OUTSTANDING PROBLEMS
IN GEOMETRIC
CONSTRAINT SOLVING FOR
CAD
Meera Sitharam,
University of Florida
Partially supported by NSF grants
CCR 99-02025, EIA 00-96104
ORGANIZATION

CAD motivation and state of the art

Suite of Formal Problems

Our contribution-- FRONTIER

Unsolved Problems
CAD MOTIVATION 1/4
Variational constraint representation and feature hierarchy
CAD MOTIVATION 2/4
Another Assembly constraint representation and subassembly
hierarchy
CAD MOTIVATION 3/4



A geometric (variational) constraint representation with
feature hierarchy is:
Generated declaratively.
Easily updated and maintained.
Minimal, complete.
CAD MOTIVATION 4/4
The Catch: implicit representation. How to
 Want explicit geometric realization(s):
 Navigate conformation of each feature consistent with subfeatures.
 Derive implied geometric properties/invariants.
 Eliminate inconsistencies in requirements.
 Independently manipulate features and interface with other
representations.
STATE OF THE ART 1/3
 2 dimensions :Small, simple, no feature hierarchy, standalone.
 3 dimensions : 2d views; CSG; history of sweeps,
extrusions; parametric constraint solving
Hoffman et al (EREP), Bruderlin et al, Bronsvoort et al, Kramer et al,
Michelucci et al,Owen et al (D-cubed), Latham, Middleditch et al
STATE OF THE ART: 3 Dimensions 2/3
Pictures of 2d views of 3d part
STATE OF THE ART: 3D 3/3
D-cubed's pipe routing
GOAL
FORMAL BASIC PROBLEM 1/7
Input1: Primitive geometric objects:
(id, type) (type chosen from repertoire)
FORMAL BASIC PROBLEM 2/7
Input2: Geometric constraints:
(object1, object2, .., objectk, type)
(type chosen from repertoire)
constraint types include some inequalities
FORMAL BASIC PROBLEM 3/7
FORMAL BASIC PROBLEM 4/7
Input3 : Feature hierarchies:
(more than one) partial order or DAG of subsets of objects
 partial realization (output) information for the nodes of
DAG.
FORMAL BASIC PROBLEM 5/7
FORMAL BASIC PROBLEM 6/7
SUITE OF FORMAL PROBLEMS 1/12


Existence: of realization
Conformation: One conformation (if it exists) for each node in feature
hierarchy, represented as a rigid transformation applied to each child's
conformation.
FORMAL BASIC PROBLEM 7/7

For conformation, need to solve polynomial system over the
reals.
d
d2=((x2-x1)2 + (y2-y1)2
(x1,y1)
(x2,y2)
Problem classification
Red: Algebraic; Blue: Combinatorial; Purple: Mixture
SUITE OF FORMAL PROBLEMS 2/12


Generic, parameter-free version of existence
Approached combinatorially using only the geometric
constraint graph, object and constraint types.
SUITE OF FORMAL PROBLEMS 3/12

Generic answer holds

For all but a small set of forbidden parameter values that
satisfy discriminant/resultant (in)equalities.
SUITE OF FORMAL PROBLEMS 4/12

Generic Classification: some
information on how many
conformations exist?

finitely many (rigid or
wellconstrained)

infinitely many(flexible or
underconstrained)

none(inconsistently
overconstrained)
SUITE OF FORMAL PROBLEMS 5/12

Navigation: A well-defined set of conformations for
each node in feature hierarchy, represented as a set of
transformations applied to each child's set of
conformations?

Meaning of well-defined: complete in some formal sense,
systematically navigable.

Invariant: Does a given geometric property hold for all
conformations?
SUITE OF FORMAL PROBLEMS 6/12
SUITE OF FORMAL PROBLEMS 7/12
SUITE OF FORMAL PROBLEMS 8/12
SUITE OF FORMAL PROBLEMS 9/12
SUITE OF FORMAL PROBLEMS 10/12

Generic Overconstraint correction: a well-defined set
of removable constraint-sets for each node in feature
hierarchy.
SUITE OF FORMAL PROBLEMS 11/12

Generic underconstraint navigation: a well-defined set
of addable constraint-sets for each node in feature
hierarchy.
SUITE OF FORMAL PROBLEMS 12/12

Combinatorial complete generic solution: Big open question.
Gives rise to a combinatorial theory of rigidity. Whiteley et al.

Laman's theorem: complete combinatorial classification for 2D
points and distances. Simple dof analysis.
OUR CONTRIBUTIONS 1/12

(1) Formalizing decomposition problem and performance
measures.
OUR CONTRIBUTIONS 2/12
A Decomposition-Recombination plan (DR-plan) for an
input constraint system G, consistent with an input feature
hierarchy F is a DAG:

nodes are subsets of primitive objects of G such that their
induced subsystems are well-over-constrained 1

nodes include the nodes of F

each leaf/source is a primitive object in S;

each root/sink represents a maximal well-over-constrained
subsystem of G 1
1
more generally, they possess atmost a specified number of degrees of freedom
OUR CONTRIBUTIONS 3/12
OUR CONTRIBUTIONS 4/12
Other performance measures on DR-planners
An optimal DR-planner minimizes the maximum fan-in (size
of the largest subsystem in DR-plan)
OUR CONTRIBUTIONS 5/12

(2) Partial-generic characterization of DR-plan based on
degree of freedom analysis of constraint graphs:
minimal dense subgraph usually corresponds to wellover-constrained subsystem.

eAG



w(e) 
 w(v)   D
vAG
Algorithm for construction of DR-plan: using
network flows to iteratively find the minimal dense
subgraphs in current graph
graph transformations that repeatedly simplify them.
OUR CONTRIBUTIONS 6/12
OUR CONTRIBUTIONS 7/12
OUR CONTRIBUTIONS 8/12
Optimal DR Planning problem (Partial-generic version)
Already
finding smallest well-constrained graph is
NP-complete. Polynomial time algorithms known for
special cases. Approximation status unknown.
OUR CONTRIBUTIONS 9/12

(3) Towards a more complete generic solution
OUR CONTRIBUTIONS 10/12
OUR CONTRIBUTIONS 11/12

(4) Decomposition gives partial-generic solution to:

Existence
Classification
Overconstraint Correction
Generic underconstraint Navigation
Dealing with mixed representations, multiple input feature
hierarchies









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(5) Plus additional work on equation and conformation management
gives:
Well-constrained Conformations
Well-constrained Navigation
Easy updates of constraint repertoire
Easy updates of constraint representation, feature hierarchy and
realizations
Online constraint solving
OUR CONTRIBUTIONS 12/12

(6) Software architecture and implementation
REITERATING UNSOLVED PROBLEMS 1/3

Isolation of Conformation: Chirality, Semi-global
constraints, Symmetries, Forces.

Efficiently solving polynomial systems for rigid
transformations : physically based semi-numerical
algorithms are welcome.

Invariant problem.

Inverse problem of finding minimal constraint
representation
REITERATING UNSOLVED PROBLEMS 2/3

Underconstrained Conformation and Navigation: in
addition to addable constraint sets, need forbidden
parameter regions.
REITERATING UNSOLVED PROBLEMS 3/3

Complete generic solution to original problems-combinatorial geometry, geometric graphs.

Approximation algorithm for Optimal DR-plan problem,
even the partial-generic version based on dof analysis.

Complexity of existence problem
NP-hard; not known to be in NP; in DNPR (partial
algebraic version of NP); not known to be DNPR-hard.

Algebraic description of generic describe the semialgebraic set of forbidden parameter values when
generic solution does not hold