Geometric transformation

In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function whose domain and range are sets of points – most often a real coordinate space,

R

2

{\displaystyle \mathbb {R} ^{2}}

or

R

3

{\displaystyle \mathbb {R} ^{3}}

– such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations, such as in transformation geometry.

Classifications

Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve:

Displacements preserve distances and oriented angles (e.g., translations);

Isometries preserve angles and distances (e.g., Euclidean transformations);

Similarities preserve angles and ratios between distances (e.g., resizing);

Affine transformations preserve parallelism (e.g., scaling, shear);

Projective transformations preserve collinearity;

Each of these classes contains the previous one.

Möbius transformations using complex coordinates on the plane (as well as circle inversion) preserve the set of all lines and circles, but may interchange lines and circles.

Conformal transformations preserve angles, and are, in the first order, similarities.

Equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case. and are, in the first order, affine transformations of determinant 1.

Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points.