CGT 511 Procedural Methods

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1 3D object representation CGT 511 Procedural Methods Volume representation Voxels 3D object representation Boundary representation Wire frame Procedural Fractals Bedřich Beneš, Ph.D. Purdue University Department of Computer Graphics Technology Oct tree CSG Surfaces Polygonal Free form Bézier surfaces Particle systems Grammars NURBS Implicit surfaces Procedural Techniques Three classes: fractals particle systems grammars Used when shape cannot be represented as a surface (fire, water, smoke, flock of birds, explosions, model of mountain, grass, clouds, plants, etc.) Used for Simulation of Natural Phenomena Procedural Techniques Model is generated by a piece of code Model is not represented as data! The generation can take some time and of course the data can be precalculated 1

$The Mandelbrot Set the big logo The Mandelbrot set discovered in 1970 by Benoit Mandelbrot It is a non linear deterministic fractal It is graph of a solution of a dynamic system The Mandelbrot Set$

2 The Mandelbrot Set the big logo The Mandelbrot set discovered in 1970 by Benoit Mandelbrot It is a non linear deterministic fractal It is graph of a solution of a dynamic system The Mandelbrot Set Take the equation z n+1 = z n2 +c where: z i and c are complex numbers and z 0 =0+i0 Explore c complex numbers from the complex plane Measure the speed of divergence of the z n i.e., measure when z n > 2 (predefined value) The Mandelbrot Set Explore c complex numbers from the complex plane Measure the speed of divergence of the z n i.e., measure when z n > 2 (predefined value) there are two kinds of points: z n < 2 for n >, (stable points) z n > 2 for certain n and n greater (unstable points) The Mandelbrot Set a) stable points displayed in black b) unstable points color=f(n) for every point c in the plane < 2 2i>, <2+2i> do set z = 0+i0 set n=0 while (n<max) and ( z <2) do z=z 2 +c end of while if (n==max) Draw Point(Black) else Draw Point(n) end of for 2

3 The Mandelbrot Set Zooming into the Mandelbrot set The Mandelbrot Set Zooming into the Mandelbrot set The Mandelbrot Set Zooming into the Mandelbrot set The Mandelbrot Set Zooming into the Mandelbrot set 3

The Mandelbrot Set Zooming into the Mandelbrot set Dimensions Dimension how long is the coast of Corsica? A stick of 500m will give 700km (not very precise) A stick of 100m will give 1200km (?

we have applied n sticks of length N() times the total length is K stick of the length 1m and = 10cm=1/10m we need N() = 10 pieces if =0.

4 The Mandelbrot Set Zooming into the Mandelbrot set Dimensions Dimension how long is the coast of Corsica? A stick of 500m will give 700km (not very precise) A stick of 100m will give 1200km (?) What if we use length equal to dl? The length will be Dimensions How long is the coast of Corsica? we have applied n sticks of length N() times the total length is K stick of the length 1m and = 10cm=1/10m we need N() = 10 pieces if =0.5m we need N()=2 K= N() But in the case of Corsica we need to use formula K= D N() to get reasonable results What is this D? (Richardson) Dimensions How long is the coast of Corsica? Having N() = 1/ we get K= D / with >0 (i.e., getting the stick shorter) we get D K lim 0 For the line segment we need to set D=1 it the case of Corsica we need to set non integer The D is so called Hausdorff dimension Sometimes called fractal dimension Fractals do not have dimensions 0,1,2,... 4

5 Fractals Fractals fractal is a set that is self similar, A i ( A) i1 it is a set that is copy of itself here i is transformation scales down scale, denoted by s must be 0 < s < 1 i.e., a contraction Fractal is set that has Hausdorff s dimension greater than its dimension topological Fractals Fractal is a set that is geometrically complex and its structure is given by repetition of a certain shape at different scales. Fractals Fractals Linear Non linear Deterministic Non deterministic Deterministic or exact fractals Non deterministic or random fractals (Based on the type of self similarity (exact, random)) Linear rotation, scale, translation, shear Non linear the others Fractals Fractals Linear Non linear Deterministic Non deterministic Four classes: 1. Linear deterministic, 2. Non linear deterministic, 3. Non linear non deterministic 4. Linear non deterministic In CG we mostly use the last mentioned 5

$Dimensions Special cases If we use just ONE transformation all the time with fixed and n repetitions the fractal dimension is determined as D=log n/log Linear Deterministic Fractals The Cantor s$

