterizing a curve - using the arclength as the parameter.
- Geometric definition of curvature.
- The TNB frame (Tangent - Normal - Binormal unit vectors).
- Tangent and Normal components of acceleration
Arclength
You know some special-case formulas for calculating the distance along a straight line between 2 points, or the perimeter of a circle.
But now that you know how to describe *any* continuous curve in space with parametric equations, we'll develop a general way to integrate to find the distance along *any* continuous curve in space.
$\myv r(t)$ is a vector function.
$$\myv r(t)=\langle f(t),g(t),h(t)\rangle\nonumber;
\ \ t_i\leq t\leq t_f.$$
where $\myv r(t_i)=\myv a$ and $\myv r(t_f)=\myv b$
The tip of the position vector $\myv r(t)$ traces a curve (trajectory) in space which
looks schematically like:
- The arclength is the distance along the curve from $\myv a$ to $\myv b$.
- Arclength is a scalar.
- It is not the distance "as the crow flows" of a vector starting at $\myv a$ and terminating at $\myv b$.
- $\myv a\equiv \myv r(t_i)$ and $\myv b\equiv \myv r(t_f)$.
The arclength from $\bf{a}$ to $\bf{b}$ is approximately equal to the sum of the