Arbitrary Polygon Oscillator: Generalizing Polygonal Synthesis to Arbitrary Shapes, Morphing, and Three-Dimensional Polyhedra

Polygonal synthesis generates audio by traversing the perimeter of a polygon with a phasor; prior work uses a constant angular velocity, whereas the proposed system adopts constant arc-length (perimeter) velocity. Existing formulations operate on regular, parametrically defined polygons, producing smooth timbral transitions within a single family of shapes. This paper generalizes polygonal synthesis around a unified arc-length engine: vertex data of any origin feed the same DSP pipeline. First, we adapt the oscillator to accept arbitrary vertex configurations from an external buffer, opening the possibility for a broad class of closed polygons -- regular, irregular, or star-shaped -- to function as a waveform generator. Second, a hybrid interpolation algorithm enables smooth morphing between polygons with unequal vertex counts, passing through intermediate shapes that have no parametric description. Third, we extend the paradigm to three dimensions: a convex polyhedron rotated about three axes is sliced by a fixed horizontal plane, and the resulting cross-section yields a continuously variable polygon controlled by the solid's orientation. The system runs in RNBO (Cycling~'74) with a geometry caching strategy that avoids per-sample recomputation. Antialiasing combines a four-point polyBLAMP correction derived from runtime B'ezier tangents with adaptive oversampling, adapting the correction geometrically to general vertex configurations without per-shape analytical derivation.