Spiralähnlichkeiten

Spiralähnlichkeiten, also known as spiral similarities, refers to a geometric transformation in the plane that combines a rotation with a uniform scaling centered at a fixed point. Imagine a point P in the plane. A spiral similarity maps P to a new point P' such that the line segment connecting the center of the transformation to P is rotated by a fixed angle and its length is scaled by a fixed factor.

The center of the spiral similarity is the unique point that remains fixed under the transformation. The angle of rotation and the scaling factor are the two defining parameters of the transformation. If the scaling factor is 1, the transformation is purely a rotation. If the angle of rotation is 0, it is purely a dilation (uniform scaling). A spiral similarity is a combination of both.

Spiral similarities have applications in various fields, including geometry, fractals, and crystallography. For instance, they play a role in the construction of certain fractal shapes and in describing the symmetry of molecular structures. The concept is fundamental in understanding how shapes can be transformed in a way that preserves their overall form while changing their size and orientation around a specific point.