Conformal Map - Alternative Angles

Alternative Angles

A conformal map is called that because it conforms to the principle of angle-preservation. The presumption often is that the angle being preserved is the standard Euclidean angle, say parameterized in degrees or radians. However, in plane mapping there are two other angles to consider: the hyperbolic angle and the slope, which is the analogue of angle for dual numbers.

Suppose is a mapping of surfaces parameterized by and . The Jacobian matrix of is formed by the four partial derivatives of and with respect to and .

If the Jacobian g has a non-zero determinant, then is "conformal in the generalized sense" with respect to one of the three angle types, depending on the real matrix expressed by the Jacobian g.

Indeed, any such g lies in a particular planar commutative subring, and g has a polar coordinate form determined by parameters of radial and angular nature. The radial parameter corresponds to a similarity mapping and can be taken as 1 for purposes of conformal examination. The angular parameter of g is one of the three types, shear, hyperbolic, or Euclidean:

• When the subring is isomorphic to the dual number plane, then g acts as a shear mapping and preserves the dual angle.
• When the subring is isomorphic to the split-complex number plane, then g acts as a squeeze mapping and preserves the hyperbolic angle.
• When the subring is isomorphic to the ordinary complex number plane, then g acts as a rotation and preserves the Euclidean angle.

While describing analytic functions of a bireal variable, U. Bencivenga and G. Fox have written about conformal maps that preserve the hyperbolic angle.