Parametric Curves

Introduction

Functions of the form $\gamma :\mathcal{D}\subseteq \mathbb{R}\to {\mathbb{R}}^n$ have many different names. Most generally, they are called vector-valued functions. However, when their domain $\mathcal{D}$ is an interval, we often call them parametric curves or paths, because their images resemble curves in $n$-dimensional space, provided that they are also continuous. There is no agreed-upon name or definition for these functions - some authors require that they are continuous and that their domain is an interval, but many definitions and theorems are applicable even when those two conditions are not met.

Notation

It is very common to denote such functions using bold symbols like $\mathbf{r}(t)$ or $\mathbf{v}(t)$.

Definition: Endpoints

Let $\gamma :\mathcal{D}\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be a vector-valued function.

If $\mathcal{D}$ is a closed interval of the form $[a;b]$, then we call $\gamma (a)$ and $\gamma (b)$ the endpoints of $\gamma$.

Definition: Closed Parametric Curve

A closed parametric curve is a continuous vector-valued function $\gamma :[a;b]\subset \mathbb{R}\to {\mathbb{R}}^n$ on a closed interval $[a;b]$ such that

$$\gamma (a)=\gamma (b)$$

Equivalence

Definition: Reparemtrization

A reparametrization between two vector-valued functions $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\phi :I_{\phi }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is a bijective function $h_{I_{\gamma }\to I_{\phi }}:I_{\gamma }\to I_{\phi }$ with inverse $h_{I_{\phi }\to I_{\gamma }}:I_{\phi }\to I_{\gamma }$ such that

$$\begin{array}{r}\gamma (t)=\phi (h_{I_{\gamma }\to I_{\phi }}(t))\forall t\in I_{\gamma }\\ \phi (t)=\gamma (h_{I_{\phi }\to I_{\gamma }}(t))\forall t\in I_{\phi }\end{array}$$

NOTE

This is the most general definition for reparametrization. However, it is quite common to require that both $h_{I_{\gamma }\to I_{\phi }}$ and $h_{I_{\phi }\to I_{\gamma }}$ have additional properties such as continuity, continuous differentiability, etc. In those cases, when we say that a reparametrization has some property, we mean that both $h_{I_{\gamma }\to I_{\phi }}$ and $h_{I_{\phi }\to I_{\gamma }}$ have this property.

Definition: Equivalence of Parametrizations

Two vector-valued functions $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\phi :I_{\phi }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ are equivalent if they have the same image and there exists a reparametrization between them.

Note

This is the most general definition of equivalence for parametrizations. However, sometimes we require that such a reparametrization also has additional properties such as continuity, continuous differentiability, etc.. In this case, we say that $\gamma$ and $\phi$ are “equivalent up to a PROPERTY reparametrization” such as “equivalent up to a continuous reparametrization” or “equivalent up to a continuously differentiable reparametrization”.