2009 VIZ-09 Edited Paper

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2D Geometric Constraint Solving : an Overview

Samy Ait-Aoudia, Mehdi Bahriz, Lyes Salhi

ESI - Ecole nationale Suprieure en Informatique, BP 68M - Oued Smar 16270Algiers, Algeria

[email protected]

Abstract - Geometric constraint solving has applications in

many different fields, such as Computer-Aided Design,

molecular modelling, tolerance analysis, and geometric theorem

proving. Geometric modelling by constraints enables users to

describe shapes by relationships called constraints between

geometric elements. The aim is to derive automatically these

geometric elements and provide thus effort and time saving.

Moreover, users can easily modify existing designs. Many

resolution methods have been proposed for solving systems of

geometric constraints. We classify these methods in three broad

categories : algebraic, rule-oriented and graph-constructive

solvers.

Keywords - Constraints solving, geometric constraints,

algebraic, rule-oriented and graph-constructive solvers

I. INTRODUCTION

Geometric constraint solving has applications in manydifferent fields, such as Computer-Aided Design,

molecular modelling, tolerance analysis, and geometrictheorem proving. Geometric modelling by constraintsenables users to describe shapes by specifying a rough

sketch and adding to it geometric constraints i.e. a set ofrequired relations between geometric elements (see [6],

[27], [28]). The constraint solver must derive

automatically the correct shape needed. Typically, in 2D,geometric modelling by constraints specifies geometricalobjects such as points, lines, circles, conics by a set of

constraints : distances between points, points and lines,parallel lines, angles between lines, incidence relations

between points and lines, points and circles, tangency

relations between lines and circles or between circles, andso on.

Intuitively a dimensioned sketch is considered to be

well-constrained if it has a finite number of solutions for

non-degenerate configurations. Figure 1 shows a 2Dwell-constrained problem with three distance constraints

between pairs of points and two angle constrains.Similarly a dimensioned sketch is considered to be under-

constrained if it has an infinite number of solutions for

non-degenerate configurations. Finally a dimensionedsketch is considered to be over-constrained if it has no

solutions for non-degenerate configurations. Examples of

2D under-constrained and over-constrained sketches aregiven in figures 2.a and 2.b respectively.

B

D

d2

d3

C

d1

Figure 1. A simple constrained design.

(a) (b)

Figure 2. (a) an under-constrained sketch, (b) an over-constrained

sketch

Many resolution methods have been proposed forsolving systems of geometric constraints. They can be

classified in many ways. We classify these methods in

three broad categories : algebraic, rule-constructive andgraph-constructive solvers.

This paper is organized as follows. The algebraic

approach for constraints solving is explained in section II.We present in section III, the rule-oriented methods to

handle constraint problems. The graph-based

methodology is given in section IV. Section V givesconclusions.

II.ALGEBRAIC APPROACH

A.Numerical methodsIn a numerical constraint solver, the geometric

constraints are first translated into a system of algebraicequations (linear or not). This system is then solved by

applying a numerical method. For each constraint

correspond an algebraic equation. As an example of thiscorrespondence, the equation corresponding to "internal"

2009 Second International Conference in Visualisation

978-0-7695-3734-4/09 $25.00 2009 IEEE

DOI 10.1109/VIZ.2009.29

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978-0-7695-3734-4/09 $25.00 2009 IEEE

DOI 10.1109/VIZ.2009.29

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978-0-7695-3734-4/09 $25.00 2009 IEEE

DOI 10.1109/VIZ.2009.29

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and "external" tangency between two circles (see figure

3), one centred atA with radius r1, the second centred at

B with radius r2 is the following :

(xB - xA)2 + (yB - yA)

2 - (r1 r2)2= 0.

r2r1

A B

C1

C2

B

r2

C1

C2

r1

A

Figure 3. Tangency between two circles.

