Discrete Curves

A discrete curve in $\mathbb{R}^n$ is just a finite sequence $\gamma=(\gamma_0,\ldots,\gamma_n)$ of points in $\mathbb{R}^n$. Given a partition

\[a=t_0 < t_1 <t_2 <\ldots <t_n=b\]

of a closed interval $[a,b]\subset \mathbb{R}$ we can define a piecewise linear path $\hat{\gamma}: [a,b] \to \mathbb{R}^n$ by setting for $t_{j-1}\leq t \leq t_j$

\[\hat{\gamma}(t) = \gamma_{j-1}+(t-t_{j-1}) (\gamma_j-\gamma_{j-1}).\]

We call $\hat{\gamma}$ a parametrization of $\gamma$. Any partitioning of another interval $[\tilde{a},\tilde{b}]$ into $n$ sub-intervals leads to another parametrization $\tilde{\gamma}$ of $\gamma$.

$\gamma$ is called embedded if $\hat{\gamma}$ is injective. It is called regular if $\hat{\gamma}$ is at least locally injective, i.e. for each $t\in [a,b]$ there is an $\epsilon >0$ such that $\hat\gamma |_{(t-\epsilon,t+\epsilon)}$ is injective. It is easy to see that these definitions do not depend on the choice of the parametrization $\hat{\gamma}$.

The length of a discrete curve is defined as

\[L(\gamma_0,\ldots,\gamma_n)=\sum_{j=1}^n \,|\gamma_n-\gamma_{j-1}| .\]

A parametrization $\hat{\gamma}$ of $\gamma$ is called a parametrization by arclength if each edge has the same length as the corresponding parameter interval:

\[|\gamma_n-\gamma_{j-1}|=t_j-t_{j-1}.\]

Closed discrete curves are defined in a way similar to to the smooth case: They are pairs $(\gamma,n)$ where $\gamma$ is an $n$-periodic sequence of points in $\mathbb{R}^n$. For practical purposes they are the same thing as a discrete curve $(\gamma_1,\ldots,\gamma_n)$ together with the convention that indices are to be counted modulo $n$.

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