A New Potential Function for Self Intersecting Gielis Curves With Rational Symmetries

Abstract

Abstract:We present a new potential field equation for self-intersecting Gielis curves with rational rotational symmetries. In the literature, potential field equations for these curves, and their extensions to surfaces, impose the rotational symmetries to be integers in order to guarantee the unicity of the intersection between the curve/surface and any ray starting from its center. Although the representation with natural symmetries has been applied to mechanical parts modeling and reconstruction, the lack of a potential function for Rational symmetry Gielis Curves (RGC) remains a major problem for natural object representation, such as flowers and phyllotaxis. We overcome this problem by combining the potential values associated with the multiple intersections using R-functions. With this technique, several differentiable potential fields can be defined for RGCs. Especially, by performing N-ary R-conjunction or R-disjunction, two specific potential fields can be generated: one corresponding to the inner curve, that is the curve inscribed within the whole curve, and the outer -or envelope-that is the curve from which self intersections have been removed.

From Superellipses to Superformula and Technology

“…A further development was by using R-functions to combine supershapes, allowing to define controllable potential fields on the shapes (Fig. 18) [31,32,33]. R-functions [34] form a natural alliance with superellipses [35,36].…”