conformalaffine

Conformalaffine is a term used in some mathematical discussions to denote a concept that combines conformal and affine ideas, but it is not a standard, widely adopted term in mainstream geometry. As a result, its precise definition can vary depending on the author or context.

One interpretation envisions conformalaffine geometry as a framework in which a space carries both a conformal class of metrics (preserving angles up to a scale) and an affine connection that defines parallelism, with compatibility conditions linking the two structures. In this sense, a conformalaffine transformation is one that preserves this combined structure.

A more operational usage refers to the conformalaffine group: the set of maps that are simultaneously conformal and affine. In Euclidean space R^n, the full conformal group includes inversions and non-affine maps, while the affine group consists of maps x -> Ax + b with A in GL(n). The intersection of these two groups—the conformalaffine group—reduces to the similarities: maps of the form x -> s Q x + b, where s > 0 is a scale, Q is an orthogonal matrix, and b is a translation. Including reflections, Q can lie in O(n). In this sense, conformalaffine transformations are precisely uniform scaling combined with rotations (or reflections) and translations.

Properties and structure: the conformalaffine (similarity) group is a Lie group with dimension n(n−1)/2 + n + 1. It acts transitively on R^n and preserves angles up to a global scale factor.

Applications: discussions of conformalaffine transformations appear in differential geometry, computer vision, and physics when studying symmetries that allow scaling alongside rigid motions without distorting shapes beyond a uniform scale. See also conformal geometry, affine geometry, Möbius transformations, and similarity transformations.