Sidesremain

Sidesremain is a concept used in geometry and computational geometry to describe the number of sides a polygon has after applying a defined simplification or edge-removal operation. It serves as a measurement of how much a polygon is reduced under a specified process, such as removing redundant vertices, collapsing collinear sequences, or pruning small deviations in a polygonal chain.

Definition: Let P be a simple polygon with n vertices and E edges. Let A be a fixed simplification or edge-removal algorithm. After applying A to P, the resulting polygon P' has m sides. Then sidesremain(P, A) = m. If the operation is iterated, sidesremain can be extended to sidesremain(P, A, k) = m_k, with m_{k+1} ≤ m_k in most practical cases.

Examples: A common instance is removing all collinear consecutive vertices, which reduces the number of sides by the number of collapsed runs. In digitized boundaries, smoothing algorithms such as Visvalingam-Whyatt or Douglas-Peucker can yield a new polygon with fewer sides by eliminating minor deviations.

Properties: Sidesremain values depend on both the input geometry and the chosen algorithm. In general, increasing the strength or aggressiveness of the simplification does not increase the number of sides (monotonicity). Some inputs or algorithms may leave the polygon unchanged, resulting in the same number of sides.

Applications: Sidesremain is used in geographic information systems to simplify map polygons, in computer graphics for mesh decimation, and in pattern recognition when extracting streamlined contour outlines.

See also: polygon, polygon simplification, vertex collapse, collinearity, polyline simplification algorithms.