axialschen

Axialschen is a theoretical construct in geometry and applied mathematics describing a class of transformations and surfaces with axial symmetry around a fixed axis. In this framing, an axialschen transformation preserves the angular coordinate around the axis while allowing the radial and axial coordinates to change in a controlled way. The term is used in neutral discussions of axisymmetric deformation models and is presented here as a hypothetical concept.

Definition: In cylindrical coordinates (r, theta, z), an axialschen map is given by f(r, theta, z) = (g_r(r,z), theta, g_z(r,z)), where theta is unchanged. This captures pure axial symmetry with potentially nonlinear radial and axial distortions.

Properties: Axialschen maps commute with rotations around the axis; they may be smooth and invertible (diffeomorphisms) in the diffeomorphic subclass. If g_r(r,z) = r and g_z(r,z) = z, the map is the identity. Other variants allow g_r and g_z to depend on r or z separately or jointly.

Variants and examples: Axialschen surfaces can be generated by rotating a profile curve and applying a radial deformation; an axialschen dilation uses g_r(r,z) = a r with a>0 and g_z(r,z) = z. An anisotropic axial bulge uses g_r(r,z) = r(1 + a(z)) so that the radius changes with height.

Applications: In theoretical studies, axialschen models help analyze axisymmetric material deformations, fluid flows, or metamaterial design. In computer graphics, they provide efficient representations of axisymmetric objects, and in architecture they inspire forms with stable axial symmetry and variable thickness.

Etymology: The name combines the concept of an axis with a Germanic-sounding suffix to indicate a class of symmetry-based transformations.

See also: axial symmetry, rotational surface, axisymmetric deformation, diffeomorphism.