Sinusoidal

Type:: section
Appearance:: optional
Excludes:: Arc, Spline

A Sinusoidal curve replaces an original straight polygon segment by a sinusoidal deflection. It is an alternative to Spline and Arc; exactly one of the three definitions is selected for each CurvedSegment. Its endpoints remain the original polygon vertices.

For a selected segment from $\mathbf{p}_0$ to $\mathbf{p}_1$ with length $L$ , unit tangent $\mathbf{t}$ , and right-hand unit normal $\mathbf{n}$ , the curve is

$\begin{eqnarray*} \gamma(s) = \mathbf{p}_0 + s\mathbf{t} + A \sin\!\left(\pi H \frac{s}{L}\right)\mathbf{n}, \qquad 0 \leq s \leq L , \end{eqnarray*}$

where $A$ is Amplitude and $H$ is HalfPeriods. Since $H$ is a positive integer, the deflection is zero at both endpoints, so the curve meets the neighbouring polygon segments exactly.

The normal follows the orientation of the selected segment. For the counter-clockwise boundaries of a polygon, a positive amplitude points outwards. A sinusoidal curve can only replace an original straight segment.