Editing Differentiable curve (section)

== Re-parametrization and equivalence relation ==
{{See also|Position vector|Vector-valued function}}

Given the image of a parametric curve, there are several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain reparametrizations. A suitable [[equivalence relation]] on the set of all parametric curves must be defined. The differential-geometric properties of a parametric curve (such as its length, its [[#Frenet frame|Frenet frame]], and its generalized curvature) are invariant under reparametrization and therefore properties of the [[equivalence class]] itself. The equivalence classes are called {{math|''C''<sup>''r''</sup>}}-curves and are central objects studied in the differential geometry of curves.

Two parametric {{math|''C''<sup>''r''</sup>}}-curves, <math>\gamma_1 : I_1 \to \mathbb{R}^n</math> and <math>\gamma_2 : I_2 \to \mathbb{R}^n</math>, are said to be {{em|equivalent}} if and only if there exists a [[bijective]] {{math|''C''<sup>''r''</sup>}}-map {{math|''φ'' : ''I''<sub>1</sub> → ''I''<sub>2</sub>}} such that
<math display="block">\forall t \in I_1: \quad \varphi'(t) \neq 0</math>
and
<math display="block">\forall t \in I_1: \quad \gamma_2\bigl(\varphi(t)\bigr) = \gamma_1(t).</math>
{{math|''γ''<sub>2</sub>}} is then said to be a {{em|re-parametrization}} of {{math|''γ''<sub>1</sub>}}.

Re-parametrization defines an equivalence relation on the set of all parametric {{math|''C''<sup>''r''</sup>}}-curves of class {{math|''C''<sup>''r''</sup>}}. The equivalence class of this relation simply a {{math|''C''<sup>''r''</sup>}}-curve.

An even ''finer'' equivalence relation of oriented parametric {{math|''C''<sup>''r''</sup>}}-curves can be defined by requiring {{mvar|φ}} to satisfy {{math|''φ''{{prime}}(''t'') > 0}}.

Equivalent parametric {{math|''C''<sup>''r''</sup>}}-curves have the same image, and equivalent oriented parametric {{math|''C''<sup>''r''</sup>}}-curves even traverse the image in the same direction.