Outline. Solid models and fractals. Constructive solid geometry. Constructive solid geometry COM3404. Richard Everson

Transcription

1 Outline Solid models and fractals COM School of Engineering, Computer Science and Mathematics University of Exeter Constructive solid geometry Fractals Dimension s Landscape generation L-systems R.M.Everson@exeter.ac.uk References Fundamentals of D Computer Graphics. Watt. Chapters & Computer Graphics: Principles and Practice. Foley et al (995). Fractal Geometry of Nature. Mandelbrot. (9). / / Constructive solid geometry Models constructed from primitives using volumetric oolean operations Primitives Cuboids, spheres, cylinders, prisms, cones. Operations union, intersection, subtraction Special algorithms (Goldfeather, SCS) for rendering the CSG without directly computing the D object representation. solid geometry / Constructive solid geometry Representation Stored in an attributed tree Nodes Combining operation Leaves Primitives Logic lteration Limitations The logic underlying the model is preserved in the tree. Dimensions of primitives are easily altered. Limited range of operations to construct models Detailed modifications to a local region may be difficult because oolean operations are global: e.g., blending at joins is difficult. Specific rendering algorithms or conversion to polygon form 5 /

2 Implicit functions and metaballs r = (x, y, z) Define a function f (r) on all of the modelling space. Surface of a model is the contour or iso-surface f (r) = const Sources Define potential function as a sum of sources with location and strength f (x, y) = (x x ) + (y y ) f (x, y) = c defines a circle of radius c centred at (x, y ) x =, y =.5, contour f (x, y) = f (x, y) = exp { [(x + 5) + (y + 5) ]/ } Thick contour at f (r) = exp { [(x 5) + (y 5) ]/ }. 7 / /. Negatively weighted sources can be used to sculpt concavities... f (x, y) = exp { [(x + 5) + (y + 5) ]/ } +.5 exp { [(x 5) + (y 5) ]/ } exp { [x + (y ) ]/5 } Thick contour at f (r) =.. 9 / /

3 Fractals Fractals Objects that are self similar on a wide range of scales Self similar When a component part is appropriately magnified it appears identical to the whole. Koch curve Sierpinski gasket pproximately self similar Many naturally occurring objects are statistically self-similar over a range of scales Coast line of Great ritain Clouds lveoli in lungs / Generating fractals: Koch curve Line Dimension of a line is because if it is divided into N parts each looks like the original scaled down by N Generator t each stage the initiator is replace by a copy of the generator N = L = (/) N = L = N N L = = / Dimension Initiator The initial shape Length at each stage is: N = L = Plane Divide a square into N: each looks like the original scaled by N Koch curve Each of the pieces of the Koch curve looks like the original scaled down by a factor. Dimension d /d = d = log()/ log() =... limn L(N) = Fractals have non-integer dimension / 5 /

4 s D Sierpinski gaskets Fractal fern Julia set Generated by repeated affine transformations. Iterated function systems: Michael arnsley Fractals Everywhere, cademic Press (9). / Recursive subdivision for meshes 7 / Fractal terrain Naive methods tend to produce creases ; alleviated by using overlapping hexagonal grids. Divide generator in two and displace vertically by a random amount In real landscapes the statistical properties of hills and valleys differ. Choose displacements from an asymmetric distribution so that small displacements are more probable. Landscapes may have a many polygons; problematic for rendering algorithms. Divide triangular facet edges into two, forming smaller facets Vertically displace the midpoints by random amounts / 9 /

5 Fractal terrain L-systems Lindenmayer systems Grammar to describe branching processes lphabet Production {,, [, ], (, ) } []() []() [[]()]([]()) Fractal terrain rendered with POV-Ray Generated by fracplanet See also terragen ssociate [...] with a left branch ssociate (...) with a right branch / L-systems / L-system trees [[]()]([]()) []() Rich alphabets Symbols only become effective at later stages of production Randomised variations in geometric interpretation of symbols, e.g. branching angles, width of cylinders forming trunks. / /