Fractals :Fractals Infinite detail at every point Self similarity between parts and overall features of the object Zoom into Euclidian shape Zoomed shape see more detail eventually smooths Zoom in on fractal See more detail Does not smooth Model Terrain, clouds water, trees, plants, feathers, fur, patterns General equation P1=F(P0), P2 = F(P1), P3=F(P2)… P3=F(F(F(P0)))


Self similar fractals :Self similar fractals Parts are scaled down versions of the entire object use same scaling on subparts use different scaling factors for subparts Statistically self-similar Apply random variation to subparts Trees, shrubs, other vegetation


Fractal types :Fractal types Statistically self-affine random variations SxSySz terrain, water, clouds Invariant fractal sets Nonlinear transformations Self squaring fractals Julia-Fatou set Squaring function in complex space Mandelbrot set Squaring function in complex space Self-inverse fractals Inversion procedures


Julia-Fatou and Mandelbrot :Julia-Fatou and Mandelbrot x=>x2+c x=a+bi Complex number Modulus Sqrt(a2+b2) If modulus 1 Squaring falls towards infinity If modulus=1 Some fall to zero Some fall to infinity Some do neither Boundary between numbers which fall to zero and those which fall to infinity Julia-Fatou Set Foley/vanDam Computer Graphics-Principles and Practices, 2nd edition Julia-Fatou


Julia Fatou and Mandelbrot con’d :Julia Fatou and Mandelbrot con’d Shape of the Julia-Fatou set based on c To get Mandelbrot set – set of non-diverging points Correct method Compute the Julia sets for all possible c Color the points black when the set is connected and white when it is not connected Approximate method Foreach value of c, start with complex number 0=0+0i Apply to x=>x2+c Process a finite number of times (say 1000) If after the iterations is is outside a disk defined by modulus>100, color the points of c white, otherwise color it black. Foley/vanDam Computer Graphics-Principles and Practices, 2nd edition


Constructing a deterministic self-similar fractal :Constructing a deterministic self-similar fractal Initiator Given geometric shape Generator Pattern which replaces subparts of initiator Koch Curve Initiator generator First iteration


Fractal dimension :Fractal dimension D=fractal dimension Amount of variation in the structure Measure of roughness or fragmentation of the object Small d-less jagged Large d-more jagged Self similar objects nsd=1 (Some books write this as ns-d=1) s=scaling factor n number of subparts in subdivision d=ln(n)/ln(1/s) [d=ln(n)/ln(s) however s is the number of segments versus how much the main segment was reduced I.e. line divided into 3 segments. Instead of saying the line is 1/3, say instead there are 3 sements. Notice that 1/(1/3) = 3] If there are different scaling factors Skd=1 K=1 n


Figuring out scaling factorsI prefer: ns-d=1 :d=ln(n)/ln(s) :Figuring out scaling factorsI prefer: ns-d=1 :d=ln(n)/ln(s) Dimension is a ratio of the (new size)/(old size) Divide line into n identical segments n=s Divide lines on square into small squares by dividing each line into n identical segments n=s2 small squares Divide cube Get n=s3 small cubes Koch’s snowflake After division have 4 segments n=4 (new segments) s=3 (old segments) Fractal Dimension D=ln4/ln3 = 1.262 For your reference: Book method n=4 Number of new segments s=1/3 segments reduced by 1/3 d=ln4/ln(1/(1/3))


Sierpinski gasket Fractal Dimension :Sierpinski gasket Fractal Dimension Divide each side by 2 Makes 4 triangles We keep 3 Therefore n=3 Get 3 new triangles from 1 old triangle s=2 (2 new segments from one old segment) Fractal dimension D=ln(3)/ln(2) = 1.585


Cube Fractal Dimension :Cube Fractal Dimension Apply fractal algorithm Divide each side by 3 Now push out the middle face of each cube Now push out the center of the cube What is the fractal dimension? Well we have 20 cubes, where we used to have 1 n=20 We have divided each side by 3 s=3 Fractal dimension ln(20)/ln(3) = 2.727 Image from Angel book


Language Based Models of generating images :Language Based Models of generating images Typical Alphabet {A,B,[,]} Rules A=> AA B=> A[B]AA[B] Starting Basis=B Generate words Represents sequence of segments in graph structure Branch with brackets Interesting, but I want a tree B A[B]AA[B] AA[A[B]AA[B]]AAAA[A[B]AA[B]] A A A B B A A A B AA B A A A A A B AA B


Language Based Models of generating images con’d :Language Based Models of generating images con’d Modify Alphabet {A,B,[,],(,)} Rules A=> AA B=> A[B]AA(B) [] = left branch () = right branchStarting Basis=B Generate words Represents sequence of segments in graph structure Branch with brackets B A[B]AA(B) AA[A[B]AA(B)]AAAA(A[B]AA(B)) A A A B B A A A B AA B A A A A A B AA B


Language Based models have no inherent geometry :Language Based models have no inherent geometry Grammar based model requires Grammar Geometric interpretation Generating an object from the word is a separate process examples Branches on the tree drawn at upward angles Choose to draw segments of tree as successively smaller lengths The more it branches, the smaller the last branch is Draw flowers or leaves at terminal nodes A A A B AA B A A A A A B AA B


Grammar and Geometry :Grammar and Geometry Change branch size according to depth of graph Foley/vanDam Computer Graphics-Principles and Practices, 2nd edition


Particle Systems :Particle Systems System is defined by a collection of particles that evolve over time Particles have fluid-like properties Flowing, billowing, spattering, expanding, imploding, exploding Basic particle can be any shape Sphere, box, ellipsoid, etc Apply probabilistic rules to particles generate new particles Change attributes according to age What color is particle when detected? What shape is particle when detected? Transparancy over time? Particles die (disappear from system) Movement Deterministic or stochastic laws of motion Kinematically forces such as gravity


Particle Systems modeling :Particle Systems modeling Model Fire, fog, smoke, fireworks, trees, grass, waterfall, water spray. Grass Model clumps by setting up trajectory paths for particles Waterfall Particles fall from fixed elevation Deflected by obstacle as splash to ground Eg. drop, hit rock, finish in pool Drop, go to bottom of pool, float back up.


Physically based modeling :Physically based modeling Non-rigid object Rope, cloth, soft rubber ball, jello Describe behavior in terms of external and internal forces Approximate the object with network of point nodes connected by flexible connection Example springs with spring constant k Homogeneous object All k’s equal Hooke’s Law Fs=-k x x=displacement, Fs = restoring force on spring Could also model with putty (doesn’t spring back) Could model with elastic material Minimize strain energy


“Turtle Graphics” :“Turtle Graphics” Turtle can F=Move forward a unit L=Turn left R=Turn right Stipulate turtle directions, and angle of turns Equilateral triangle Eg. angle =120 FRFRFR What if change angle to 60 degrees F=> FLFRRFLF Basis F Koch Curve (snowflake) Example taken from Angel book


Using turtle graphics for trees :Using turtle graphics for trees Use push and pop for side branches [] F=> F[RF]F[LF]F Angle =27 Note spaces ONLY for readability F[RF]F[LF]F [RF[RF]F[LF]F] F[RF]F[LF]F [LF[RF]F[LF]F] F[RF]F[LF]F