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.