Example :

A constrained scheme is shown in figure 4. It consists

of three points and three distance constraints. The point

P1 lies at the origin and P3 on the positive x-axis. The

corresponding set of equations is given below :

eq1 : x1 = 0.eq

2: y

1= 0.

eq3

: y3

= 0.

eq4

: y2

y1

d1

= 0.

eq5

: (x3 x

2)

2

+ (y3

y2)

2

d3

2

= 0.

eq6

: x3

x1

d2

= 0.

Sutherland [55], Hillyard and Braid [24], Borning [7]

Borning and Duisberg [8] and Gosling [22] have used therelaxation method to solve the constrained problem.

Relaxation is a quite slow method and sometimes

convergence can't be achieved. Barford [5] used a

projective method.

Many systems have used the Newton-Raphsonsiteration to solve the set of geometric constraints. This

"popular" numerical method was used, among many

others, by Lin et al. [41], Light et al. [40], Lee et al. [37],

Nelson [45], Prusinkiewicz et al. [49], Rocheleau et al.

[50], Gossard et al. [23], Serrano [51], Perez et al [47],

Anderl et al. [4]. This method needs an initial guess,

typically given by the sketch of the desired geometric

scheme.

(x2,y2)

(x3,y3)(x1,y1)

d1

d2

d3

x

y

P1 P3

P2

Figure 4. A constraint problem.

However, there is a well-known problem. If Newton-

Raphsons method often works well, sometimes it does

not converge or it converges to an unwanted solution. Inthis last case, the user changes his initial guess until

Newton-Raphsons method works if it does.

An alternative method to Newton-Raphson for

geometric constraint solving is homotopy or continuation

[3]. Lamure et al. [34] have tested several configurationsusing homotopy with success and where Newton-Raphson's method fails. Nonetheless, homotopy, is

slower than Newton's method. Homotopy was also used

for solving 3D constraints by Durand et al. [14]. Foufou

et al. [16] recently used efficient numerical methods to

solve systems of constraints.

B.Symbolic methodsAs in the numerical solvers, the constraints are again

formulated as a system of algebraic equations. However,

instead of applying numerical techniques to determine a

solution, general symbolic computations are undertaken

to find the solution to the system of equations. Methods

such as Grbner basis or Wu-Ritt [32] techniques can be

applied to find symbolic expressions for the solutions.

Many systems have used a symbolic resolution of the

system of equations. We can non exhaustively mention

works done by Ericson et al. [15], Kondo [31], Buchanan

et al. [12], Chou et al. [13] and Gao et al [18].

III. RULE-CONSTRUCTIVE APPROACH

A.Problem formulationRule-based solvers rely on the predicates

formulation. The constraints are expressed as facts and aninference engine is used to derive the solution byexhaustively applying rules.

Some examples of expressing constraints as facts aregiven below :

coord(P1, [X, Y])coordinates of point P1 are X and Y.

dist(P1, P2, d)d is the distance between points P1 et P2.

slope(P1, P2, a)a is the slope of the line P1P2.

angle(P1, P2, P2, b)b is the angle P1P2P3.

slope(P1, P2) = slope(P3, P4)line P1P2 is parallel to line P3P4.

A typical example of rule used by the inferenceengine is given hereafter :

[coord(P1, [X1, Y1]), coord (P2, [X2, Y2]), dist(P1, P3, R1),dist(P2, P3, R2)] coord(P3, intersection(circle(c(P1), R1), circle(c(P2),

R2)))]

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It states that if points P1and P2are known and thedistance values R1 and R2between points (P1,P3) and(P2,P3) respectively are given then we can compute thecoordinates of the point P3 by intersecting two circles,one centered atP1with radius d13 the other centered atP2with radius d23.

B.Rule-Oriented solversWe will review now some works done in the rule-

oriented field. The majority of rule-constructive solvershave been implemented at the end of the eighties andearly nineties.

Brderlin ([10], [11]), has developed a system takinginto account 2D and 3D constraints. This system wasimplemented using Prolog. The class of 2Dconfigurations solved was not very large and the 3D casesolves very simple schemes.

