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ContentsContents What is Modeling? Model Construction
- Interactive modeling tools- Scanning tools- Computer vision- Procedural generation • Sweeps • Fractals • Grammars • Particle System • Physically Based Modeling
Summary
ModelingModelingHow 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 ConstructionModel ConstructionInteractive 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 ToolsInteractive Modeling Tools User constructs objects with drawing
programs- Menu commands, direct manipulation, …- CSG, parametric surfaces, quadrics, …
Scanning ToolsScanning ToolsAcquire geometry of objects with
active sensors- CAT / MRI- Laser range scanner- Magnetic sensor- Robotic arm …
Procedural ModelingProcedural ModelingGoal
Describe 3D models algorithmicallyBest for models result from …
- Repeating processes- Self-similar processes- Random processes
Advantages- Automatic generation- Concise representation- Parameterized classes of models
SweepSweepUseful for constructing 3D objects that
possess - translational- rotational- other symmetries
P(u) P(u,v)
u
v
v
u
P(u,v)
Generating curve around helico-spiral axis- Model is parameterized Helico-spiral : • z0,λz,r0,Nθ,Δθ
Generating curve : shape, • Nc, λc
definitions Θi+1= Θi + ΔΘ zi+1 = ziλz
SweepSweep
GoalFor construction of natural objectswith realistic description- mountain,cloud,tree …
Euclidean equation 보다 빠르고 , 쉽다 .특성
- Infinite detail at every point- Self-similar with infinite resolution
FractalsFractals
Useful for describing natural 3D phenomenon- Terrain- Plants- Clouds- Water- Feathers- Fur …
FractalsFractals
Deterministic self-similar fractals- Parts are scaled copies of original
Statistical self-similar fractals- Parts have same statistical properties as original
FractalsFractals
General procedure- Initiator : start with a shape- Generator : replace subparts with scaled copy of original
Deterministic FractalsDeterministic Fractals
initiator generator
nsD = 1s : scaling factorn : number of subparts for subdivision of unit segment.D : fractal similarity dimension
Deterministic FractalsDeterministic Fractals
D =ln(1/s)ln n
Deterministic FractalsDeterministic Fractals
1DE = 1
1/n s = 1/n, n = 2ns1 = 1
DE = 2 s = 1/n1/2, n = 4ns2 = 1
A
A’ = A/n
DE = 3 s = 1/n1/3, n = 8ns3 = 1
Fractal dimension of an object is alwaysgreater than the Euclidean dimension.
Deterministic FractalsDeterministic Fractals
Euclidean dimensionCurve = 1
Fractal dimensionD > 1
D ≈ 1 → smooth curveD = 2 → Peaco curve : 2 dimension2<D<3 → self-intersect
General procedure- Initiator : start with a shape- Generator : replace subparts with a self- similar random pattern
Statistical FractalsStatistical Fractals
Geometric Construction MethodAffine Fractal-Construction MethodRandom Midpoint-Displacement MethodSelf-Squaring Fractal Method
- using to complex space- Mandelbrot set
Self-Inverse Fractal Method
Statistical FractalsStatistical 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 FractalsStatistical Fractals
Affine Fractal-Construction Method- using fractional Brownian motion- random direction and rendom length- Terrains(mountain,valley,ocean,…)- D ≈ 2.15 → realistic mountain
Statistical FractalsStatistical Fractals
Brownian motion(random walk) in the xy plane
x
y
Random Midpoint-Displacement Method- Fractional Brownian motion → time-consuming
Statistical FractalsStatistical Fractals
a b
y(a)
y(b)y
xa b
y(a)
y(b)y
xymid
a+b/2
ymid = 1/2 [y(a) + y(b)] + rR : 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, i2 = -1
Statistical FractalsStatistical Fractals
Self-Inverse Fractal Method- geometric inversion transformation
Statistical FractalsStatistical Fractals
p
p’
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
GrammarsGrammars
Applying different rules at each step ofthe transformation from initial to final object
Geometric substitution rules to altering
Shape GrammarsShape Grammars
Rule 1 Rule 2 Rule 3
Using for creating plants- object : sum of partsTree(a trunk) → Branches → Leaves
L-Grammars(graftals)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 byspecified forces such as a gravity field
Particle SystemParticle System
Particle SystemParticle Systemexamples
a clump of grass Waterfall hitting a stone An object disintegratinginto a cloud of particles
Nonrigid object- rope,cloth,rubber ball,…
Interaction of external and internal forcesModeling method
- network of point nodes with flexible connections between the nodes
Spring network
Physically Based ModelingPhysically Based Modeling