mathematics-physics-wiki/docs/en/physics/mathematical-physics/vector-analysis/curves.md

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.

??? note "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}.

??? note "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}.