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Solid models and fractalsCOM3404
Richard Everson
School of Engineering, Computer Science and MathematicsUniversity of Exeter
http://www.secamlocal.ex.ac.uk/studyres/COM304
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Outline
1 Constructive solidgeometry
2 Metaballs3 Fractals
DimensionExamplesLandscape generation
4 L-systems
References
Fundamentals of 3D Computer Graphics. Watt. Chapters 1 & 2Computer Graphics: Principles and Practice. Foley et al (1995).Fractal Geometry of Nature. Mandelbrot. (1988).
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Constructive solid geometry
Models constructed from primitives using “volumetric Boolean” operations
Primitives Cuboids, spheres,cylinders, prisms, cones.
Operations union,intersection, subtraction
Special algorithms(Goldfeather, SCS) forrendering the CSG withoutdirectly computing the 3Dobject representation.
http://en.wikipedia.org/wiki/Constructive solid geometry
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Constructive solid geometry
Representation Stored in an attributed tree
Nodes Combining operationLeaves Primitives
Logic The logic underlying the model is preserved in the tree.
Alteration Dimensions of primitives are easily altered.
Limitations
Limited range of operations to construct modelsDetailed modifications to a local region may bedifficult because Boolean operations are global:e.g., blending at joins is difficult.Specific rendering algorithms or conversion topolygon form
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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
Examplef (x , y) = (x − x0)
2 + (y − y0)2
f (x , y) = c2 defines a circle ofradius c centred at (x0, y0)
x0 = 1, y0 = 0.5, contour f (x , y) = 82
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
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Metaballs
SourcesDefine potential function as asum of ‘sources’ with locationand strength.
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
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0.2
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Example
f (x , y) = exp{−[(x + 5)2 + (y + 5)2]/60
}+1.5 exp
{−[(x − 5)2 + (y − 5)2]/60
}Thick contour at f (r) = 0.8.
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Metaballs
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Metaballs
Negatively weighted sourcescan be used to sculptconcavities
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
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1
Example
f (x , y) = exp{−[(x + 5)2 + (y + 5)2]/60
}+1.5 exp
{−[(x − 5)2 + (y − 5)2]/60
}− exp
{−[x2 + (y − 4)2]/5
}Thick contour at f (r) = 0.8.
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Metaballs
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Fractals
Fractals
Objects that are self similar on a wide range of scales
Self similarWhen a component part isappropriately magnified itappears identical to the whole.
Koch curveSierpinski gasket
Approximately self similarMany naturally occurring objectsare statistically self-similar overa range of scales
Coast line of Great BritainCloudsAlveoli in lungs
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Generating fractals: Koch curve
Initiator The initial shape
Generator At each stage theinitiator is replace by a copy of thegenerator
Length at each stage is:
N = 1 L =(
43
)N = 2 L = (4/3)× 1
3 × 4 =(
43
)2
N = 3 L =(
43
)3
N L =(
43
)N
limN→∞ L(N) = ∞
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Dimension
Line Dimension of a line is 1because if it is divided into N partseach looks like the original scaled
down by N11
Plane Divide a square into N: each
looks like the original scaled by N12
Koch curve Each of the 4 pieces ofthe Koch curve looks like theoriginal scaled down by a factor 3.Dimension d
41/d = 3
d = log(4)/ log(3) = 1.26...
Fractals have non-integer dimension
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Examples
3D Sierpinski gaskets Julia set
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Fractal fern
Generated by repeated affine transformations.
Iterated function systems: Michael Barnsley “Fractals Everywhere”,Academic Press (1988).
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Recursive subdivision for meshes
Divide generator in two and displace vertically by a random amount
Divide triangular facet edges into two, forming 4 smaller facetsVertically displace the midpoints by random amountsRichard Everson 18 / 24
Fractal terrain
Naive methods tend to produce‘creases’; alleviated by usingoverlapping hexagonal grids.
In real landscapes the statisticalproperties of hills and valleysdiffer. Choose displacementsfrom an asymmetric distributionso that small displacements aremore probable.
Landscapes may have a manypolygons; problematic forrendering algorithms.
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Fractal terrain
Fractal terrain rendered with POV-Ray
Generated by fracplanet
See also terragen
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L-systems
Lindenmayer systems
Grammar to describe branching processes
Alphabet { A, B, [, ], (, ) }Production
A → AAB → A[B]AA(B)
1 A
2 AA
3 AAAA
1 B
2 A[B]AA(B)
3 AA[A[B]AA(B)]AAAA(A[B]AA(B))
Associate [. . .] with a left branch
Associate (. . .) with a right branch
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L-systems
B
B
A
A
A
B
B
B
B
B
A
A
A
A
A
A
A
A
AA
A
A
A[B]AA(B)
AA[A[B]AA(B)]AAAA(A[B]AA(B))
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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.
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