Curves

Definition: a curve is a continuous vector-valued function of one real-valued parameter.

• A closed curve \(\mathbf{c}: \mathbb{R} \to \mathbb{R}^3\) is defined by \(\mathbf{c}(a) = \mathbf{c}(b)\) with \(a \in \mathbb{R}\) the begin point and \(b \in \mathbb{R}\) the end point.
• A simple curve has no crossings.

Definition: let \(\mathbf{c}: \mathbb{R} \to \mathbb{R}^3\) be a curve, the derivative of \(\mathbf{c}\) is defined as the velocity of the curve \(\mathbf{c}'\). The length of the velocity is defined as the speed of the curve \(\|\mathbf{c}'\|\).

Proposition: let \(\mathbf{c}: \mathbb{R} \to \mathbb{R}^3\) be a curve, the velocity of the curve \(\mathbf{c}'\) is tangential to the curve.

Proof:

Will be added later.

Definition: let \(\mathbf{c}: \mathbb{R} \to \mathbb{R}^3\) be a differentiable curve, the infinitesimal arc length \(ds: \mathbb{R} \to \mathbb{R}\) of the curve is defined as

\[ ds(t) := \|d \mathbf{c}(t)\| = \|\mathbf{c}'(t)\|dt \]

for all \(t \in \mathbb{R}\).

Theorem: let \(\mathbf{c}: \mathbb{R} \to \mathbb{R}^3\) be a differentiable curve, the arc length \(s: \mathbb{R} \to \mathbb{R}\) of a section that start at \(t_0 \in \mathbb{R}\) is given by

\[ s(t) = \int_{t_0}^t \|\mathbf{c}'(u)\|du, \]

for all \(t \in \mathbb{R}\).

Proof:

Will be added later.

Arc length parameterization

To obtain a speed of unity everywhere on the curve, or differently put equidistant arc lengths between each time step an arc length parameterization can be performed. It can be performed in 3 steps:

1. For a given curve determine the arc length function for a given start point.
2. Find the inverse of the arc length function if it exists.
3. Adopt the arc length as variable of the curve.

Obtaining a speed of unity on the entire defined curve.

For example consider a curve \(\mathbf{c}: \mathbb{R} \to \mathbb{R}^3\) given in Cartesian coordinates by

\[ \mathbf{c}(\phi) = \begin{pmatrix} r \cos \phi \\ r \sin \phi \\ \rho r \phi\end{pmatrix}, \]

for all \(\phi \in \mathbb{R}\) with \(r, \rho \in \mathbb{R}^+\).

Determining the arc length function \(s: \mathbb{R} \to \mathbb{R}\) of the curve

\[ \begin{align*} s(\phi) &= \int_0^\phi \|\mathbf{c}'(u)\|du, \\ &= \int_0^\phi r \sqrt{1 + \rho^2}du, \\ &= \phi r \sqrt{1 + \rho^2}, \end{align*} \]

for all \(\phi \in \mathbb{R}\). It may be observed that \(s\) is a bijective mapping.

The inverse of the arc length function \(s^{-1}: \mathbb{R} \to \mathbb{R}\) is then given by

\[ s^{-1}(\phi) = \frac{\phi}{r\sqrt{a + \rho^2}}, \]

for all \(\phi \in \mathbb{R}\).

The arc length parameterization \(\mathbf{c}_s: \mathbb{R} \to \mathbb{R}^3\) of \(\mathbf{c}\) is then given by

\[ \mathbf{c}_s(\phi) = \mathbf{c}(s^{-1}(\phi)) = \begin{pmatrix} r \cos (\phi / r\sqrt{a + \rho^2}) \\ r \sin (\phi / r\sqrt{a + \rho^2}) \\ \rho \phi / \sqrt{a + \rho^2}\end{pmatrix}, \]

for all \(\phi \in \mathbb{R}\).