Curve

A curve, technically a parametric curve, is a continuous function that maps a one-dimensional interval to a higher NN-dimensional space: γ:I→RNγ : I → \mathbb{R}^N, t→γ(t)t\to \gamma(t) where II is a real interval, tt is called the parameter of the curve and N≥2N\geq2. The image γ(I)γ(I) is called the support of the curve.

Properties

  • A curve is said to be bounded (unbounded) if its support is a bounded (unbounded) set.
  • A curve is said to be closed if I=[a,b]I = [a,b] is a closed and bounded interval and γ(a)=γ(b)\gamma(a) = \gamma(b) holds.
  • A curve is said to be simple if it is injective except at most at the endpoints, that is, if and only if
γ(t1)=γ(t2)  ⇒  t1=t2  o  t1=a,  t2=b\gamma(t_1)=\gamma(t_2)\;\Rightarrow\;t_1=t_2\;o\;t_1=a,\;t_2=b
  • A curve is said to be regular if it is of class C1C^1 and γ′(t)≠0  ∀t∈I˚\gamma'(t)\neq0\;\forall t\in \mathring{I} holds. It is said to be piecewise regular if there exists a partition PP of the interval II defined as a=x0<x1<x2<⋯<xN=ba=x_0<x_1<x_2<\cdots<x_N=b such that the restriction to [xi−1,xi][x_{i-1},x_i] is a regular curve.
  • Two regular curves γ1:I1→RN\gamma_1:I_1\rightarrow\mathbb{R}^N and γ2:I2→RN\gamma_2:I_2\rightarrow\mathbb{R}^N are said to be equivalent if they have the same support and there exists a class C1C^{1} diffeomorphism φ:I1→I2\varphi:I_1\rightarrow I_2 such that γ1=γ2∘φ\gamma_1=\gamma_2\circ\varphi. The two curves have the same orientation if φ′(s)>0  ∀s∈I1\varphi'(s)>0\;\forall s\in I_1.

The tangent vectors τ^\hat{\tau} of a curve are given by

τ^:I˚→RNτ^(t)=γ′(t)∣∣γ′(t)∣∣\hat{\tau}:\mathring{I}\rightarrow\mathbb{R}^{N}\quad\hat{\tau}(t)= \frac{\gamma'(t)}{||\gamma'(t)||}