Robustness in CAD geometric construction

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Robustness in CAD geometric construction

• Pascal Schreck
• LSIIT - Strasbourg
• French CNRS UPRES-A 7005
• Strasbourg University
• (France)

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Geometric construction in CADExample
Declarative specification of the figure
Sketch
Construction program fix point B fix straight
line D define A on D and at 8 cm of B define
D  passing through B doing an angle
100 with D define C on D  at 14 cm of
B define D2 ? bisectors of D and D  define D3 ?
bisectors of D and (AC) define O intersection
of D2 and D3 define H orth projection of O onto
D define G circle with centre O passing
through H ...
Dimensions
Graphical figure (e solution)
gt figure(s) which respects the dimensions
or a way to construct it
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Plan
1. Example 2. Prior approaches 3. Correctness,
completeness, robustness 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
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Prior approaches (1)
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion

• Numerical iterative approaches
• Newton-Raphson (Nelson, Light et al.)
• Continuation method (Lamure Michelucci)
• Relaxation (Sutherland, Borning)
• Optimisation (Ge et al.)

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Prior approaches (2)
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion

• Combinatorial approaches
• Sunde (88) Verroust et al. (CD and CA-sets)
• Owen (92) (graph decomposition)
• Hoffman et al. (93) (graph agregation)
• Ait et al. (93) (perfect matching)

One solution
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Prior approaches (3)
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion

• Symbolic approaches
• Algebraic
• Ritt principle (Chou, Wu)
• Hörner Basis (Hendricson)
• Lebesgue method (Chen)
• Geometric
• without decomposition (Aldefeld,
  Brüderlin)
• with decomposition (Mathis and Schreck)

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Correctness, completeness(1)

• Correctness only solutions
• Completeness all the solutions
• Robustness all the solutions in all the cases

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Correctness, completeness(2)
Given two parallel lines D1 and D2, a point A on
D1, a point B on D2 and a point M, construct a
line d passing through M and cuting D1 at X and
D2 at Y such as AX BY k
-- Existence of different cases
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Correctness, completeness(3)
Introduce Z / AZ k AXBY (XZBY, XZYB
parallelogram)
M
If XZYB parallelogram then define O
midpoint of BZ define d (MO) -- (if M ?
O) Verify If XZBY parallelogram then
define dir direction of BZ define d
passing through M with direction dir Verify
A
Z
X
O
define O midpoint of BZ define d (MO)
B
Y
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Geometric universe
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
(1) Heterogeneous signature

• Sorts
• predicative symbols
• functional symbols

point, line, circle, length
is-onl point line is-onc point circle
length length point point
dist point point -gt length intercc circle
circle -gt point sline point point -gt line ccr
point length -gt circle
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Geometric universe
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
(2) Problems

• multi-functions

intersection(s) of 2 circles bisectors...
intercc circle circle -gt point bis line
line -gt line
construction of the midpoint of (A,B) ?
intercc1circle circle -gt pointintercc2circle
circle -gt point
mid(A,B)interll( sline(A,B),
sline(intercc1(ccp(A,B)),
intercc2(ccp(B,A))) )
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Geometric universe
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
(3) More problems

• pseudo-function

mid(A,B)interll( sline(A,B),
sline(intercc1(ccp(A,B)),
intercc2(ccp(B,A))) )
intersection of 2 circlesline passing trough 2
points...
? preconditional axiom
? A point, B point, L line (A ? B)? A is-on
L ? B is-on L ? L sline(A, B)
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Construction program(1) Basic considerations
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
Given a construction statement

• Constructiblity of x Is there (at least) one
  solution ?
• Construction of x What are the solutions ?

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Construction program(2) Refinements
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
All the solutions ?
In all the cases ?
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Construction program(3) From logic to program
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
? and ?
if then else
? for multi-function
list and iteration
? for multi-scenario
case of
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Construction program(4) Small example
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion
Intersection of two circles
Note ?(c1, c2) ? 0 lt r1-r2 ? dist(O1,O2) ?
r1r2
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Implementation (1)
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion

• partially implemented 2 prototypes

CAE Progé knowledge based system Written in
Prolog correctness, completeness
CAD Yams multi-agent system Written in
C resolution, efficiency
Not yet implemented
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Implementation (2)
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion

• taking preconditions into account

• using the guard associated with the basic
  constructors (atributed signature)
• examining the cases where preconditions are
  satisfied and the cases where they are not

• a geometric prover able to make hypothesis

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Conclusion
1. Example 2. Prior approaches 3. Correctness,
completeness ... 4. Geometric universe 5.
Construction program 6. Implementation 7.
Conclusion

• a logical correct framework
• correctness, completeness and robustness of
  constructions
• partially implemented into two geometric
  symbolic solvers
• such a study should be useful within other
  methods