A data Type for Computational Geometry & Solid Modelling Abbas Edalat Andr é Lieutier Imperial College Dassault Systemes

A data Type for A data Type for Computational Geometry &Computational Geometry &

Solid Modelling Solid Modelling

Abbas Edalat André LieutierImperial College Dassault Systemes

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AimAim

i. Mathematically sound, realistic

ii. Bridges theory and practice

To develop a data type for CG & SM

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Why do we need a data type for solids?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 linesThe Intersection of two lines

With floating point arithmetic, find the point P of the intersection L1 L2. Then: min_dist(P, L1) > 0, min_dist(P, L2) > 0.

1L

2LP

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The Convex Hull AlgorithmThe Convex Hull Algorithm

A, B & C nearly collinear A

B

C

With floating point we can get:

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A, B & C nearly collinear


A

B

C


(i) AC, or

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A

B

C


(i) AC, or(ii) just AB, or

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A

B

C


(i) AC, or(ii) just AB, or(iii) just BC, or

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(i) AC, or(ii) just AB, or(iii) just BC, or(iv) none of them.

The quest for robust algorithms is the most fundamental unresolved problem in solid modelling and computational geometry.


A

B

C

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• Subset A X topological space.

Membership predicate A : X {tt, ff }

is continuous iff A is both open and closed.

Axff

Axttx

A Fundamental Problem in Topology and GeometryA Fundamental Problem in Topology and Geometry

• 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 basic building blocks of classical geometry are not continuous and hence not computable in Rn.

The Discontinuity of the Membership PredicateThe Discontinuity of the Membership Predicate

True

x

• Example: The point is in the box.x

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x

False

• The basic building blocks of classical geometry are not continuous and hence not computable in Rn.

The Discontinuity of the Membership PredicateThe Discontinuity of the Membership Predicate

• Example: The point is in the box.x

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• There is a discontinuity if x goes through the boundary. x

Non-computability of the Membership PredicateNon-computability of the Membership Predicate

• This predicate is not computable:

x If x is on the boundary, we cannot in general determine if it is in or out at any finite stage of computation.

FalseTrue

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Non-computable Operations in Classical CG & SMNon-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 cubesIntersection of two 3D cubes

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17


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This is Really Ironical!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/GeometryElements 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.

• Redefine: A : X {tt, ff}

with the Scott topology on {tt, ff}.

Ax

IntAxff

IntAxtt

x c

tt ff

open

open

observable observable

Non-observable

A is now a continuous function.

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Elements of a Computable Topology/GeometryElements 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 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 XThe 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:

21

212

1

11

),(

otherwise

}){},{(

)( )(

OO

OOOxff

Oxtt

x

fffttff

XSXS O

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Properties of the Solid DomainProperties 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 O1

c.

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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ExamplesExamples

• A = {xR2 |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 = {xR2 |x| ≤ 1} ⃒�represented by({x | ⃒� x| < 1} , {x | ⃒� x| > 1}) Maximal (SR2, ⊑)is a regular solid object.

A

B

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Boolean operations and predicatesBoolean operations and predicates

Theorem All these operations and predicates are Scott continuous.

),()),( , ),((

:

22112121 BABABBAA

SXSXSX

),(),(

:

1221 OOOO

SXSX

),()),( , ),((

:

22112121 BABABBAA

SXSXSX

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Subset InclusionSubset Inclusion

otherwise.

)),(),,((

},{:

21

12

2121 BAff

XBAtt

BBAA

ffttSXXSb

• Subset inclusion is Scott continuous.

.)},{( compact} |),{( 221 cb ASXAAXS• Let

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General Minkowski operatorGeneral 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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• (A, B) is a computable partial solid object if there exists a total recursive function ß such that

An effectively given solid domainAn 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) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )

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Computing a Solid ObjectComputing 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 DomainComputable 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 ConvergenceQuantative 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 semi-computable, but not computable.

• We will enrich the notion of domain-theoretic computability to include a quantitative measure of convergence.

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• 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

Hausdorff ComputabilityHausdorff Computability

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Hausdorff computabilityHausdorff 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 ComputabilityLebesgue 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 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.

Hausdorff and Lebesgue Computable ObjectsHausdorff and Lebesgue Computable Objects

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Comparison of Various Notions of ComputabilityComparison of Various Notions of Computability

Partial solid

in Rn

Distance to a point

Boolean operators

Lebesgue measure

Computablesemi-

computablecomputable

non-computable

Hausdorff computable

computablenon-

computablenon-

computable

Lebesgue computable

semi-computable

computable computable

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The data typeThe 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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39


40


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• 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 TIR2 has vertices T1,T2,T3,T4.

Im(x)=Int({Hm((Tif(i)))1im) | f:NmN4})

where Nk={1,2,…,k}

The Convex Hull mapThe Convex Hull map

• For x=(T1,T2,…,Tm)(IR2)m, define:

Cm : (IR2)m SR2

Cm(x)=(Im(x),Em(x)) with

Em(x)=(H4m((Ti1,Ti

2,Ti3,Ti

4))1im)c

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The Convex Hull is Computable!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 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 WorkFurther 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 computabilityHausdorff computability

)1,0( )1,1(

)0,1(

nQ

0r 1r 1nr nr r

21

121n

n21

0

.computableleft

rational,

nn

nn

QQ

rrr

However:

Q([1,0] [0, –1])= [r,1] {0} R2

is not Hausdorff computable.

is Hausdorff computable.

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Hausdorff and Lebesgue computabilityHausdorff and Lebesgue computability

• Lebesgue computable Hausdorff computable

Let 0 < rn Q with rn ↗ r, left computable, non-computable 0 < r < 1.

• Then [r,1] {0} R2 is Lebesgue computable but not Hausdorff computable.

10r nr r0

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Hausdorff and Lebesgue computabilityHausdorff and Lebesgue computability

• Hausdorff computable Lebesgue computableComplement of a Cantor set with Lebesgue measure

1– r with r =lim rn : left computable, non-computable.

• start with

• stage 1

• stage 2

• At stage n remove 2n open mid-intervals of length sn/2n.

nmnnn rsrrs

n

0m1 ,

0 1

s0

21s

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• 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

Hausdorff ComputabilityHausdorff Computability

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Documents

A data Type for Computational Geometry & Solid Modelling Abbas Edalat Andr é Lieutier Imperial College Dassault Systemes