Differential Geometry of Curves - Reparametrization and Equivalence Relation

Reparametrization and Equivalence Relation

Given the image of a curve one can define several different parameterizations of the curve. Differential geometry aims to describe properties of curves invariant under certain reparametrizations. So we have to define a suitable equivalence relation on the set of all parametric curves. The differential geometric properties of a curve (length, Frenet frame and generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class.The equivalence classes are called Cr curves and are central objects studied in the differential geometry of curves.

Two parametric curves of class Cr

and

are said to be equivalent if there exists a bijective Cr map

such that

and

γ₂ is said to be a reparametrisation of γ₁. This reparametrisation of γ₁ defines the equivalence relation on the set of all parametric Cr curves. The equivalence class is called a Cr curve.

We can define an even finer equivalence relation of oriented Cr curves by requiring φ to be φ‘(t) > 0.

Equivalent Cr curves have the same image. And equivalent oriented Cr curves even traverse the image in the same direction.