6 Dimensions Special cases If we use just ONE transformation all the time with fixed and n repetitions the fractal dimension is determined as D=log n/log Linear Deterministic Fractals The Cantor s discontinuum (published in 1883) 1) take a line segment 2) scale it 1/3 3) shift it to the left and right. 4) do it recursively (Georg Cantor ) Linear Deterministic Fractals What is the dimension of the Cantor s discontinuum? scale = 3 n=2 new pieces for this we can use formula D=log n/log D= log 2/ log 3= 0.63 this object is something between a point and a line Linear Deterministic Fractals The Koch s snowflake 1) take a line segment 2) erase the mid third (similar to Cantor) 3) create equilateral triangle 4) repeat this recursively Scale = 3, n = 4, D=log 4 /log 3 = 1.26 Properties: does not have derivative! infinite length zero area the area inside is non zero and non infinity (January 25, 1870 March 11, 1924) 6

Linear Deterministic Fractals The Sierpinsky gasket (Wacław Franciszek Sierpiński

n = 3 D=log 3 /log 2 = 1.59 not plane, not line!

a cube 2) erase mid cross of cubes 3) repeat this recursively Scale = 3 n = 20

Linear Deterministic Fractals The Peano curve (Guiseppe Peano 1858 1932) 1) take a

each line segment =3 n=9 D=log 9/ log 3= 2 space filling curve Non Linear

7 Linear Deterministic Fractals The Sierpinsky gasket (Wacław Franciszek Sierpiński ) 1) take a triangle 2) erase mid triangle 3) repeat this recursively = 2 n = 3 D=log 3 /log 2 = 1.59 not plane, not line! Linear Deterministic Fractals The Menger sponge (Karl Menger ( )) 1) take a cube 2) erase mid cross of cubes 3) repeat this recursively Scale = 3 n = 20 D=log 20 /log 3 = 2.73 Dust of points! It is invisible! Linear Deterministic Fractals The Peano curve (Guiseppe Peano ) 1) take a line segment 2) substitute it by the lines from the image 3) repeat recursively on each line segment =3 n=9 D=log 9/ log 3= 2 space filling curve Non Linear Deterministic Fractals Quaternions if we apply the formula z n+1 = z n2 +c in quaternion space (hypercomplex numbers 4D) we will get something like this: 7

Linear Non Deterministic Fractals random numbers and linear transformations motivation: What would happen if increasing the speed of playing music? Noise?

Linear Non Deterministic Fractals The way to model Nature is capturing the self similarity with stochastic processes Brownian Motion Brownian motion (Bm) Particles

8 Linear Non Deterministic Fractals random numbers and linear transformations motivation: What would happen if increasing the speed of playing music? Noise? Change of the speed is scaling Linear Non Deterministic Fractals Another motivation: How large is this stone? 10cm? 1m? 10m? Linear Non Deterministic Fractals The way to model Nature is capturing the self similarity with stochastic processes Brownian Motion Brownian motion (Bm) Particles of pollen in water watched under microscope The key for these models are noise functions Noise is (usually) self similar moving because of random hits of molecules of water This is also called the random walk 8

9 Brownian Motion 1D Bm simulation: Random hits have Gaussian random number distribution e x 2 1 ( ) 2 f ( x) 2 2 So we need them... the rand() function generates 0,1,,A (A=2 15 1) suppose we have n of such numbers d i the Gaussian random number W n 1 12 is: W d A n i1 i 3 n Brownian Motion 1D Bm simulation (contd.): let s have func. W( that kicks a point in the y direction it has Gaussian random numbers distribution it is called the white noise the Bm (Brownian function) X( is then simply said, we accumulate the perturbations W( X( X ( W ( s) t s Brownian Motion 1D Bm simulation (contd.): The fractal dimension of this curve is D=1.5 Bm scaling if we scale the Bm in the axis x by coefficient r we have to scale in the y axis by r H, where H is so called Hurst exponent then the fractal dimension of the curve is D=2 H this curve is called Fractional Brownian Motion or fbm H is 0<H<1 so the D is <1,2> (this is NOT Fractal Brownian Motion) Fractional Brownian Motion (fbm) fbm H=0.5 > fbm has D=1.5 i.e., it is Bm H>0.5 > fbm has 1<D<1.5 H<0.5 > fbm has 1.5<D<2 higher D means wilder the curve is autocorrelated H=0.5 autocorrelation is 0 H<0.5 autocorrelation is positive H>0.5 autocorrelation is negative 9

10 Fractional Brownian Motion (fbm) Linear Non Deterministic Fractals In order to generate nice fractals, we need fbm The above described function is good But does not provide adaptive results The Midpoint Displacement Algorithm the MDP is the key algorithm for fractals 1) Take a line segment 2) Find its midpoint (center) 3) Move it randomly in the y direction 4) Apply this step to all lines recursively The Midpoint Displacement Algorithm The random numbers distribution has its and. Scale down the by two in every iteration. (the second level of recursion has /2, the third has /4, the fourth has /8, etc.) Divide the Gaussian random numbers by two (if we divide in each step by 2 H, H is the Hurst exponent we will get fbm with dimension D=2 H) 10

