anglepreserving

Angle‑preserving refers to transformations or mappings that maintain the magnitude of angles between intersecting curves or vectors. In mathematics such maps are called conformal; they preserve the shape of infinitesimal figures though not necessarily their size. In Euclidean geometry a linear transformation is angle‑preserving exactly when it is a similarity: a composition of a uniform scaling and an orthogonal transformation (rotation or reflection). Consequently any orthogonal matrix multiplied by a positive scalar yields an angle‑preserving linear map.

In the complex plane the term is most often applied to holomorphic functions with non‑zero derivative. By the Cauchy–Riemann equations these functions are locally conformal, so they preserve angles and orientation at points where the derivative does not vanish. Möbius transformations, rational maps of the form \((az+b)/(cz+d)\) with \(ad-bc\neq0\), are global examples: they map circles and lines to circles or lines while keeping angles intact.

In differential geometry a smooth map between Riemannian manifolds is called conformal if it multiplies the metric tensor by a positive scalar function; such maps preserve angles between tangent vectors at each point. The notion extends to higher dimensions, where Liouville’s theorem states that, for dimensions greater than two, any sufficiently smooth conformal map is a composition of translations, dilations, rotations, and inversions.

Angle‑preserving transformations are central in fields such as complex analysis, geometric function theory, and physics, where they model isotropic scaling and the behavior of fields under local rotations. They also underpin practical applications like cartography, where conformal map projections preserve local shapes at the expense of area distortion.