A case study in geometric constructions

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A case study in geometric constructions

• Pascal Schreck
• Étienne Schramm

LSIIT - Strasbourg French CNRS UPRES-A
7005 Strasbourg University (France)
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geometric constraint solving in CAD
sketch
construction program fix point A to (0,0) fix
straight line D to Ox B point on D at 8 cm from
A D  straight line passing through A
doing an angle of 100 with D C point on D  at
14 cm from A let D2 ? bisectors of D and D  let
D3 ? bisectors of D and (AC) O intersection of
D2 with D3 H orthogonal projection of O on D G
circle with centre O passing
through H ...
dimensions
gt figure(s) which respects the dimensions
or a way to construct it
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A simple framework
polygons
constraints on the edges
- length
- angle
it's a toy (but not trivial) example of geometric
constructions in CAD
Note the method presented is NOT the ultimate
perfect and absolute geometric solver!
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Plan
Some definitions
Classical approaches
Our simple geometric universe (polygons)
Algorithm, implementation
Conclusion
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Some definitions

• system of constraints

• solution of S

- v(C) is nearly true (v(X) is at e from a true
solution)

• S is well-constrained

the set of solutions is finite and not empty

• S is structurally well-constrained

combinatorial notion which meets the previous one
under some conditions
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Some definitions

• (geometric) solver

A piece of software able to compute one or all
the solutions of a (geometric) constraint system

• correctness (of a solver) or e-correctness

The objects yield by the solver are really (or
e-) solutions.

• completeness

All the solutions are found.

• robustness

Whatever the numeric values of the parameters of
a parametric system, all the solutions are found.
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Classical works

• Numerical iterative approaches
• Newton-Raphson method (Nelson, Light et
  al.)
• Continuation method (Lamure Michelucci)
• Relaxation (Sutherland, Borning)
• Interval calculus (B. Neveu et al.)

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Classical works

• Combinatorial approaches
• Sunde (88) Verroust et al. (CD and CA-sets)
• Owen (92) (graph decomposition)
• Hoffmann et al. (93, 2001) (graph
  aggregation or decomposition)
• Ait et al. (93), D. Michelucci (perfect
  matching, maximum flow)

general if associated with one of the above
numerical methods
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Classical works
(symbolic, constructive )
Formal approaches

• algebraic

(formal calculus)
Ritt-Wu decomposition, Gröbner bases (automatic
proof)
Lebesgue's method (Chen et al. 92)

• geometric

(knowledge based systems)
Education Buthion (74), Schreck (92)
CAD Aldefeld, Brüderlin (88)
Mathis, Schreck and Dufourd (95, 97)
? yield exact solutions but slow
? generality ?
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Description of our geometric universe
Polygons
- Mathematical point of view
- Constructive point of view
description of a polygon by some lengths and
angles
- polygon constructive functions
- triangle functions
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Description of our geometric universe
Polygon constructive functions
Permutation function
n1
gn
Note this is NOT the parallelogram rule of
Verroust
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Description of our geometric universe
Triangle functions
t1 (l1, l2, l3) ?
- invariance by isometry
- invariance by direct isometry
(displacement or rigid body
motion)
- multi functions several results
- pseudo-functions l1-l2 ? l3 ? l1l2
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Description of our geometric universe
Simply constrained polygons
one constraint an ordered pair (n, l)
lengths
a set of ordered pairs L(n1, l1), , (nk, lk)
coherent (n, l) ? L and (m, k)? L ? k l
angles
one constraint a triple (n, m, a)
a set of triples i.e. a graph G
G is coherent if it has no circuit
constraint scheme S (L, G, m)
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Properties
In our case
a constraint scheme is well-constrained modulo
the displacements group if and only if it is
coherent and has 2n-3 constraints. We call it a
simply constrained polygon
at most n length constraints
at most n-1 angle constraints
we have three main cases
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Properties
In our case
a constraint scheme is well-constrained modulo
the displacements group if and only if it is
coherent and has 2n-3 constraints. We call it a
simply constrained polygon
at most n length constraints
at most n-1 angle constraints
we have three main cases considering the first
four edges
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Constructions
(example with the last case)
S
P''
g3(g2(P''))
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Constructions
algorithm
proc_1(S) simplify_1(S)
solve( )
build_1( )
coef_1(S),
computation of l' and "elimination" of an edge of
S
simplify_1(S)
application of f1,a,l to P (1, a, l coef)
build_1(coef, P)
computation of a "and l' "
coef_1(S)

proc_1(S) build_1(coef_1(S), solve(simplify_1(S)
))
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Correctness, complexity and implementation
Correctness the previous algorithm is correct
and complete wrt the simply
constrained polygons.
The previous algorithm has been efficiently
implemented using a stack in order to eliminate
the recursivity and a forest in order to
implement the angle graph.
The complexity is then linear.
The algorithm can be easily extended in order to
perform formal constructions.
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Conclusion
I presented a "toy" case study in geometric
construction
- it seems to us that it is a nice introduction
to the problem of geometric constructions in CAD.
But
- this framework introduce some of the basic
notions in this domain, and it points out some
fundamental difficulties.
- it leads to a correct, complete and efficient
solving algorithm for the simply constrained
polygons.
- it has never been the goal of this study, but
this framework could be extended using
"assembly" of polygons