Sunde in [53] uses a rule-constructive method. Heintroduces the notion of CA (Constraint Angle) and CD(Constraint Distance) sets. The method tries to merge allthe initial CA and CD sets in a single CD set. If themerging operation succeeds the figure solution isconstructed.

Aldefeld in [2] uses a forward chaining inferencemechanism, where the notion of direction of lines isimposed by introducing additional rules, and thusrestricting the solution space.

A similar method was presented by Suzuki et al. [54],where handling of over-constrained and under-constrained problems is given special consideration.

Verroust et al [56] have used the same theoreticalfoundations as Sunde [53]. They added some rules toextent the class of problems solved.

Rule-constructive approach provide a qualitativestudy of geometric constraints. Although it is a goodapproach for prototyping and experimentation, theextensive computations involved in the exhaustivesearching and matching make it inappropriate for real

world applications.IV. GRAPH-BASED APPROACH

A.Graph representationIn graph-based methods, an undirected graph

G=(V,E) where |V|=n and |E|=m represents the

constraint problem. The geometric elements are

represented by the graph nodes and the constraints are the

graph edges. The class of configurations solved by these

methods is typically the ruler and compass constructive

problems. In Computer-Aided Design, the graph-based

approach has become dominant.

Example:

A dimensioning scheme defining a constraint problemis shown in figure 5. It involves six points and six lines.

The constraints are six point-point distances, three line-

line angles, twelve point-line implicit distances that are

zero. The corresponding constraint graph is shown in

figure 6. The edges are labeled with the values of the

distance and angle dimensions. The unlabeled edges

correspond to the implicit point-line distances.

A

B

C

D

E

F

d1 d2

d3

d4d5

d6

Figure 5. A constraint problem.

A

B

C

D

E

F

AB BC

CD

DEEF

AF

d1 d2

d3

d4d5

d6

Figure 6. The constraint graph.

B.Graph structure analysisGraph-constructive solvers are stemming from graph

theory. They are based on an analysis of the structure of

the constraint graph. The graph constructive approachprovides means for developing sound and efficient

algorithms. Let us now give some definitions concerning

the structural properties of the constraint graph when the

degree of freedom of each geometric object is two and

each constraint 'fixes' one degree of freedom.

Definition 1

A constraint graph G=(V,E) where |V|=n and |E|=m

is structurally well-constrained if and only ifm=2*n-3

and m2*n-3 for any induced sub-graph G=(V,E)where |V|=nand |E|=m(see [33]).

Definition 2

A constraint graph G=(V,E) contains a structurallyover-constrained part if there is an induced sub-graph

G=(V,E) having more than 2*n-3 edges.

Definition 3

A constraint graph G=(V,E) is structurally under-

constrained if it is not over-constrained and the

number of edges is less than 2*n-3.

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We must mention that the Laman's theorem [33] is

proved only for point to point distance constraints and

that the extension of his definitions may imply incorrect

cases. A typical example of such cases is given by a

triangle constrained with three angle constrains. In figure

7.left, a triangle is constrained with three angle

constraints (with ++=180). The triangle is

geometrically under-constrained but the constraint graphrelated to (figure 7.right) is well constrained.

Figure 7. An under-constrained problem (left), and its associated

constraint graph (right).

The graph constructive approach uses only thestructure of the constraint graph and forgets numerical

information. A constraint graph can be structurally well-

constrained but numerically under-constrained. The

dimensioned scheme shown in figure 8 (left) is

numerically under-constrained (the edges are labeled with

distance arguments). Its corresponding constraint graph

shown in figure 8 (right) is structurally well-constrained.

Nonetheless, with the graph constructive approach, such

cases will be detected during the construction phase.

A B

C

1

0

D

1

1

1

D

A B

C

Figure 8. A numerical under-constrained problem (left), and its

associated constraint graph (right).

C.Resolution principleGraph-based algorithms for solving geometric

constraint problems have two phases : an analysis phase

and a construction phase. These algorithms are also called

decomposition recombination methods.

During the first phase the graph of constraints is

analyzed and decomposed in small sub-problems.

