Implisoidulla

Implisoidulla is a hypothetical construct in geometry and computer graphics used to describe a family of shapes produced by a controlled morphing process that interpolates between a regular polygon and a circle. The term appears in speculative discussions of shape interpolation and morphological design, serving as a simple model for studying how angular features soften during continuous deformation.

Consider a regular n-gon inscribed in a circle of radius R. Let r_poly(θ) denote its radial function, the distance from the center to the polygon boundary at angle θ. For s in [0,1], define the implisoidulla_s boundary by r_s(θ) = (1 - s) r_poly(θ) + s R. The curve traced by r_s as θ varies is a convex, closed curve. When s=0, r_0 = r_poly and the boundary is the polygon; when s=1, r_1 = R is the circle. Intermediate s produce rounded, polygon-like shapes that preserve the original symmetry.

Extensions include using a smooth weight function w(θ) to vary the interpolation across angles, generating a family with tunable corner sharpness; or replacing the simple linear interpolation with a B-spline-based morph to achieve smoother transitions. In computational terms, implisoidulla can be generated from distance fields or support functions, facilitating gradient-based optimization for shape fitting.

Because implisoidulla is a theoretical construct used for illustration, it appears in speculative literature on shape morphing and design education rather than as a standard mathematical object. It provides a minimal, parameterized example of how angular features fade under morphing and how morphing parameters influence area, perimeter, and curvature.