88 lines
3 KiB
Markdown
88 lines
3 KiB
Markdown
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# Curves
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> *Definition*: a curve is a continuous vector-valued function of one real-valued parameter.
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>
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> * 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.
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> * A simple curve has no crossings.
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<br>
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> *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 $\|\mathbf{c}'\|$.
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<br>
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> *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.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *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
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>
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> $$
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> ds(t) := \|d \mathbf{c}(t)\| = \|\mathbf{c}'(t)\|dt
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> $$
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>
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> for all $t \in \mathbb{R}$.
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<br>
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> *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
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>
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> $$
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> s(t) = \int_{t_0}^t \|\mathbf{c}'(u)\|du,
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> $$
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>
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> for all $t \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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## Arc length parameterization
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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:
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1. For a given curve determine the arc length function for a given start point.
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2. Find the inverse of the arc length function if it exists.
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3. Adopt the arc length as variable of the curve.
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Obtaining a speed of unity on the entire defined curve.
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For example consider a curve $\mathbf{c}: \mathbb{R} \to \mathbb{R}^3$ given in Cartesian coordinates by
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$$
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\mathbf{c}(\phi) = \begin{pmatrix} r \cos \phi \\ r \sin \phi \\ \rho r \phi\end{pmatrix},
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$$
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for all $\phi \in \mathbb{R}$ with $r, \rho \in \mathbb{R}^+$.
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Determining the arc length function $s: \mathbb{R} \to \mathbb{R}$ of the curve
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$$
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\begin{align*}
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s(\phi) &= \int_0^\phi \|\mathbf{c}'(u)\|du, \\
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&= \int_0^\phi r \sqrt{1 + \rho^2}du, \\
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&= \phi r \sqrt{1 + \rho^2},
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\end{align*}
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$$
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for all $\phi \in \mathbb{R}$. It may be observed that $s$ is a bijective mapping.
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The inverse of the arc length function $s^{-1}: \mathbb{R} \to \mathbb{R}$ is then given by
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$$
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s^{-1}(\phi) = \frac{\phi}{r\sqrt{a + \rho^2}},
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$$
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for all $\phi \in \mathbb{R}$.
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The arc length parameterization $\mathbf{c}_s: \mathbb{R} \to \mathbb{R}^3$ of $\mathbf{c}$ is then given by
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$$
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\mathbf{c}_s(\phi) = \mathbf{c}(s^{-1}(\phi)) = \begin{pmatrix} r \cos (\phi / r\sqrt{a + \rho^2}) \\ r \sin (\phi / r\sqrt{a + \rho^2}) \\ \rho \phi / \sqrt{a + \rho^2}\end{pmatrix},
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$$
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for all $\phi \in \mathbb{R}$.
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