Sequences of construction steps are derived. During the

second phase the recombination is carried out and theconstruction steps are processed to place the geometric

elements. Figure 9 shows an example of decomposition

of a constrained scheme.

The placement steps correspond to standardized

geometric construction steps (place a point at given

distances from two points, place a line at prescribed angle

from another line through a point, rotate and translate a

sub-structure, ).

Figure 9. Constrained Design decomposition.

Figure 10 shows an example of a placement step. The

point p3 is to be constructed from two known points p1

andp2. dij is the value of the constraint distance between

points pi an pj. To obtain point p3, we intersect two

circles, one centered at p1 with radius d13 the other

centered at p2 with radius d23. If the point to pointdistance constraint between p1 and p2 is the first

constraint "pinned", we assume that p1 is placed at the

origin and p2on the positive x-axis at distance d12 from

p1.

These approaches are fast and more methodical. In

this methods the constraints are satisfied in a constructive

manner, which makes the constraint solving process

natural for the user and suitable for interactive debugging.

Figure 10. Placing a point relative to two known points.

D.Graph based solversIn the early nineties, Owen [46] was the first to give

an efficient and sound graph-based algorithm. This

algorithm is quadratic and performs well on ruler and

compass constructive problems. Following that, based on

Owen's work, was developed a commercial systemnamed D-Cubed. Since that and until recently, a lot of

researches were conducted in this promising way. Among

these works we can mention those done by Hoffmann's

"constraint team" ([9], [17], [42], [43], [44], [25], [26],

[19]), Lamure et al. [35], Lee et al. [38], Ait-Aoudia et al.

[1], Jermann et al. [29], Gao et al [20], Lee et al. [39],

Sitharam et al [52], Gao et al. [21], Zhang et al. [57],

Joan-Arinyo et al. [30], Podgorelec et al. [48].

xp1 p2

p3

p3

d12

d13 d23

y

A

B C

AB

AC BC

AB

C

1

2

3

6

5

7

4

9

8

10

11

12

13

14

16

15

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V.CONCLUSION

We have classified the geometric constraint solvers inthree broad categories. we summarise hereafter someadvantages and drawbacks for each approach.

Numerical methods are O(n3) or worse. Thesemethods suffer from the lack of "geometric explanation"during the resolution process. Also, most numerical

methods have difficulties for handling over-constrainedor under-constrained schemes. The advantage of thesemethods is that they have the potential to solve largenonlinear system that may not be solvable using any ofthe other methods. Almost all existing solvers switch toalgebraic methods when the given configuration is notsolvable by the native method. Algebraic methods handleeasily the 3D case. Speeding up the resolution must beconsidered further.

Symbolic methods are typically exponential in time

and space. They can be used only for small systems.

According to Lazard [36], computing the Grbner bases

of an irreducible system of degree two in ten unknowns is

a hopeless case.

Rule-based solvers rely on the predicates formulation.

Although they provide a qualitative study of geometricconstraints, the "huge" amount of computations needed

(exhaustive searching and matching) make them

inappropriate for real world applications.Graph-based methods can be very efficient. In

Computer-Aided Design, the graph-based approach hasbecome dominant. However, these methods are onlyapplicable to particular kinds of problems, typically rulerand compass constructive problems. The graph-basedapproach to handle efficiently the 3D case deservesfurther studies. The theoretical framework is no longerapplicable. The extension of Laman theory (sectionIV.B)to the 3D case will give: a constraint graph G=(V,E)where |V|=n and |E|=m is structurally well-constrained ifand only if m=3*n-6and m3*n-6 for any induced

sub-graph G=(V,E) where |V|=n and |E|=m. The3D constrained design shown in figure 11 involves eight

points and eighteen point to point distances. This wellknown example verifies the formula given above but it isnot structurally well constrained. The two sub-partsdefined by the points {A,B,C,D,E} and {A,G,H,F,E}canrotate around theAEaxis.

Figure 11. A 3D constrained design.

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2009 VIZ-09 Edited Paper