The Midpoint Displacement Algorithm Properties very easy to implement it is interpolation of two points, so it is also called the Fractal interpolation just division by two and random numbers call

11 The Midpoint Displacement Algorithm Properties very easy to implement it is interpolation of two points, so it is also called the Fractal interpolation just division by two and random numbers call quite realistic The Midpoint Displacement Algorithm Two dimensional fbm works with 2D arrays it is called fbm surface dimension is D=3 H, where 0<H<1 so the D is <2,3> D=2.5 D=2.8 The Midpoint Displacement Algorithm 2D fbm on quadrilaterals (diamond square) having four points P 0,P 1,P 2,P 3 1) Evaluate the midpoint 2) Evaluate points on the edges 3) Evaluate points in the middle 4) Ad 2) etc... The Midpoint Displacement Algorithm 2D fbm on quadrilaterals (diamond square) 11

the limit of this process is fbm Random Faults Random Faults Implementation Data: 2D array Algorithm: Do this many times: 1) Generate random line (get two

12 Random Faults Another method for generating fbm 1. take an object 2. divide it into two parts 3. increase the elevation of one and decrease the other one (change color, etc.) 4. apply this step many times 5. in each decrease the intensity of changes 2x 6. the limit of this process is fbm Random Faults Random Faults Implementation Data: 2D array Algorithm: Do this many times: 1) Generate random line (get two random points) 2) For each point in the 2D array I. Check if it is on the left or right (dot produc II. Increase/decrease the value Random Faults Variations Use any 2D object (circle, square) Randomly alter the area inside/outside 12

Random Faults on a Sphere 1) Take sphere and divide it randomly into two

recursively, but decrease the intensity in each step with 1/2 H Random

(molecules) are floating in the water 2) When a particle approaches an

particles Diffusion Limited Aggregation DLA The important part is the way

13 Random Faults on a Sphere 1) Take sphere and divide it randomly into two hemispheres assign to each hemisphere different color 2) Apply the step 1 recursively, but decrease the intensity in each step with 1/2 H Random Faults on a Sphere Diffusion Limited Aggregation DLA 1) Particles (molecules) are floating in the water 2) When a particle approaches an condensation center it is aggregated 3) The process is repeated for many particles Diffusion Limited Aggregation DLA The important part is the way the particles travel i.e., random walk Corals, lightings, frozen ice on a window, etc. 13

Diffusion Limited Aggregation DLA Diffusion Limited Aggregation DLA 3D object representation 3D object representation Volume representation Boundary representation Voxels Wire frame Oct tree Surfaces

14 Diffusion Limited Aggregation DLA Diffusion Limited Aggregation DLA 3D object representation 3D object representation Volume representation Boundary representation Voxels Wire frame Oct tree Surfaces CSG Polygonal Free form Bézier surfaces NURBS Procedural Fractals Particle systems Grammars L systems Lindenmayer's systems Lindenmayer s system or D0L systems [d zero l system] (parallel string rewriting systems) L-system is an ordered triple : L=<V,P,w>, where: V is a finite non-empty set called an alphabet P is a finite set of production rules in the form A->a, where A V and a V* * denotes reflexive transitive closure w V is so called axiom (starting symbol) Aristid Lindenmayer ( ) Implicit surfaces 14

15 L systems Example: F >F+F F+F +, ~ 60 o generates: F >F+F F+F=> => F+F F+F + F+F F+F F+F F+F + F+F F+F => etc. Interpretation makes the Koch s snowflake L systems can generate linear deterministic fractals Peano curve, Sierpinsky gasket, etc. L systems Example: L= < {F,[,],+, }, {F >F[+F]F, [ >[, ] >], + > +, > }, F > products: F > >F[+F]F=> F[+F]F [+ F[+F] F ] F[+F] F=> =>etc //the alphabet //rewriting rules //axiom L systems string is interpreted by the turtle graphics turtle is an abstract geometrical automaton having its state (position and orientation) able to interpret commands symbol from the alphabet has assigned a command usually F go forward + turn right turn left [ save state onto stack ] pop state from the stack bracketed L systems (Prusinkiewicz) L systems the first string from the example is interpreted: F [ + F ] F 15

