Diagonaalid

Diagonaalid, or diagonals, are line segments that connect nonadjacent vertices within a polygon or polyhedron, or, in the context of matrices, elements that lie along the diagonals of a square array. The concept is used across several areas of geometry and linear algebra.

In a polygon with n vertices, the number of diagonals is n(n-3)/2. This counts all straight connections between pairs of nonconsecutive vertices. In convex polygons, all diagonals lie inside the shape; in concave polygons, some diagonals lie outside the boundary. Diagonals in regular polygons come in distinct lengths, and their intersections create characteristic patterns, especially in polygons with many sides.

Diagonals vary in length; given the coordinates of the vertices, the length of a diagonal is the distance between its endpoints. In regular polygons, a small set of diagonal lengths often occurs, determined by the central angles between vertices. The arrangement of diagonals inside a polygon can influence properties such as symmetry and area decomposition.

In three dimensions, a space diagonal connects two vertices that do not share an edge. For example, a cube has four space diagonals. The number and arrangement of space diagonals depend on the specific polyhedron.

In matrices, the main diagonal runs from the top-left to the bottom-right, containing the elements aii. Diagonals parallel to the main diagonal are called superdiagonals (above) and subdiagonals (below). Diagonals have practical roles in algorithms, the computation of the matrix trace, and various numerical methods used to solve linear systems or study eigenvalues.