Rectifiable complex contours, reversal, concatenation, closedness, and orientation

A complex contour is a continuous rectifiable path $\gamma :[a,b]\to \mathbb{C}$, $a\le b$, in the sense of Polygonal length and rectifiability of complex paths. It is closed when $\gamma (a)=\gamma (b)$. Its reversal is ${\gamma }^{-}(t)=\gamma (a+b-t)$ on the same interval.

For arbitrary maps $\alpha ,\beta :[0,1]\to \mathbb{C}$ satisfying $\alpha (1)=\beta (0)$, define

$(\alpha \ast \beta )(t)=\{\begin{array}{ll}\alpha (2t),& 0\le t\le \frac{1}{2},\\ \beta (2t-1),& \frac{1}{2}\le t\le 1.\end{array}$

Endpoint matching makes the two expressions agree at $1\mathrm{/}2$, so this formula defines a map even when the factors are not paths. If both factors are complex contours, their continuity and endpoint matching make the concatenation continuous, including at the join. In this case the concatenation is rectifiable and its length is $L_{\mathbb{C}}(\alpha )+L_{\mathbb{C}}(\beta )$: inserting $1\mathrm{/}2$ into a partition can only increase its sum by the triangle inequality; the refined sum splits into two sums of the factors, while combining partitions from both factors gives the reverse supremum inequality. Taking factor sums within $\epsilon \mathrm{/}2$ of their finite suprema proves equality. This argument includes constant pieces. Reversal preserves rectifiability and length by Complex path length is invariant under monotone surjective reparametrization.

A complex contour is piecewise $C^1$ when its coordinate path is continuous piecewise $C^1$ as specified in Complex path length equals coordinate length and piecewise speed integral; its length is the speed integral supplied there, including corners and zero-speed pieces. Singleton paths have length zero.

An increasing continuous bijection of compact parameter intervals preserves orientation; a decreasing one reverses orientation, using the order convention of Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences. The length is unchanged by either, and more generally by every continuous surjective nondecreasing or nonincreasing reparametrization, including pauses, by Complex path length is invariant under monotone surjective reparametrization.