Reunakeskukset

Reunakeskukset are the edge centers in geometry, referring to the midpoints of the edges of a polygonal or polyhedral element. In a planar polygon with vertices v1, v2, ..., vn, the i-th reunakeskus is the midpoint ei = (vi + v_{i+1})/2 (indices taken modulo n). In three-dimensional space, each edge AB has a reunakeskus at its midpoint (A + B)/2. The collection of all edge centers forms the edge-midpoint data set of the shape.

In two-dimensional polygons, connecting consecutive edge centers yields the edge-midpoint polygon. For triangles this produces the medial triangle, which is similar to the original triangle and scaled by a factor of 1/2. For general polygons, the edge-midpoint polygon provides a natural, related geometry that can differ in shape from the original polygon.

Applications and relevance include computational geometry, mesh generation and refinement, and finite element methods. Edge centers appear as nodes in higher-order elements, for example quadratic elements on edges use mid-edge nodes. They also play a role in subdivision schemes in computer graphics, such as methods that refine polygons by inserting midpoints on edges to produce smoother surfaces.

In the context of 3D mesh processing, edge centers contribute to dual constructions and refinement strategies, where the distribution of midpoints influences mesh quality and connectivity. The Finnish term reunakeskus is a direct translation of edge center or edge midpoint, and in Finnish texts it is used to denote this geometric concept.