conformal: one construct, many systems

In complex analysis a conformal map is a transformation that preserves the angle and the sense of rotation between any two curves meeting at a point, wherever it is locally invertible. The key result is that every holomorphic function with a non-vanishing derivative is conformal, which ties the analytic property of differentiability to a purely geometric one. Ahlfors's Complex analysis develops the theory, including the Riemann mapping theorem that any simply connected proper subset of the plane can be mapped conformally onto a disc. The referent is a locally angle-preserving (but generally not distance-preserving) transformation of the plane.[1]

Unit

Whether local angles are preserved (equivalently, a non-zero derivative)

Recovery means

Not applicable

Measured by

Checked via the Cauchy–Riemann equations and a non-vanishing derivative at each point

In general relativity and differential geometry a conformal transformation multiplies the metric tensor by a positive, position-dependent factor, so that measured lengths change but angles and the null-cone (causal) structure of spacetime are left unchanged. Because the light-cones are preserved, conformal relations let physicists rescale and compactify spacetimes to study their causal structure and boundaries. Wald's General relativity treats conformal transformations and their role in causal structure and asymptotics. The referent is a scalar rescaling of the metric that holds angles and null directions invariant while altering scale.[2]

Unit

The conformal factor (positive scalar field) relating the two metrics

Recovery means

Not applicable

Measured by

Identified when two metrics differ by a positive scalar multiplier, leaving the null cones intact

In cartography a conformal (or orthomorphic) projection is one that preserves local angles and the shapes of small features, meaning the scale at any point is identical in every direction even though it varies from place to place across the map. The Mercator and the Lambert conformal conic are the classic cases, valued because a constant compass bearing plots as a straight line and because small shapes stay faithful. Snyder's Map projections: A working manual sets out the mathematics of these and other projections. The referent is a map transformation that sacrifices area fidelity to keep local angles and shapes correct. It is the same angle-preserving construct as in complex analysis, here realised on the curved Earth.[3]

Unit

Local angular and shape fidelity at a point on the map

Recovery means

Not applicable

Measured by

Confirmed when scale is equal in all directions at a point, so the Tissot indicatrix is a circle