1
1
1
1
1
1
1
1
1
1
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1.7
1.5
√
2
√
2
Figure 1: (
left
) Wellconstrained System;
middle left:
underconstrained system;
middle right:
overconstrained
system; (
right
) consistently overconstrained system
1
Rigidity
Geometric constraint systems have been studied in the context of variational constraint solving in CAD for
nearly 2 decades.
[65, 40, 64, 50, 34, 36, 39, 48, 13, 60, 31, 5, 49, 61, 17, 62, 63] [6, 11, 28, 29, 20, 21, 1]
[23, 26, 27, 25] [30, 19, 25, 14, 15, 43, 3, 4] [44, 7, 42, 47, 2, 45, 46] [18, 37, 41, 52]. For recent reviews of the
extensive literature on geometric constraint solving more elaborate descriptions and examples for the definitions
below, see, e.g, [23, 33, 12, 53].
1.1
Definitions
A
geometric constraint system
consists of a finite set of primitive geometric objects such as points, lines, planes,
conics etc. and a finite set of geometric constraints between them such as distance, angle, incidence etc. The
constraints can usually be written as algebraic equations and inequalities whose variables are the coordinates
of the participating geometric objects. For example, a distance constraint of
d
between two points (
x
1
, y
1
) and
(
x
2
, y
2
) in 2D is written as (
x
2

x
1
)
2
+ (
y
2

y
1
)
2
=
d
2
.
In this case the distance
d
is the
parameter
associated
with the constraint. Most of the constraint solvers so far deal with 2D constraint systems. With the exception
of work [22, 24, 26, 27], [25], [18, 38, 37, 41, 51, 58, 56, 57], related to the FRONTIER geometric constraint
solver [52], to the best of our knowledge, work on standalone 3D geometric constraint solvers is relatively sparse
[7, 42].
A
solution or realization
of a geometric constraint system is the (set of) real zero(es) of the corresponding
algebraic system. In other words, the solution is a class of valid instantiations of (the position, orientation and
any other parameters of) the geometric elements such that all constraints are satisfied. Here, it is understood
that such a solution is in a particular geometry, for example the Euclidean plane, the sphere, or Euclidean 3 di
mensional space. A constraint system can be classified as
overconstrained
,
wellconstrained
, or
underconstrained
.
Wellconstrained systems have a finite, albeit potentially very large number of
rigid
solutions; i.e., solutions that
cannot be infinitesimally flexed to give another nearby solution: the solution space (modulo rigid body transfor
mations such as rotations and translations) consists of isolated points  it is zerodimensional. Underconstrained
systems have infinitely many solutions; their solution space is not zerodimensional. Overconstrained systems
do not have a solution unless they are
consistently overconstrained
. In that case, they could be embedded within
overall underconstrained systems, see Figure 1. Systems that are not underconstrained are called
rigid
systems.
A geometric constraint graph
G
= (
V, E, w
) corresponding to geometric constraint system is a weighted
graph with vertex set (representing geometric objects)
V
and edge set (representing constraints)
E
;
w
(
v
) is the
weight of vertex
v
and
w
(
e
) is the weight of edge
e
, corresponding to the number of degrees of freedom available
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 Fall '08
 Staff
 Geometry, Computeraided design, Constraint satisfaction, Algebraic geometry, Constraint satisfaction problem, Constraint programming

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