Using the recursive pursuit relative motions we have investigated the beauty of trajectories of motions on pattern formation. Research findings suggest that the concepts for recursion and pursuit relative motions can be used as an interesting way to generate complicated geometric trajectories and graphic patterns. When the recursive levels are somewhat increased, the final pattern seems to be unpredictable and sometimes even peculiar. To manipulate the numbers of recursive levels, the recursive pursuit radius, and slight variations for dynamic range of angular increments might result in immeasurable variations of the trajectories and whole different outcomes. It is apparent that numerous other patterns can be easily generated with these manipulations by appropriately selecting the various parameter values.

A combination of pure recursion and pursuit relative motions should complicate procedures for iteration of the relative motions. In the first case studies of the numbers of recursive levels, more complicated patterns can be created with complex recursive levels. In the second case studies of dynamic variations of recursive radius, new trajectories of movement massively vary with constant recursive procedure. In the third case studies of dynamic range of angular increments, slight variations for ceiling and floor limited would result in the strikingly diversities of trajectories. With such variations of dynamic angular increments can generate a rich variety of patterns with intricate geometric structure. In conclusions, recursion is a concept for manipulates the framework of relative motions to present the self-reference, and complicates the iteration of motions. Further explorations will include manipulating the other different kinds of relative motions, transferring the simple relative motions into the complex recursive frameworks, and experimenting a series of case studies to create fascinated patterns.

[1] http://www.pa.msu.edu/people/horvatin/Astronomers/astronomers_e_pre.htm

[2] Hansen Vagn Lundsgaard. Geometry in Nature. A K Peters, Ltd., Wellesley, Massachusetts, US (1993).

[3] http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Greek_astro.html

[4] http://www.scienceu.com/observatory/articles/retro/retro.html

[5] http://alpha.lasalle.edu/~smithsc/Astronomy/retrograd.html

[6] John M. A Little Book of Coincidence. Walker & Company, New York, US (2001).

[7] http://www.hps.cam.ac.uk/starry/hipparchus.html

[8] Harold A (ed). Turtle Geometry - The Computer as a Medium for Exploring Mathematics. MIT Press Series, Cambridge, US (1981).

[9] Amanda A (ed). Logo for the Macintosh - An Introduction through Object Logo. MIT Press, Cambridge, US (1993).

[10] Jame C. Visual Modeling with LOGO - A Structured Approach to Seeing. MIT Press, Cambridge, US (1998).

[11] Sokolnikoff I S. Mathematics of Physics and Modern Engineering. McGraw-Hill Book Company, New York, US (1967).

[12] Erwin K. Advanced Engineering Mathematics, John Wiley & Sons, Columbus, US (1988).

[13] Rogers D. (ed) Mathematical Elements for Computer Graphics. McGraw-Hill Book Company, New York, US (1990).

[14] Sharp J. In Pursuit of Pursuit Curves. In ISAMA 99, San Sebasti, Spain (1999).

[15] Sun CW. Relative Motions and Complex Shapes. In ISAMA 99, San Sebasti, Spain (1999).

[16] Liao GZ and Sun CW. An Aesthetic Exploration of Relative Motions. Visual Mathematices Art and Science Electronic Journal, Vol 6. No.2 (2004).

Contents: [Index] |
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Concluding Remarks |