1. Introduction to Implicit Modelling
Come and hear
Introduction to
Implicit Modelling
by Brian Wyvill
With a little help
from his students

2. Overview Introduction to Implicit Surfaces Blending, Warping, CSG
Some Problems
The BlobTree
Blending
Texturing
Animation
Hierarchical Implicit Surfaces
Building Models

3. Introduction to Implicit Surfaces
Implicit Definition
f(x,y) = x2 + y2 - r2 = 0
e.g. r = 1
f(0,0) = < 0 inside
f(0,0) = > 0 outside
implies search space to find
x,y to satisfy: f(x,y) = 0
iso-surface: f(x,y) - c = 0
Parametric Definition
x = r sin(a)
y = r cos(a)
0 c a c 2p

4. The Geoff Function Field Function Proximity Blending:
Click Me
Proximity Blending:
Add contributions from
generating skeletal elements
in the neighbourhood
Field Function

5. Blending and The Soft Train
1986

7. B

9. Polygonization Algorithm

11. Or a numeric technique can be used to find the gradient by sampling the field around the point p:,

19. Warping

24. Barr Operators

25. 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. - + + - - - + -
Sphere
Cylinder
Plane
Boolean expression (u= union, d= difference, i= intersection)
d( sphere, cylinder) u( sphere, i( i( cylinder, plane1), plane2) )
The cylinder is infinite in extent
it is first intersected with two
half space planes.

26. CSG Tree CSG Implementation
Boolean Expressions are usually represented as a binary tree.

27. CSG Intersections with Voxels

28. CSG Intersection Value
Boolean Operations
Union and intersection of primitives, A and B may be
respectively 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 )

29. CSG - Min and Max Union P0 P1 P2 Field = 0
fA|B(p0)=Max(fA(p0), fB(p0))
Depending on position of p0
Field = 0
Object A
Object B P0 P1 P2
fA(p1) =Max(fA(p1), fB(p1))
fB(p2)=Max(fA(p2), fB(p2))

30. CSG - Min Intersection P0 P1 P2 Field = 0 fA|B(p0)=Min(fA(p0), fB(p0))
Depending on position of p0
Field = 0
Object A
Object B P0 P1 P2
fB(p1) =Min(fA(p1), fB(p1))=0
fA(p2)=Min(fA(p2), fB(p2))=0

31. CSG - Min Difference P0 P1 P2 Field = 0
Object A
Object B P1 P2 P0
Min(fA(p0), fB(p0)) = 1 - fA|B(p0)
Depending on position of p0
Min(fA(p1), 1 - fB(p1))=0
Min(fA(p1), 1 - fB(p1))=fA(p1)

32. Polygonization Problems
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
CSG operations produce sharp contours between primitives. X is the true intersection point for C1 and C2 If DIFF is used by combine then P1 and P2 result. 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
0 = f 1(x) = f 1 (P12 + n12) = f1(P12) + (n12 , t f1( P12 ))
0 = f2(x) = f2 (P12 + n12) = f2(P12) + (n12, t f2( P12 ))

33. Iterating to the Surface
l1 = so
n1 =
and similarly
n2 =
- f1
( tf1, tf1)
- f1tf1
( tf1, tf1)
- f2tf2
( tf2, tf2)

34. Adaptive Polygonisation

35. CSoftG Wheels
Csoft Wheel before and after removal of artifacts

36. Canmore Coffee Grinder

37. Ray Traced Canmore Coffee Grinder by Kees van Overveld and
Brian Wyvill

38. Building the Piano Parametric Bounding Curve
Cylinders intersected with bounding plane and parametric curve

39. Model of 9ft. Steinway Concert Grand
Plant by Dr. Prusinkiwicz

40. American Type 4-4-0

41. The BlobTree

42. A BlobTree Example

43. More BlobTree Nodes
Note the inclusion
of the
texture node.

44. Traversing The BlobTree
N - indicates a node in the BlobTree
L (N ) - left child R (N ) - right child
function 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