Abstract
From this chapter on, the book concentrates on the methods of computer graphics rather than the mathematical tools previously described, although the precise division is of necessity blurred. A few new mathematical topics will be introduced, but most of the remainder concentrates on applications of what has been introduced previously. There have been occasional hints in previous chapters as to how mathematical concepts can be used. This is where these start coming home to roost.
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References
M. Mäntylä (1988) Introduction to Solid Modeling, Computer Science Press, Rockville, MD.
G. Wyvill, C. McPheeters and B. Wyvill describe their animation method in Animating Soft Objects, The Visual Computer, August 1986, 2(2), 235-242. These three have been responsible for much development of isosurface methods.
You will find this in B.B. Mandelbrot (1983) The Fractal Geometry of Nature, W.H. Freeman, New York, p. 15.
First published by Waclaw Sierpinski in 1916, found by the author in W. Sierpinski, (1916, republished 1975) Sur Une Courbe dont tout point est un point de ramification, in W. Sierpinski: Oeuvres Choisis, Tome II, PWN - Polish Scientific Publishers, Warsaw, Poland.
In M.F. Barnsley (2000) Fractals Everywhere (3 rd edn), Morgan Kauffnan.
Comment in C.B. Boyer and U.A. Merzbach (1991) The History of Mathematics (2 nd edn), Wiley, New York, p. 598.
These concepts are thoroughly discussed in B. Mandelbrot (1983) The Fractal Geometry of Nature, W.H. Freeman, New York. Only a brief overview is given here.
Richardson's results are described in detail in in B.B. Mandelbrot (1983) The Fractal Geometry of Nature, W.H. Freeman, New York.
Again, a comment from C.B. Boyer and U.A. Merzbach (1991) The History of Mathematics (2 nd edn), Wiley, New York, p. 621. Neighbourhoods were explained by Hausdorff in his publication of 1914, Basic Features of Set Theory.
Two books played a huge part in explaining and popularizing images of these fractal forms: H.-O. Peitgen and P.H. Richter (1986) The Beauty of Fractals, Springer-Verlag, Berlin
H.-O. Peitgen and D. Saupe (1988) The Science of Fractal Images, Springer-Verlag, New York. The first of these has an excellent ‘how to do it’ section, giving ranges of values within which to seek interesting Mandelbrot set outlines.
Biomorphs', forms that look like zoological cellular structures are shown in C.A. Pickover (1986) Biomorphs: Computer displays of biological forms generated from mathematical feedback loops, Computer Graphics Forum, 5, 313-316.
in R.A. Earnshaw, A.R. Crilly and H. Jones (eds), Fractals and Chaos, Springer, New York, pp. 35-42.
P. Prusinkiewicz and A. Lindenmayer (1990) The Algorithmic Beauty of Plants, Springer-Verlag, New York. This is a beautifully illustrated classic, reissued in paperback form in 1996.
This is after a method of Prusinkiewic and his collaborators in one of their many excellent follow-up papers to Prusinkiewic and Lindenmayer (1990), the book previously cited. The full reference is P. Prusinkiewicz, M. James and R. Mëch (1994) Synthetic topiary. Proceedings of ACM SIGGRAPH ’94, 351-358.
A summary of these methods is found in H. Jones, A. Tunbridge and P. Briggs (1997) Modelling of growing biological processes using parametric L-systems, in Visualization and Modeling, R.A. Earnshaw, H. Jones and J. Vince (eds), pp 303-317, Academic Press, San Diego.
J.A. Kaandorp (1994) Fractal Modelling: Growth and Form in Biology. Springer.
M.F. Barnsley (2000) Fractals Everywhere (3 rd edn), Morgan Kauffman.
Y. Fisher (1994) Fractal Encoding: Theory and Applications to Digital Images, Springer
N. Lu (1997) Fractal Imaging, Academic Press.
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Jones, H. (2001). Geometric Modelling and Fractals: Building Descriptions of Objects. In: Computer Graphics through Key Mathematics. Springer, London. https://doi.org/10.1007/978-1-4471-0297-7_8
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DOI: https://doi.org/10.1007/978-1-4471-0297-7_8
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