Model Construction 김 성 남

Contents What is Modeling? Model Construction - Interactive modeling tools - Scanning tools - Computer vision - Procedural generation Sweeps Fractals Grammars Particle System Physically Based Modeling Summary

Modeling How do we … - Represent 3D objects in a computer? - Construct such represents quickly and/or automatically with a computer? - Manipulate 3D objects with a computer?

Model Construction Interactive modeling tools - CAD programs - Subdivision surface editors Scanning tools - CAT, MRI, laser, magnetic, robotic arm, … Computer vision - Stereo, motion, … Procedural generation - Sweeps, Fractals, Grammars

Interactive Modeling Tools User constructs objects with drawing programs - Menu commands, direct manipulation, … - CSG, parametric surfaces, quadrics, …

Scanning Tools Acquire geometry of objects with active sensors - CAT / MRI - Laser range scanner - Magnetic sensor - Robotic arm …

Computer Vision Infer 3D geometry from images - Stereo - Motion - Constraints

Procedural Modeling Goal Describe 3D models algorithmically Best for models result from … - Repeating processes - Self-similar processes - Random processes Advantages - Automatic generation - Concise representation - Parameterized classes of models

Procedural Modeling Sweeps Fractals Grammars

Sweep Useful for constructing 3D objects that possess - translational - rotational - other symmetries P(u)P(u,v) u v v u

Generating curve around helico-spiral axis - Model is parameterized Helico-spiral : z 0,λ z,r 0,N θ,Δ θ Generating curve : shape, N c, λ c definitions Θ i+1 = Θ i + ΔΘ z i+1 = z i λ z Sweep

Generate different shells by varying parameters Sweep

Goal For construction of natural objects with realistic description - mountain,cloud,tree … Euclidean equation 보다 빠르고, 쉽다. 특성 - Infinite detail at every point - Self-similar with infinite resolution Fractals

Useful for describing natural 3D phenomenon - Terrain - Plants - Clouds - Water - Feathers - Fur … Fractals

Deterministic self-similar fractals - Parts are scaled copies of original Statistical self-similar fractals - Parts have same statistical properties as original Fractals

General procedure - Initiator : start with a shape - Generator : replace subparts with scaled copy of original Deterministic Fractals initiator generator

Apply generator repeatedly Deterministic Fractals

ns D = 1 s : scaling factor n : number of subparts for subdivision of unit segment. D : fractal similarity dimension Deterministic Fractals D = ln(1/s) ln n

Deterministic Fractals 1 D E = 1 1/n s = 1/n, n = 2 ns 1 = 1 D E = 2 s = 1/n 1/2, n = 4 ns 2 = 1 A A’ = A/n D E = 3 s = 1/n 1/3, n = 8 ns 3 = 1

Fractal dimension of an object is always greater than the Euclidean dimension. Deterministic Fractals Euclidean dimension Curve = 1 Fractal dimension D > 1 D ≈ 1 → smooth curve D = 2 → Peaco curve : 2 dimension 2<D<3 → self-intersect

General procedure - Initiator : start with a shape - Generator : replace subparts with a self- similar random pattern Statistical Fractals

Geometric Construction Method Affine Fractal-Construction Method Random Midpoint-Displacement Method Self-Squaring Fractal Method - using to complex space - Mandelbrot set Self-Inverse Fractal Method Statistical Fractals

Geometric Construction Method - In general - choose a generator randomly at each step from a set of predefined shapes. - using the twisting,scaling function, … Statistical Fractals

Affine Fractal-Construction Method - using fractional Brownian motion - random direction and rendom length - Terrains(mountain,valley,ocean, … ) - D ≈ 2.15 → realistic mountain Statistical Fractals Brownian motion(random walk) in the xy plane x y

Random Midpoint-Displacement Method - Fractional Brownian motion → time-consuming Statistical Fractals ab y(a) y(b) y x ab y(a) y(b) y x y mid a+b/2 y mid = 1/2 [y(a) + y(b)] + r R : a value from a Gaussian distribution 0 <= r <= |b-a| 2H H = 2 – D, D > 1 is fractal dimension

Self-Squaring Fractal Method - transformation function to point in complex space - z = x + iy, i 2 = -1 Statistical Fractals

Self-Inverse Fractal Method - geometric inversion transformation Statistical Fractals p p’

Natural Objects : Terrain,cloud,plant, … Statistical Fractals

Shape grammars - sets of production rules applied to an initial object to add layers of detail - harmonious with the original shape L-grammars(graftals) - object : sum of parts - describing plants Grammars

Applying different rules at each step of the transformation from initial to final object Geometric substitution rules to altering Shape Grammars Rule 1 Rule 2 Rule 3

Using for creating plants - object : sum of parts Tree(a trunk) → Branches → Leaves L-Grammars(graftals) Tree = T[T[TL]T[TL][TL]T[T[TL]T[TL][TL][TL]

Display Flude-like property - change over time by flowing,billowing,spattering,expanding, … - clouds,smoke,fire,fireworks,waterfall, water spray,clumps of grass, … Particle shape can be - spheres,ellipsoids,boxes, … Particel movement may be controlled by specified forces such as a gravity field Particle System

examples a clump of grassWaterfall hitting a stone An object disintegrating into a cloud of particles

Nonrigid object - rope,cloth,rubber ball, … Interaction of external and internal forces Modeling method - network of point nodes with flexible connections between the nodes Spring network Physically Based Modeling

Examples

Summary Interactive Modeling Tools - CAD programs - Subdivision surface editors … Scanning Tools - CAT,MRI,Laser,Magnetic,Robotic arm, … Computer Vision - Stereo,motion, … Procedural generation - Sweeps - Fractals - Grammars - Particle Systems - Physically Based Modeling