Transcript A New Foundation for Computational Geometry and Solid

A data Type for
Computational Geometry &
Solid Modelling
Abbas Edalat
André Lieutier
Imperial College
Dassault Systemes
Aim
To develop a data type for CG & SM
i. Mathematically sound, realistic
ii. Bridges theory and practice
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Why do we need a data type for solids?
Answer: To develop robust algorithms!
Lack of a proper data type and use of real
RAM in which comparison of real numbers
is decidable give unreliable programs in
practice!
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The Intersection of two lines
L1
L2
P
With floating point arithmetic, find the point
P of the intersection L1  L2. Then:
min_dist(P, L1) > 0,
min_dist(P, L2) > 0.
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The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
B
C
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The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
B
C
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The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
(ii) just AB, or
B
C
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The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
(ii) just AB, or
(iii) just BC, or
B
C
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The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
(ii) just AB, or
(iii) just BC, or
(iv) none of them.
B
C
The quest for robust algorithms is the most fundamental
unresolved problem in solid modelling and computational
geometry.
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A Fundamental Problem in Topology and Geometry
• Subset A  X topological space.
Membership predicate A : X  {tt, ff }
 tt
x
 ff
x A
x A
is continuous iff A is both open and closed.
• In particular, for A  Rn, A  , A  Rn
A : Rn  {tt, ff } is not continuous.
• Most engineering is done, however, in Rn.
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The Discontinuity of the Membership Predicate
• The basic building blocks of classical geometry are
not continuous and hence not computable in Rn.
• Example: The point
xis in the box.
True
x
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The Discontinuity of the Membership Predicate
• The basic building blocks of classical geometry are
not continuous and hence not computable in Rn.
• Example: The point
xis in the box.
False
x
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Non-computability of the Membership Predicate
• There is a discontinuity if x
goes through the boundary.
True
False
x
• This predicate is not
computable:
If x is on the boundary, we
cannot in general determine if
it is in or out at any finite
stage of computation.
x
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Non-computable Operations in Classical CG & SM
• A: Rn  {tt, ff} not continuous means it is not
computable, even for simple objects like A=[0,1]n.
• x  A is not decidable even for simple objects:
for A = [0,)  R, we just have the undecidability
of x  0.
• The Boolean operations such as  is not
continuous, hence noncomputable, wrt any natural
notion of topology on subsets. For example:
 : C(Rn)  C(Rn)  C(Rn), where C(Rn) is
compact subsets with the Hausdorff metric.
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Intersection of two 3D cubes
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Intersection of two 3D cubes
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Intersection of two 3D cubes
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This is Really Ironical!
• Topology and geometry have been developed to
study continuous functions and transformations
on spaces.
• The membership predicate and the binary
operations for  and  are the fundamental
building blocks of topology and geometry.
• Yet, these fundamental functions are not
continuous in classical topology and geometry.
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Elements of a Computable Topology/Geometry
• A : X  {tt, ff} fails to be continuous on A, the
boundary of A.
• For any open or closed set A, the predicate x  A is
non-observable, like x = 0.
open
open
tt
• Redefine: A : X {tt, ff}
tt

x   ff


x  IntA
ff
observable
observable
x  IntAc
x  A
with the Scott topology on {tt, ff}.
• A is now a continuous function.

Non-observable
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Elements of a Computable Topology/Geometry
• Note that A=B iff int A=int B & int Ac=int Bc,
i.e. sets with the same interior and exterior have the
same membership predicate.
• We now change our view: In analogy with classical set
theory where every set is completely determined by its
membership predicate, we define a (partial) solid
object to be given by any continuous map
f : X  { tt, ff }
• Then:
f –1{tt} is open; it’s called the interior of the object.
f –1{ff} is open; it’s called the exterior of the object.
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Partial Solid Objects
• We have now introduced partial solid objects, since
X \ (f –1{tt}  f –1{ff})
may have non-empty interior.
• We partially order the continuous functions:
f, g : X  {tt, ff } f ⊑ g  x  X . f(x) ⊑ g(x)
• f ⊑ g  f –1{tt}  g –1{tt} & f –1{ff}  g –1{ff}
Therefore, f ⊑ g means g has more information about an
idealized real solid object.
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The Solid Domain of X
• The solid domain S (X) of X is the partial order
(X  {tt, ff }, ⊑ )
• S(X) is isomorphic to the poset SO(X) of pairs of disjoint
open sets (O1,O2) ordered componentwise by inclusion:
S(X )

f

( f 1{tt}, f 1{ ff })

