Trajectory Extension

This extension can be used to predefine trajectories that are based on mathematical functions and geometrical primitives.
The renderer can evaluate those functions at runtime according to its own update rate.
Consequently, the accuracy of the trajectory is preserved according to the quality/granularity of the renderer.
This approach can be compared to differences between vector graphics (e.g., .eps) and raster graphics (e.g., .bmp).

Suggested Functions

Taxonomy of trajectories suggest five different kinds

Style Curve Type
Constant Point
Periodic Lines, Curves, Lissajous Figures, Bresenham’s Algorithm
Quasi-Periodic Invariant Tori, Spirals, Spirographs, Turtle Graphics
Chaotic Strange Attractors, Webs and Wreaths, Mazes, Space Filling Curves, L-Systems
Stochastic Random Walks, Noise, Jitter

Periodic Curves

Closed Curves Open Curves
Broken Unbroken
Bicorn Swastika Curve Lituus Curve
Lemniscate Devil’s Curve Witch of Agnesi
Scarabaeus Plateau Curve
Rose Curve Cissoid of Diocles
Butterfly Curve
Gear Curve
Lissajous Curve and in 3D
Limaçon Curve
Epitrochoid, Epicycloid
Circular Motion



Stochastic Trajectories


  • James, S. G. (2005). Developing a flexible and expressive realtime polyphonic wave terrain synthesis instrument based on a visual and multidimensional methodology. Master’s thesis, Edith Cowan University, Perth, Australia. [pdf]
  • Nara Hahn, Keunwoo Choi, Hyunjoo Chung, and Koeng-Mo Sung (2010). Trajectory Sampling for Computationally Efficient Reproduction of Moving Sound Sources, 128th AES Convention, Preprint 8080.
  • Fernando Lopez-Lezcano: A dynamic spatial locator ugen for CLM, SMC 2008 [pdf]
  • Garry Kendall. Composing from a geometric model:” five-leaf rose”. Computer Music Journal, 5(4):66–73, 1981.