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 \(\mathcal{C}\).
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 Norms obtained from the tangent vector:
Notation
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.
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
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 Norms obtained from \(\gamma\)'s normal vector:
Notation
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 \(t\):
Notation
Equivalence of Parametrizations#
Theorem: Equivalence of Regular Injective Parametrizations
Let \(\gamma: I_{\gamma} \subseteq \mathbb{R} \to \mathbb{R}^n\) and \(\varphi: I_{\varphi} \subseteq \mathbb{R} \to \mathbb{R}^n\) be parametrizations of the same curve \(\mathcal{C}\).
If \(\gamma\) and \(\varphi\) are \(C^1\)-[[TODO|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 \(\varphi\) are differentiable with non-vanishing derivatives and are also equivalent up to a continuously differentiable reparametrization \(\{h_{I_{\gamma} \to I_{\varphi}}, h_{I_{\varphi} \to I_{\gamma}}\}\) with a non-vanishing derivative, then exactly one of the following is true for their unit tangent vectors:
- Case (I): \(\mathbf{T}_{\varphi}(t) = \mathbf{T}_{\gamma}(h_{I_{\varphi} \to I_{\gamma}}(t))\) for all \(t \in I_{\varphi}\)
- Case (II): \(\mathbf{T}_{\varphi}(t) = -\mathbf{T}_{\gamma}(h_{I_{\varphi} \to I_{\gamma}}(t))\) for all \(t \in I_{\varphi}\)
Proof
By definition, we have \(\varphi(t) = \gamma(h_{I_{\varphi} \to I_{\gamma}}(t))\). The chain rule gives us
The unit tangent vector \(\mathbf{T}_{\varphi}\) of \(\varphi\) is given by \(\frac{\varphi'}{||\varphi'||}\) which combined with the above yields
We can use the property of the norm to transform the above expression into the following:
Since \(h_{I_{\varphi} \to I_{\gamma}}'(t)\) is continuously differentiable with a non-vanishing derivative, it must be the case that either \(h_{I_{\varphi} \to I_{\gamma}}'(t) \gt 0\) or \(h_{I_{\varphi} \to I_{\gamma}}'(t) \lt 0\) for all \(t \in I_{\varphi}\). Moreover, we notice that \(\frac{h_{I_{\varphi} \to I_{\gamma}}'(t)}{|h_{I_{\varphi} \to I_{\gamma}}'(t)|}\) is \(1\) if and only if \(h_{I_{\varphi} \to I_{\gamma}}'(t) \gt 0\) and \(\frac{h_{I_{\varphi} \to I_{\gamma}}'(t)}{|h_{I_{\varphi} \to I_{\gamma}}'(t)|}\) is \(-1\) if and only if \(h_{I_{\varphi} \to I_{\gamma}}'(t) \lt 0\) and so either \(\frac{h_{I_{\varphi} \to I_{\gamma}}'(t)}{|h_{I_{\varphi} \to I_{\gamma}}'(t)|} = 1\) for all \(t \in I_{\varphi}\) or \(\frac{h_{I_{\varphi} \to I_{\gamma}}'(t)}{|h_{I_{\varphi} \to I_{\gamma}}'(t)|} = -1\) or \(t \in I_{\varphi}\).
Definition: Orientation of Parametrizations
We say that \(\gamma\) and \(\varphi\) 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 \(\varphi\) are equivalent up to an
- orientation-preserving reparametrization in the first case;
- orientation-reversing reparametrization in the second case;