Parametrization

Parametrizations

Definition: Curve Parametrization

A parametrization of a curve $\mathcal{C}$ in ${\mathbb{R}}^n$ is a continuous function $\gamma :I\subseteq \mathbb{R}\to {\mathbb{R}}^n$ on an interval $I$ whose image is $\mathcal{C}$.

$$\gamma (I)=\mathcal{C}$$

Note: Parametric Curves

Parametrizations are often called parametric curves.

The same curve can have many different parametrizations.

Example: Different Parametrizations of the Same Curve

TODO

More over, not all parametrizations are created equal. A single curve can have multiple parametrizations but some of them will be more useful than others because they have certain properties.

Tangent Vectors

Differentiable parametrizations allows us to define the notion of “tangentiality” for Curves.

Definition: Tangent Vector

Let $\gamma :I\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be a curve parametrization which is differentiable at $t\in I$.

The tangent vector of $\gamma$ at $t$ is the derivative $\dot{\gamma }(t)$ of $\gamma$ there.

NOTE

The tangent vector is also known as $\gamma$‘s velocity and its magnitude as $\gamma$‘s speed.

Definition: Unit Tangent Vector

The unit tangent vector is the Unit Vector obtained from the tangent vector:

$$\frac{1}{||\dot{\gamma }(t)||}\dot{\gamma }(t)$$

NOTATION

$$\mathbf{T}(t)$$

It is apparent from the above definition that there is no unique tangent vector which is intrinsic to a curve’s geometry. Instead, tangent vectors depend on the parametrization in question. However, it turns out that the tangent vectors of all parametrizations at the same point, if they exist, are collinear.

Theorem: Collinearity of Tangent Vectors

Let $\mathcal{C}$ be a curve in ${\mathbb{R}}^n$ and let $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\phi :I_{\phi }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be parametrizations of $\mathcal{C}$. Let $\mathbf{p}\in \mathcal{C}$ and let $t\in I_{\gamma }$ and $s\in I_{\phi }$ be such that $\gamma (t)=\mathbf{p}$ and $\phi (s)=\mathbf{p}$.

If $\gamma$ is differentiable at $t$ and $\phi$ is differentiable at $s$, then there exists some $\lambda \in \mathbb{R}$ such that

$$\dot{\gamma }(t)=\lambda \dot{\phi }(s)$$

PROOF

TODO

The above theorem tells us that all tangent vectors at a point of a curve line on a Straight Line

Definition: Tangent Line

The tangent line to a curve $\mathcal{C}$ at $\mathbf{p}\in \mathcal{C}$ is the Straight Line

$$\{\mathbf{p}+k\mathbf{t}\mid k\in \mathbb{R}\},$$

where $\mathbf{t}$ is any non-zero tangent vector of a parametrization of $\mathcal{C}$ at $\mathbf{p}$.

Normal Vectors

Definition: Normal Vector

Let $\gamma :I\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be a curve parametrization which is twice differentiable at $t\in I$.

The normal vector of $\gamma$ at $t$ is the derivative of its tangent vector there, i.e. $\gamma$‘s second derivative $\ddot{\gamma }(t)$.

Definition: Unit Normal Vector

The unit normal vector is the Unit Vector obtained from $\gamma$‘s normal vector:

$$\frac{1}{||\dot{\mathbf{T}}(t)||}\dot{\mathbf{T}}(t)$$

NOTATION

$$\mathbf{N}(t)$$

Binormal Vectors

Definition: Binormal Vector

Let $\gamma :I\subseteq \mathbb{R}\to {\mathbb{R}}^3$ be a curve parametrization which is twice differentiable at $t\in I$.

The binormal vector of $\gamma$ at $t$ is the cross product of its unit tangent vector and its unit normal vector at $t$:

$$\mathbf{T}(t)\times \mathbf{N}(t)$$

NOTATION

$$\mathbf{B}(t)$$

Equivalence of Parametrizations

Definition: Reparametrization

Let $\mathcal{C}$ be a curve in ${\mathbb{R}}^n$ and let $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\phi :I_{\phi }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be parametrizations of $\mathcal{C}$.

A reparametrization between $\gamma$ and $\phi$ 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 or smoothness. In this case, 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

Let $\mathcal{C}$ be a curve in ${\mathbb{R}}^n$.

