Instantsides

Instantsides are a geometric construct used in computational geometry to describe a particular feature of polygon offsets. Given a polygon P in the plane and a nonnegative offset distance r, the offset P_r is the set of all points at distance at most r from P. The boundary ∂P_r consists of straight segments and circular arcs. An instantside is a maximal straight segment of ∂P_r that runs parallel to an edge of P. For many polygons, each edge of P contributes a corresponding instantside on ∂P_r, though the relationship can be more complex for nonconvex shapes.

Definition and properties. In the common case of a convex polygon, there is a direct correspondence: every edge of P yields one instantside of ∂P_r that is parallel to that edge. In nonconvex polygons, instantsides may arise from portions of multiple edges or from more intricate vertex configurations, and some straight portions can disappear or merge as r changes. The length of an instantside typically grows with r, starting from the original edge length when r = 0 and increasing as the offset distance increases; the exact dependence is influenced by the adjacent interior angles at the corresponding vertices.

Computation and applications. Instantsides appear naturally in standard polygon offsetting or dilation algorithms, such as those used to compute the boundary of P_r via Minkowski sums with a disk. They are relevant in practical contexts such as toolpath generation in CAD/CAM, collision detection in robotics and graphics, and certain morphological operations in image processing, where straight boundary portions provide useful structural information.

Examples and related concepts. For a simple square, there are four instantsides, one parallel to each side, whose lengths expand with r while the rounded corners are formed by arcs. See also offset, Minkowski sum, dilation, and straight skeleton.