Distance along a Curve

Overview


The distance, or arc length, along a curve is defined to be the following integral
{% s = \int_0^t |\dot{\vec{p}}(t)| dt %}

Parameterization by Distance


Given a curve {% \gamma %}, a new curve {% \alpha %} parameterized by arc length is defined as
{% \alpha = \gamma \circ s^{-1} : I \rightarrow \mathbb{R}^3 %}

Frenet


Given a curve {% \alpha %}, parameterized by arc length, we define the following.

  • Unit Tangent Vector
    {% \vec{t}(s) = \frac{d \alpha(s)}{ds} %}
    {% |t(s)| = \sqrt{\frac{dx}{dt}^2 + \frac{dy}{dy}^2} \frac{dt}{ds} = \frac{ds}{dt}\frac{dt}{ds} = 1 %}
  • Curvature - the curvature of the curve is defined as
    {% \kappa(s) = || \vec{t}'(s) || %}