Introduction to Boolean Operations on Free-form Solids

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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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### Introduction toBoolean 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
• Many artificial objects can be constructed out of simple parts - cylinders, rectangular blocks, spheres, etc.
• Very common in CAGD
Constructive Solid Geometry (CSG)
• Solids as expressions of Boolean operations of primitive solids
• Algorithms implemented directly on the representation
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.
• 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

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

S. Krishnan

And

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

BOOLE: Algorithm - stage 1

Computing bounding box overlap for each patch and Pruning

BOOLE: Algorithm - stage 2

Paring remained patches

BOOLE: Algorithm - stage 5

Partitioning the boundary into components

BOOLE: Algorithm - stage 6

Classifying components

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’

* Reference:

SHANKAR KRISHNAN and DINESH MANOCHA,

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

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

### 2001 SIGGRAPHApproximate Boolean Operationson Free-form Solids

Henning Biermann

Daniel Kristjansson

Denis Zorin

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
• Compute an approximate intersection curve, finding its images in each of the two parametric domains of the original surfaces.
Approximate Operations : Algorithm step 2
• Construct the connectivity of the control mesh for the result
Approximate Operations : Algorithm step 3
• Optimize the parameterization of the result over the original domains
Approximate Operations : Algorithm step 4
• Determine geometric positions for the control points of the result using hierarchical fitting
Approximate Operations
• Subtracting a cylinder from the mannequin head
* 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