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Come and hear
Introduction to Implicit Modelling
by Brian Wyvill
With a little helpfrom his students
University of Victoria Island Graphics Lab. Project CSC 305 page 2
Overview
••Introduction to Implicit SurfacesIntroduction to Implicit Surfaces
••Blending, Warping, CSGBlending, Warping, CSG
••Some ProblemsSome Problems
••The The BlobTreeBlobTree
••BlendingBlending
••TexturingTexturing
••AnimationAnimation
••Hierarchical Implicit SurfacesHierarchical Implicit Surfaces
••Building ModelsBuilding Models
University of Victoria Island Graphics Lab. Project CSC 305 page 3
Introduction to Implicit Surfaces
Parametric Definition
x = r sin(α)y = r cos(α)0 c α c 2π
Implicit Definition
f(x,y) = x2 + y2 - r2 = 0e.g. r = 1f(0,0) = 0 + 0 - 1 < 0 insidef(0,0) = 1 + 1 - 1 > 0 outside
implies search space to find x,y to satisfy: f(x,y) = 0
iso-surface: f(x,y) - c = 0
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The Geoff Function
Field Function
Proximity Blending:Add contributions fromgenerating skeletal elementsin the neighbourhood
Click Me
University of Victoria Island Graphics Lab. Project CSC 305 page 5
Blending and The Soft Train
PolygonizerInfo.
1986
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B
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Polygonization Algorithm
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Warping
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Barr Operators
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Constructive Solid Geometry (CSG)
Primitives are combined using boolean set operations: Union, Intersection, Difference. Each primitive represents a half space, ie the set of points defining the half space
E.g.
Boolean expression (u= union, d= difference, i= intersection)d( sphere, cylinder) u( sphere, i( i( cylinder, plane1), plane2) )
-- +
Sphere
-- +
Cylinder-
+
Plane
The cylinder is infinite in extentit is first intersected with twohalf space planes.
University of Victoria Island Graphics Lab. Project CSC 305 page 26
CSG Tree
CSG Implementation
Boolean Expressions are usually represented as a binary tree.
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CSG Intersections with Voxels
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CSG Intersection Value
Boolean Operations
Union and intersection of primitives, A and B may berespectively defined as a composition of the field values, FA , FB
FA » FB = max(FA, FB)
FA … FB = min(FA, FB)
Difference use -min (FA, FB)(- in this case inverts inside and outside )
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CSG - Min and Max
fB(p2)=Max(fA(p2), fB(p2))fA(p1) =Max(fA(p1), fB(p1))
fA|B(p0)=Max(fA(p0), fB(p0))Depending on position of p0
Field = 0
Object A
Object B
P1 P2P0
Union
University of Victoria Island Graphics Lab. Project CSC 305 page 30
CSG - Min
fA(p2)=Min(fA(p2), fB(p2))=0fB(p1) =Min(fA(p1), fB(p1))=0
fA|B(p0)=Min(fA(p0), fB(p0))Depending on position of p0
Field = 0
Object A
Object B
P1 P2P0
Intersection
University of Victoria Island Graphics Lab. Project CSC 305 page 31
CSG - Min
Min(fA(p1), 1 - fB(p1))=0
Min(fA(p1), 1 - fB(p1))=fA(p1)
Min(fA(p0), fB(p0)) = 1 - fA|B(p0)Depending on position of p0
Difference
Field = 0
Object A
Object B
P1 P2P0
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0 = f 1(x) = f 1 (P12 + n12) = f1(P12) + (n12 , f1( P12 ))0 = f2(x) = f2 (P12 + n12) = f2(P12) + (n12, f2( P12 ))
X is the true intersection point for C1 and C2
Segment P1 P2 is far from x.
Estimate for x s. t. f 1 (x) = f2 (x) = 0
We can apply a first order Taylor expansion to the difference : n12 = x - P12
Polygonization Problems
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λ1 =
so
n1 =
and similarly
n2 =
− f1
( f1, f1)
Iterating to the Surface
− f1 f1
( f1, f1)
− f2 f2
( f2, f2)
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Adaptive Polygonisation
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CSoftG Wheels
Csoft Wheel before and after removal of artifacts
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Canmore Coffee Grinder
Ray TracedCanmoreCoffeeGrinder
by
Kees van OverveldandBrian Wyvill
University of Victoria Island Graphics Lab. Project CSC 305 page 38
Parametric Bounding Curve
Cylinders intersected with bounding plane and parametric curve
Building the Piano
University of Victoria Island Graphics Lab. Project CSC 305 page 39
Model of 9ft. Steinway Concert Grand
Plant by Dr. Prusinkiwicz
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American Type 4-4-0
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The BlobTree
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A BlobTree Example
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More BlobTree Nodes
Note the inclusionof thetexture node.
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N - indicates a node in the BlobTreeL (N ) - left child R (N ) - right childfunction F returns the field value for the node N at the point M
function F(N , M)1. Primitive: F( M)2. Warp: F( L (N ) , w( M)) (warp is a unary operator)3. Blend: F( L (N ) , M)+ F( R (N ) , M))4. Union: max( F( L (N ) , M), F( R (N ) , M))5. Intersection: min( F( L (N ) , M), F( R (N ) , M))6. Difference: min( F( L (N ) , M), -F( R (N ) , M))
end
Traversing The BlobTree