(O1 , O2 )
O1  O2  
tt

x   ff


x  O1
x  O2
otherwise
SO ( X )
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Properties of the Solid Domain
• Theorem For a (second countable) locally compact
Hausdorff space X (e.g. Rn), S(X) is (–) continuous.
and (U1, U2) << (V1, V2) iff U1  V1 & U2  V2
• Theorem If X is Hausdorff, then:
(O1, O2)  Maximal (S X, ⊑) iff
O1= int O2c &
O2= int O1c.
This is a regular solid object: O1=int O1 & O2=int O2
• Definition
(O1, O2) , O1 ≠ ∅ ≠ O2 , is a classical object if O1 ∪ O2 = X.
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Examples
A
• A = {xR2 ⃒ |x| ≤ 1}  [1, 2]
represented in the model by
(int A, int Ac) = ( {x ⃒ |x| < 1}, R2 \ A )
is a classical (but non-regular) solid object.
B
• B = {xR2 ⃒ |x| ≤ 1}
represented by
({x ⃒ |x| < 1} , {x ⃒ |x| > 1})  Maximal (SR2, ⊑)
is a regular solid object.
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Boolean operations and predicates
 : SX  SX
(O1 , O2 )  (O2 , O1 )
 : SX  SX  SX
(( A1 , A2 ) , ( B1 , B2 ))  ( A1  B1 , A2  B2 )
 : SX  SX  SX
(( A1 , A2 ) , ( B1 , B2 ))  ( A1  B1 , A2  B2 )
Theorem All these operations and predicates are Scott continuous.
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Subset Inclusion
• Let Sb X  {( A1, A2 )  SX | A2 compact}{(, )}.
c
 : Sb X  SX  {tt , ff }
tt

(( A1 , A2 ), ( B1 , B2 ))   ff


A2  B1  X
A1  B2  
otherwise.
• Subset inclusion is Scott continuous.
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General Minkowski operator
• For smoothing out sharp corners of objects.
• SbRn = { (A, B)  SRn | Bc is bounded} ∪{(∅,∅)}.
All real solids are represented in SbRn.
• Define:
__ :
SRn  SbRn  SRn
((A,B) , (C,D)) ↦ (A ⊕ C , (Bc ⊕ Dc)c)
where A ⊕ C = { a+c | a A, c C }
• Theorem __ is Scott continuous.
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An effectively given solid domain
• SX can be given effective structure for any locally
compact second countable Hausdorff space, e.g.
Rn, Sn, Tn, [0,1]n.
• Consider X=Rn. The set of pairs of disjoint open
rational polyhedra of the form K = (L1 , L2), gives
a basis for SX.
• Let Kn= (π1 ( K n ) , π2 ( K n) ) be an enumeration
of basis.
• (A, B) is a computable partial solid object if there
exists a total recursive function ß such that
(A , B) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )
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Computing a Solid Object
• In this model, a solid object is
represented by its interior and
exterior.
• The interior and the exterior
are approximated by two
nested sequence of rational
polyhedra.
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Computable Operations on the Solid Domain
• F: (SX)n  SX
or
F: (SX)n  { tt, ff }
is computable if it takes computable sequences of
partial solid objects to computable sequences.
• Theorem All the basic Boolean operations and
predicates are computable wrt any effective
enumeration of either the partial rational polyhedra
or the partial dyadic voxel sets.
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Quantative Measure of Convergence
• In our present model for computable solids,
there is no quantitative measure for the
convergence of the basis elements to a
computable solid.
• Example: The minimum distance from say a
rational point to a computable solid is semicomputable, but not computable.
• We will enrich the notion of domain-theoretic
computability to include a quantitative measure
of convergence.
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Hausdorff Computability
• We strengthen the notion of a computable solid by using
the Hausdorff distance d between compact sets in Rn.
•
d(C,D) = min{ r | C  Dr & D  Cr }
where
Dr = { x |  y  D. |x-y|  r }
• (A , B)  S [–k, k]n is Hausdorff computable
iff there exists an effective chain Kß(n) of basis elements
with ß a total recursive function such that:
(A , B) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )
with
d (π1 ( K ß(n) ) , A ) < 1/2n , d (π2 ( K ß(n) ) , B) < 1/2n
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Hausdorff computability
• Two solid objects which have a small Hausdorff
distance from each other are visually close.
• The Hausdorff distance gives a natural quantitative
measure for approximation of solid objects.
• However, the intersection or union of two Hausdorff
computable solid objects may fail to be Hausdorff
computable.
• Examples of such failure are nontrivial to construct.
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Lebesgue Computability
(A , B)  S [–k, k]n is Lebesgue computable iff
there exists an effective chain K ß(n) of basis elements
with ß a total recursive functions such that:
(A , B) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )
µ(A) - µ(π1 ( K ß(n) ) ) < 1/2 n
and
µ(B) - µ(π2 ( K ß(n) ) ) < 1/2 n .
The definition can be extended to SRn.
Theorem Boolean operations are Lebesgue computable.
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Hausdorff and Lebesgue Computable Objects
• Hausdorff computable  Lebesgue computable
• Lebesgue computable  Hausdorff computable
• Theorem: A regular solid object is computable iff
it is Hausdorff computable.
• However: A computable regular solid object may
not be Lebesgue computable.
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Comparison of Various Notions of Computability
Partial solid
in Rn
Distance to
a point
Computable
Boolean
operators
Lebesgue
measure
semicomputable
computable
noncomputable
Hausdorff
computable
computable
noncomputable
noncomputable
Lebesgue
computable
semicomputable
computable
computable
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The data type
• We use pairs of disjoint dyadic open polyhedra
with either the Hausdorff or the Lebesgue
measure for approximating the idealized solid
object.
((A , B) , 1/2 m )
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The Convex Hull Algorithm
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The Convex Hull Algorithm
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The Convex Hull Algorithm
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The Convex Hull map
• Let Hm: (R2)m  C(R2) be the classical convex Hull
map, with C(R2) the set of compact subsets of R2.
• Let (IR2,) be the domain of rectangles in R2.
• Each rectangle TIR2 has vertices T1,T2,T3,T4.
• For x=(T1,T2,…,Tm)(IR2)m, define:
Cm : (IR2)m  SR2
Cm(x)=(Im(x),Em(x)) with
Em(x)=(H4m((Ti1,Ti2,Ti3,Ti4))1im)c
Im(x)=Int({Hm((Tif(i)))1im) | f:NmN4})
where Nk={1,2,…,k}
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The Convex Hull is Computable!
• Theorem: The map Cm : (IR2)m  SR2
is Hausdorff and Lebesgue computable.
• Complexity:
1. Em(x) is O(m log m).
2. Im(x) is O(m3) but there is a good approximation
in O(m log m), the complexity of the classical
convex hull algorithm.
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Conclusion
Our model satisfies:





