Parametric Cuspoidal Closure Operator: Unified Generation of N-Cusp Families from a Single Sector Curve

We present a parametric cuspoidal closure operator that explicitly separates the local law of a curve from its global closure. The method starts from a sector-defined generating curve and produces closed cuspoidal curves by imposing only the sector angle alpha and discrete rotational closure. In this way, shapes historically treated as separate cases (for example the four-cusp astroid closure) become outputs of a single parametric scheme. We verify numerically that, with an astroid-like generating function, the case alpha = pi/2 reconstructs the standard astroid, while alpha = 2pi/3 generates a coherent, symmetric tri-cuspid curve in the same generative family (an “astroid-like tricuspide”). We further introduce a second shape parameter p that deforms the sector curve while keeping cusp points fixed, yielding continuous families without losing exact closure. In general, for any integer N >= 3, choosing alpha = 2pi/N produces an N-cusp closure; the parameter p controls local morphology at fixed N. Finally, we discuss an inverse use: on data compatible with the family, discrete symmetry provides N and thus alpha, and the shape parameter can be estimated to reconstruct automatically a canonical “containing” curve of the same class. The operator is not limited to the astroid-like choice: replacing f with other cuspoidal generating laws yields other families while preserving the same closure structure. This work addresses a concrete need: turning a “seen” geometry into a controllable, reproducible, and verifiable mathematical mechanism. The core idea is to separate two levels that are often mixed. Local level: the curve law, i.e., the profile shape inside one sector.Global level: how that profile is “closed” in the plane, i.e., how many copies are required and which discrete symmetry is enforced.The operator introduced here formalizes this separation. The consequence is direct: keeping the same local law fixed, the number of cusps and the global symmetry emerge from the sector geometry and the closure rule. Cusp vs “almost asymptote”In informal language one may say the curve “tends to the axes like an asymptote”. In a cuspoidal setting the correct statement is different: the curve reaches the boundary in a finite point while the slope diverges. Therefore it is not a classical asymptote (where distance never becomes zero), but a cusp (contact with vertical or horizontal tangent).This distinction matters because it explains why the curve may look like it “never touches” yet it closes: the slope grows without bound while the endpoint remains finite.Definition of the closure operator