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

Antonio Argentieri; Francesco Scagliola

cs.SDAug 25, 2026

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

Authors: Antonio Argentieri, Francesco Scagliola

Organizations: Conservatorio Niccolò Piccinni di Bari Bari, Italy

Abstract

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 Bézier tangents with adaptive oversampling, adapting the correction geometrically to general vertex configurations without per-shape analytical derivation.

Figures & tables

Figure 1: Polygonal synthesis illustrated for an equilateral triangle (

N=3

f_{0}=

110\text{,}\mathrm{Hz}$$ ). Left: the phasor traverses the perimeter at constant velocity; the traversal point

\mathbf{P}(\phi)

is the polygon coordinate at phase

\phi

. Right: the resulting output waveform (one coordinate of

\mathbf{P}(\phi)

) over one complete traversal, the phase

\phi

running from

0

1

Panel of Figure 2 — Figure 2: Triangle, $N=3$ . (top) Waveforms ( $f_{0}=$ 110\text{,}\mathrm{Hz} $, $f_{s}=$44.1\text{\,}\mathrm{kHz}$ ): (a) angular-velocity [ 10 ] , (b) arc-length, no correction, (c) $\kappa=-0.234$ , (d) $\kappa=-0.234$ with radial weighting. (bottom) Spectra of (a) (blue dashed) vs (d) (red) at $f_{0}=$ 500\text{,}\mathrm{Hz} $, $f_{s}=$48\text{\,}\mathrm{kHz}$ ; SNR/THD deltas inset.

Figure 3: Four

M{=}5

shapes with different vertex counts share the harmonic lattice

\mathcal{H}(5)=\{(5m\pm 1)f_{0}\}

: pentagon (5v, convex), five-pointed star (10v, concave), pinwheel (10v, fan), gear (15v, teeth). Harmonics outside the lattice fall below the

-80\,\text{dB}

analysis floor regardless of vertex count.

f_{0}=

440\text{,}\mathrm{Hz}

, $f_{s}=$48\text{\,}\mathrm{kHz}

Figure 4: Interpolation strips (left) and magnitude spectrograms over the morph (right), 4 frames each. (a) Triangle ( $\tau=0$ ) to five-pointed star ( $\tau=1$ ). (b) Triangle ( $\tau=0$ ) to regular octagon ( $\tau=1$ ). $f_{0}=$110\text{\,}\mathrm{Hz}$$ , $f_{s}=$44\,100\text{\,}\mathrm{Hz}$$ . — Figure 4: Interpolation strips (left) and magnitude spectrograms over the morph (right), 4 frames each. (a) Triangle ( $\tau=0$ ) to five-pointed star ( $\tau=1$ ). (b) Triangle ( $\tau=0$ ) to regular octagon ( $\tau=1$ ). $f_{0}=$ 110\text{,}\mathrm{Hz} $, $f_{s}=$44\,100\text{\,}\mathrm{Hz}$ .

Figure 5: Triangle $\to$ square $\to$ star $\to$ arrow $\to$ pentagon $\to$ hexagon. Top: shape frames labelled by morph parameter $\tau$ . Bottom: STFT magnitude over the full sequence. Harmonic lines persist and redistribute while the fundamental stays fixed, and no discontinuity appears at the pair-switch boundaries.

Figure 6: Cube cross-section from square ( $\theta_{x}=\theta_{y}=0^{\circ}$ , blue) to near-regular hexagon ( $\theta_{x}=\theta_{y}=40^{\circ}$ , amber).

Straight edges ( $\kappa=0$ )
$f_{0}$	No AA	OS = 2	BL4 only	OS = 2+BL4
400 Hz	58.8	77.6	80.9	85.1
751 Hz	51.1	75.6	75.3	79.1
1350 Hz	43.9	67.7	66.2	70.1
Edge curvature ( $\kappa=-0.234$ )
400 Hz	57.8	78.5	79.3	83.4

Table 1: SNR (dB) for an equilateral triangle (3-vertex buffer,

f_{s}=44100

Hz). Best value per row in bold.

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