16 L systems Bracketed rules are useful for generating branching structures F >F[+F][ F]F F >F[[+F] F]F 1L(R)L systems Context sensitive Lindenmayer's systems D0L systems deterministic, zero context can be easily extended by random numbers 1LL systems include context sensitive rewriting 1LL systems one element from the left side is 1L(R)L systems Example P: A<B > A w: ABBBB ABBBB > AABBB > AAABB > AAAAB > AAAAA It simulates propagation of a signal from the root! 1RL systems propagation from leaves down 1L1RL systems both directions pl systems Parametric Lindenmayer's systems pl systems every symbol has parameter Example: F(x) > F(x)[+F(x/2)][ F(x/2)] x is the length of the step F(8) > F(8)[+F(4)][ F(4)] => F(8)[ +F(4)(+F(2)][ F(2))(F(2) ][ F(4)(+F(2)][ F(2))(F(2) ] => the lateral branches are getting smaller 16

17 pl systems Parametric Lindenmayer's systems (contd.) pl systems conditional rule rewriting F(x) (x<=1) > e F(x) (x<=10) > F(x 1) F(5) > F(4) > F(3) > F(2) > F(1) > e shortening the branch pl systems Example: P: A(x) x==0 > FA(x+1) A(x) x==3 > B A(x) x==1 > [ L]FA(x+1) A(x) x==2 > [+L]FA(x+1) w: A(0) A(0) >FA(1)=>F[ L]FA(2)=>F[ L]F[+L]FA(3)=>F[ L]F[+L]FB F ~ stem L ~ leaf B ~ blossom A ~ bud pl systems Open L systems Open L systems Can interact with surrounding environment?e(x 1, x 2, ) query modules set exogenous parameters x can be for example distance from the obstacle A(0)-> FA(1)-> F[-L]FA(2)->F[-L]F[+L]FA(3)->F[-L]F[+L]FB 17

E(3) Particle Systems Introduced by Reeves and Blau in 1983 used in the Star Trek movie for explosion of a planet Particle Systems Particle systems principle Init

18 Open L systems Example Open L systems P: A>?E(x) x<2 > F(x,y+1)A A>?E(x) x>=2 > e w: F(0,0)A?E(0) F(0,0)A?E(0) > F(0,0)F(0,1)A?E(1) => F(0,0)F(0,1)F(0,2)A?E(2) => F(0,0)F(0,1)F(0,2)?E(3) Particle Systems Introduced by Reeves and Blau in 1983 used in the Star Trek movie for explosion of a planet Particle Systems Particle systems principle Init phase: (a particle has initial speed, color, etc.) The initial values are generated randomly, by user, etc. Loop phase: 1) generate new particles 2) assign velocity, color, lifetime etc. 3) eliminate every particle which should die 4) compute position and color of existing particles 5) display the system 18

Particle Systems Particles can interact with environment Particle Systems Particles

ways, the way they move can depend on many things the way they are generated can

It is extremely versatile modeling tool Newtonian particle systems Applying

19 Particle Systems Particles can interact with environment Particle Systems Particles can interact with each other Particle Systems Particles can be displayed in many ways, the way they move can depend on many things the way they are generated can vary etc. It is extremely versatile modeling tool Newtonian particle systems Applying Newtonian physics to particles Particle is represented by its position x(, velocity v(, and mass m x( y( v( We want to know the change in time i.e., d dt y( d dt x( v( F( v( m 19

20 Newtonian particle systems We must solve ordinary differential equation. A simple way is to use the Euler's method x( t x( tv( F( v( t v( t m Newtonian particle systems Typical forces applied to particles: Gravity F = m g m particle mass, g constant 9.82 Viscosity F = v c v particle velocity, c constant Wind F = F wind F wind wind vector can vary in time F wind (v) Newtonian particle systems Newtonian particle systems Collision detection Particle moves with velocity v to the surface with normal vector n v n New velocity vector: divide velocity to the tangent and normal component v = v t + v n v n =(v*n) n v t = v v n then we change direction of the normal component, v so the new velocity vector is v = v t v n n surface surface 20

21 Newtonian particle systems When is a collision detected? When particle crosses the surface We need a point to surface detection algorithm also condition n * v < 0 must be satisfied (i.e., particle travels to the surface) Newtonian particle systems Putting it all together: Particle: x[3] (position), v[3] (velocity), m (mass), f[3](force accumulator) 1. Initialize particles (set boundary condition for the equation) 2. In every frame do: a. Detect collisions and recalculate the velocity vector b. Clear force accumulators for every particle c. Accumulate the forces d. Solve differential equation (from the forces, velocity, mass, and position evaluate new velocity and position) e. Display particles Summary When and why procedural methods are used? Code instead of data Dimensions and fractals mathematical monsters random linear fractals Bm and fbm the midpoint displacement random faults DLA grammars particles 21

CGT 581 G Procedural Methods Fractals

CGT 581 G Procedural Methods Fractals Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics Technology Procedural Techniques Model is generated by a piece of code. Model is not represented