Kantlängder

Kantlängder (edge lengths) are the lengths of the straight segments that form the sides of polygons and polyhedra. For a polygon with vertices v1, v2, ..., vn, the kantlängder are li = distance(vi, vi+1) for i = 1, ..., n−1 and ln = distance(vn, v1). The collection {l1, ..., ln} is the edge-length multiset of the polygon. The perimeter is P = l1 + ... + ln.

In two dimensions, edge lengths alone do not determine a polygon; many different shapes can share the same set of kantlängder. A basic existence criterion for a simple polygon with a given sequence of edge lengths is that the longest edge is strictly shorter than the sum of the remaining edges (the polygon inequality). When equality holds, the edges lie on a straight line and the polygon degenerates to a segment.

Triangles are the simplest case: for a triangle with sides a, b, c, the triangle inequalities a + b > c, a + c > b, and b + c > a must hold. The area can be computed from the side lengths using Heron's formula, and the cosine law relates the lengths to the interior angles: c^2 = a^2 + b^2 − 2ab cos(C).

Regular polygons have all kantlängder equal, while irregular polygons have varying edge lengths. In three dimensions and higher, edge lengths extend to polyhedra and spatial graphs; they influence rigidity and metric properties. In distance geometry, assigning lengths to the edges of a graph defines a metric on the vertices and underpins problems such as embedding and reconstruction.