lect11-fractals.pdf - CMSC 425 Dave Mount Roger Eastman CMSC 425 Lecture 11 Procedural Generation Fractals and L-Systems Reading The material on

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CMSC 425 Dave Mount & Roger Eastman CMSC 425: Lecture 11 Procedural Generation: Fractals and L-Systems Reading: The material on fractals comes from classic computer-graphics books. The material on L-Systems comes from Chapter 1 of The Algorithmic Beauty of Plants , by P. Prunsinkiewicz and A. Lindenmayer, 2004. It can be accessed online from . Fractals: One of the most important aspects of any graphics system is how objects are modeled. Most man-made (manufactured) objects are fairly simple to describe, largely because the plans for these objects are be designed “manufacturable”. However, objects in nature (e.g. mountainous terrains, plants, and clouds) are often much more complex. These objects are characterized by a nonsmooth, chaotic behavior. The mathematical area of fractals was created largely to better understand these complex structures. One of the early investigations into fractals was a paper written on the length of the coastline of Scotland. The contention was that the coastline was so jagged that its length seemed to constantly increase as the length of your measuring device (mile-stick, yard-stick, etc.) got smaller. Eventually, this phenomenon was identified mathematically by the concept of the fractal dimension . The other phenomenon that characterizes fractals is self similarity , which means that features of the object seem to reappear in numerous places but with smaller and smaller size. In nature, self similarity does not occur exactly, but there is often a type of statistical self similarity, where features at different levels exhibit similar statistical characteristics, but at different scales. Iterated Functions and Attractor Sets: One of the examples of fractals arising in mathemat- ics involves sets called attractors . The idea is to consider some function of space and to see where points are mapped under this function. An elegant way to do this in the plane is to consider functions over complex numbers. Each coordinate ( a, b ) in the real plane is asso- ciated with the complex number a + bi , where i 2 = - 1. Adding and multiplying complex numbers follows the familiar rules: ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i, and ( a + bi )( c + di ) = ac + adi + bci + bdi 2 = ( ac - bd ) + ( ad + bc ) i.Define themodulusof a complex numbera+bito be length of the corresponding vector inthe complex plane, Lecture 11 1 Spring 2018 CMSC 425  • • • 