Two parametrizations $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\phi :I_{\phi }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ of $\mathcal{C}$ are equivalent if and only if 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 or smoothness. 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 smooth reparametrization”.

Theorem: Equivalence of Regular Injective Parametrizations

Let $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\phi :I_{\phi }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be parametrizations of the same curve $\mathcal{C}$.

If $\gamma$ and $\phi$ are $C^1$-regular (i.e. continuously differentiable with a non-vanishing derivative) and injective, then they are equivalent up to a continuously differentiable reparametrization.

PROOF

TODO

Orientation

Continuously differentiable parametrizations which are equivalent up to a continuously differentiable reparametrization with a non-vanishing derivative exhibit a nice property which allows us to define orientations for them.

Theorem: Unit Tangent Vectors of Equivalent Parametrizations

Let $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and $\gamma :I_{\gamma }\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be two parametrizations of the same curve $\mathcal{C}$.

If $\gamma$ and $\phi$ are differentiable with non-vanishing derivatives and are also equivalent up to a continuously differentiable reparametrization $\{h_{I_{\gamma }\to I_{\phi }},h_{I_{\phi }\to I_{\gamma }}\}$ with a non-vanishing derivative, then exactly one of the following is true for their unit tangent vectors:

• Case (I): ${\mathbf{T}}_{\phi }(t)={\mathbf{T}}_{\gamma }(h_{I_{\phi }\to I_{\gamma }}(t))$ for all $t\in I_{\phi }$

• Case (II): ${\mathbf{T}}_{\phi }(t)=-{\mathbf{T}}_{\gamma }(h_{I_{\phi }\to I_{\gamma }}(t))$ for all $t\in I_{\phi }$

PROOF

By definition, we have $\phi (t)=\gamma (h_{I_{\phi }\to I_{\gamma }}(t))$. The chain rule gives us

$${\phi }^{\prime }(t)={\gamma }^{\prime }(h_{I_{\phi }\to I_{\gamma }}(t))\cdot h_{I_{\phi }\to I_{\gamma }}^{\prime }(t).$$

The unit tangent vector ${\mathbf{T}}_{\phi }$ of $\phi$ is given by $\frac{{\phi }^{\prime }}{||{\phi }^{\prime }||}$ which combined with the above yields

$${\mathbf{T}}_{\phi }(t)=\frac{{\phi }^{\prime }(t)}{||{\phi }^{\prime }(t)||}=\frac{{\gamma }^{\prime }(h_{I_{\phi }\to I_{\gamma }}(t))\cdot h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)}{||{\gamma }^{\prime }(h_{I_{\phi }\to I_{\gamma }}(t))\cdot h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)||}.$$

We can use the property of the norm to transform the above expression into the following:

$${\mathbf{T}}_{\phi }(t)=\frac{h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)}{|h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)|}\frac{{\gamma }^{\prime }(h_{I_{\phi }\to I_{\gamma }}(t))}{||{\gamma }^{\prime }(h_{I_{\phi }\to I_{\gamma }}(t))||}=\frac{h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)}{|h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)|}{\mathbf{T}}_{\gamma }(h_{I_{\phi }\to I_{\gamma }}(t)).$$

Since $h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)$ is continuously differentiable with a non-vanishing derivative, it must be the case that either $h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)>0$ or $h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)<0$ for all $t\in I_{\phi }$. Moreover, we notice that $\frac{h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)}{|h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)|}$ is $1$ if and only if $h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)>0$ and $\frac{h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)}{|h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)|}$ is $-1$ if and only if $h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)<0$ and so either $\frac{h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)}{|h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)|}=1$ for all $t\in I_{\phi }$ or $\frac{h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)}{|h_{I_{\phi }\to I_{\gamma }}^{\prime }(t)|}=-1$ or $t\in I_{\phi }$.

Definition: Orientation of Parametrizations

We say that $\gamma$ and $\phi$ have

• the same orientation in the first case;

• opposite orientations in the second case.

Note: Preserving and Reversing Orientation

We might also say that $\gamma$ and $\phi$ are equivalent up to an

• orientation-preserving reparametrization in the first case;
• orientation-reversing reparametrization in the second case;

Intuitively, the above theorem tells us that, under the specified conditions, the unit tangent vectorof one parametrization at each point on the curve is always either equal or exactly opposite to the unit tangent vector of the other parametrization at the same point.

Bibliography