Hoekbeeld

Hoekbeeld is a term used in Dutch geometry to denote the image of an angle under a transformation or mapping. In this sense, the hoekbeeld of an angle is what the original angle becomes when every point of the angle is mapped by a function or transformation.

Typically, an angle is defined by two rays emanating from a common vertex. Under a transformation f, the hoekbeeld is the angle formed by the images of those rays at the image of the vertex, provided the image is well-defined and the resulting entity is non-degenerate. If the transformation collapses the vertex or makes the two rays coincide, the hoekbeeld may be degenerate or undefined.

The concept is especially relevant in Euclidean and projective geometry, as well as in applications such as computer graphics and kinematics, where objects are frequently transformed and their angular relationships are analyzed. The measure of the hoekbeeld depends on the nature of the transformation: angle-preserving transformations (conformal or isometries) retain the original angle, while general affine or perspective mappings can alter angles.

In practice, the term hoekbeeld is used descriptively rather than as a single formal object. It is contrasted with the voorgrond of a transformation (often referred to as the preimage or the original angle before transformation). Understanding hoekbeelden helps describe how geometric figures and their angular relationships change under various mappings.