Bogenlängenparametrisierte

Bogenlängenparametrisierte, often translated as arc length parameterization, is a method in differential geometry and calculus of curves where a curve is described by a parameter that directly represents the distance along the curve from a fixed starting point. This means that the parameter value at any point on the curve is equal to the length of the curve from the origin to that point.

To obtain an arc length parameterized curve, one typically starts with a standard parameterization, denoted as $\mathbf{r}(t) = (x(t), y(t), z(t))$. The arc length $s$ from a starting point $t_0$ to a point $t$ is given by the integral of the magnitude of the derivative of the position vector: $s(t) = \int_{t_0}^{t} ||\mathbf{r}'(\tau)|| \, d\tau$. This integral calculates the speed along the curve and sums it up over the parameter interval.

Once the arc length function $s(t)$ is determined, it is invertible, allowing us to express the original parameter $t$ as a function of $s$, i.e., $t = t(s)$. Substituting this inverse function back into the original parameterization yields the arc length parameterized curve, $\mathbf{r}(s) = \mathbf{r}(t(s))$.

A key advantage of arc length parameterization is that the magnitude of the velocity vector, $||\mathbf{r}'(s)||$, is always equal to 1. This simplifies many calculations in differential geometry, such as curvature and torsion, and provides a natural way to measure distances along the curve. It also ensures that the speed of traversal along the curve is constant, which is useful in various physical applications like motion along a path.