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Introduction to Boolean Operations on Free-form Solids. CS284, Fall 2004 Seung Wook Kim. Boolean Operations. A natural way of constructing complex solid objects out of simpler primitives

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boolean operations
Boolean Operations
  • A natural way of constructing complex solid objects out of simpler primitives
  • Many artificial objects can be constructed out of simple parts - cylinders, rectangular blocks, spheres, etc.
  • Very common in CAGD
constructive solid geometry csg
Constructive Solid Geometry (CSG)
  • Solids as expressions of Boolean operations of primitive solids
  • Algorithms implemented directly on the representation
boundary representations b rep
Boundary Representations (B-Rep.)
  • Solids as a set of vertices, edges and faces with topological relations among them
  • Boolean operations implemented in the representation framework
boolean operations in b rep
Boolean Operations in B-Rep.
  • Polyhedral solids
    • Calculating plane-plane intersection only
    • Generating a single line
  • Free-form solids
    • Intersection between free-form surfaces
    • A high degree algebraic space curve
s krishnan and d manocha m gopi t culver j keyser

1999 International Journal of Computational Geometry & ApplicationsBOOLE: A Boundary Evaluation System for Boolean Combinations of Sculptured Solids

S. Krishnan


D. Manocha, M. Gopi, T. Culver, J.Keyser

boole algorithm stage 1
BOOLE: Algorithm - stage 1

Computing bounding box overlap for each patch and Pruning

boole algorithm stage 2
BOOLE: Algorithm - stage 2

Paring remained patches

boole algorithm stage 5
BOOLE: Algorithm - stage 5

Partitioning the boundary into components

boole algorithm stage 6
BOOLE: Algorithm - stage 6

Classifying components

boole surface intersection
BOOLE: Surface Intersection
  • Given the two parametric surfaces, eliminate two of the variables (using Dixon’s resultant)
  • Obtain the intersection curve in the plane as a matrix polynomial
  • Compute a starting point on each component of the intersection (using curve-surface intersection and loop detection algorithms)
  • Subdivide the domain of the surface into regions such that each sub-region has at most one curve
  • For each starting point, trace the intersection curve
boole surface intersection cont
BOOLE: Surface Intersection - cont’

* Reference:


An Efficient Surface Intersection Algorithm Based on Lower-Dimensional Formulation,

ACM Transactions on Graphics, Vol. 16, No. 1, January 1997, Pages 74–106.

2001 siggraph approximate boolean operations on free form solids

2001 SIGGRAPHApproximate Boolean Operationson Free-form Solids

Henning Biermann

Daniel Kristjansson

Denis Zorin

approximate boolean operations
Approximate Boolean Operations
  • Generating a control mesh for intersection of surfaces (approximating the result)
  • Optimizing the parameterization of the new surface with respect to the original surfaces
  • Minimizing the size and optimizing the quality of the new control mesh
approximate operations algorithm step 1
Approximate Operations: Algorithm step 1
  • Compute an approximate intersection curve, finding its images in each of the two parametric domains of the original surfaces.
approximate operations algorithm step 2
Approximate Operations : Algorithm step 2
  • Construct the connectivity of the control mesh for the result
approximate operations algorithm step 3
Approximate Operations : Algorithm step 3
  • Optimize the parameterization of the result over the original domains
approximate operations algorithm step 4
Approximate Operations : Algorithm step 4
  • Determine geometric positions for the control points of the result using hierarchical fitting
approximate operations
Approximate Operations
  • Subtracting a cylinder from the mannequin head
additional reference boolean operations on open set
* Additional reference: Boolean Operations on Open Set
  • Toshiaki Satoh, Boolean Oerations on Sets Using Surface Data, 1991 ACM 089791-427-9/91/0006/0119
  • Required to:
    • solve self-intersecting solid problems
    • generate offset solid objects