A well-defined notion of computability
Reflects the observable properties of real solids
Is closed under basic operations
Captures regular and non-regular sets
Supports a methodology for designing robust
algorithms
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Further Work
Work to be done:
• Implementation
 with exact real arithmetic
 with interval input as in CAD
• Theoretical work




Lebesgue computability of  for regular solids
Boundary representation
Differential properties of curves/surfaces
Solids of manifolds
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Hausdorff computability
(1,1)
(0,1)
rn rat ional,rn  r
left comput able.
Qn
1
2
1
2 n1
1
2n
r0 r1
rn 1 rn r
Q   Qn
n0
is Hausdorff
computable.
However:
Q([1,0]  [0, –1])
= [r,1]  {0}  R2
(1,0) is not Hausdorff
computable.
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Hausdorff and Lebesgue computability
• Lebesgue computable  Hausdorff computable
Let 0 < rn  Q with rn ↗ r, left computable, noncomputable 0 < r < 1.
• Then [r,1]  {0}  R2 is Lebesgue computable but not
Hausdorff computable.
0 r0
rn
r
1
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Hausdorff and Lebesgue computability
• Hausdorff computable  Lebesgue computable
Complement of a Cantor set with Lebesgue measure
1– r with r =lim rn : left computable, non-computable.
• start with
0
1
• stage 1
s0
• stage 2
s1
2
• At stage n remove 2n open mid-intervals of length sn/2n.
sn  rn  rn1 ,
n
s
m 0
m
 rn
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Hausdorff Computability
• We strengthen the notion of a computable solid by using
the Hausdorff distance d between compact sets as a
quantitative measure of convergence.
• (A , B)  S [–k, k]n is Hausdorff computable
iff there exists an effective chain Kß(n) of basis elements
with ß a total recursive function such that:
(A , B) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )
with
d (π1 ( K ß(n) ) , A ) < 1/2 n , d (π2 ( K ß(n) ) , B) < 1/2 n
d (π1 ( K ß(n) ) c, Ac) < 1/2 n , d (π2 ( K ß(n) ) c , Bc) < 1/2 n
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