Bogenlängenparametrisierung

Bogenlängenparametrisierung is a method used in differential geometry to reparameterize a curve. When a curve is given by a parametric function r(t), where t is the parameter, the arc length parameterization aims to define a new parameter, often denoted by s, such that the rate of change of the curve's position with respect to s is always a unit vector. This means the magnitude of the velocity vector is always 1.

The arc length s from a starting point t0 to a point t is defined by the integral of the magnitude of the velocity vector: s(t) = ∫[t0 to t] ||r'(τ)|| dτ. Here, r'(τ) is the derivative of the curve with respect to its parameter τ, and ||.|| denotes the magnitude.

To obtain the arc length parameterization, one first calculates the arc length function s(t). Then, the relationship between the original parameter t and the arc length parameter s is inverted to express t as a function of s, i.e., t = t(s). Substituting this into the original curve equation r(t) yields the arc length parameterized curve r(s) = r(t(s)).

The advantage of arc length parameterization is that it provides a natural way to measure distances along the curve. It simplifies many calculations in differential geometry, such as finding the curvature and torsion of a curve, as the magnitude of the velocity vector ||r'(s)|| is equal to 1, and the acceleration vector r''(s) is always orthogonal to the velocity vector r'(s). This makes the geometric properties of the curve more directly apparent.