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Domain-theoretic Models of Differential Calculus and Geometry

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iff x1, x2 ao & x1 x2 , b(x1 x2) f(x1) f(x2) b(x1 x2), i.e. ... iff f(x) Maximal (IR) for x ao (hence f continuous) and Graph(f) is. within lines of slope ... – PowerPoint PPT presentation

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Title: Domain-theoretic Models of Differential Calculus and Geometry


1
Domain-theoretic Models of Differential
Calculus and Geometry
(I) Domain Theory and Differential
Calculus (II) Domain-theoretic Solution of
Differential Equations (III) Domain-theoretic
Model of Geometry and Solid Modelling
  • Abbas Edalat
  • Imperial College London
  • www.doc.ic.ac.uk/ae
  • BRICS, February 2003
  • Contributions from Andre Lieutier, Ali Khanban,
    Marko Krznaric

2
Computational Model for Classical Spaces
  • A research project since 1993
  • Reconstruct some basic mathematics
  • Embed classical spaces into the set of maximal
    elements of suitable domains

3
Computational Model for Classical Spaces
  • Previous Applications
  • Fractal Geometry
  • Measure Integration Theory
  • Topological Representation of Spaces
  • Exact Real Arithmetic

4
Part (I) Domain Theory and Differential Calculus
  • Synthesize Differential Calculus developed by
    Newton and Leibnitz in the 17th century with
    Computer Science developed in the 20th century

5
Non-smooth Mathematics
Smooth Mathematics
  • Geometry
  • Differential Topology
  • Manifolds
  • Dynamical Systems
  • Mathematical Physics
  • .
  • .
  • All based on differential
    calculus
  • Set Theory
  • Logic
  • Algebra
  • Point-set Topology
  • Graph Theory
  • Model Theory
  • .
  • .

6
A Domain-Theoretic Model for Differential
Calculus
  • Indefinite integral of a Scott continuous
    function
  • Derivative of a Scott continuous function
  • Fundamental Theorem of Calculus for
    interval-valued functions
  • Domain of C1 functions
  • Domain of Ck functions

7
Continuous Scott Domains
  • A directed complete partial order (dcpo) is a
    poset (A, ?) , in which every directed set ai
    i?I ? A has a sup or lub ?i?I ai
  • The way-below relation in a dcpo is defined by
  • a b iff for all directed subsets ai i?I
    , the relation b ? ?i?I ai
    implies that there exists i ?I such that a ? ai
  • If a b then a gives a finitary approximation to
    b
  • B ? A is a basis if for each a ? A , b ? B b
    a is directed with lub a
  • A dcpo is (?-)continuous if it has a (countable)
    basis
  • The Scott topology on a continuous dcpo A with
    basis B has basic open sets a ? A b a for
    each b ? B
  • A dcpo is bounded complete if every bounded
    subset has a lub
  • A continuous Scott Domain is an ?-continuous
    bounded complete dcpo

8
The Domain of nonempty compact Intervals of R
  • Let IR a,b a, b ? R ? R
  • (IR, ?) is a bounded complete dcpo with R as
    bottom ?i?I ai ?i?I ai
  • a b ? ao ? b
  • (IR, ?) is ?-continuous
  • countable basis p,q p lt q p, q ? Q
  • (IR, ?) is, thus, a continuous Scott domain.
  • Scott topology has basis ?a b ao ? b

9
Continuous Functions
  • Scott continuous maps 0,1 ? IR with
    f ? g ? ?x ? R . f(x) ? g(x) is another
    continuous Scott domain.
  • ? C00,1 ? ( 0,1 ? IR), with f ?
    If is a topological embedding into a proper
    subset of maximal elements of 0,1 ? IR .

10
Step Functions
  • Lubs of finite and bounded collections of single-
    step functions
  • ?1?i?n(ai ? bi)
  • are called step function.
  • Step functions with ai, bi rational intervals,
    give a basis for 0,1 ? IR

11
Step Functions-An Example
R
b3
a3
b1
b2
a1
a2
0
1
12
Refining the Step Functions
R
b3
a3
b1
a1
b2
a2
0
1
13
Operations in Interval Arithmetic
  • For a a, a ? IR, b b, b ? IR, and ?
    , , ? we have a b xy x ?
    a, y ? b
  • For example
  • a b a b, a b

14
The Basic Construction
  • What is the indefinite integral of a single step
    function a?b ?
  • We expect ? a?b ? (0,1 ? IR)
  • For what f ? C10,1, should we have If ? ? a?b
    ?
  • Intuitively, we expect f to satisfy

15
Interval Derivative
16
Definition of Interval Derivative
  • f ? (0,1 ? IR) has an interval derivative b ?
    IR at a ? I0,1 if ?x1, x2 ? ao,
  • b(x1 x2) ? f(x1) f(x2).
  • The tie of a with b, is
  • ?(a,b) f ?x1,x2 ? ao. b(x1 x2) ?
    f(x1) f(x2)

17
For Classical Functions
Thus, ?(a,b) is our candidate for ? a?b .
18
Properties of Ties
  • ?(a1,b1) ? ?(a2,b2) iff a2 ? a1 b1 ? b2
  • ?ni1 ?(ai,bi) ? ? iff ai?bi 1? i ? n
    bounded.
  • ?i?I ?(ai,bi) ? ? iff ai?bi i?I
    bounded iff ?J ?finite I ?i?J
    ?(ai,bi) ? ?
  • In fact, ?(a,b) behaves like a?b
    we call ?(a,b) a single-step tie.

19
The Indefinite Integral
  • ? (0,1 ? IR) ? (P(0,1 ? IR), ? )
  • ( P
    the power set constructor)
  • ? a?b ?(a,b)
  • ? ?i ?I ai ? bi ?i?I ?(ai,bi)
  • ? is well-defined and Scott continuous.
  • But unlike the classical case, the indefinite
    integral ? is not 1-1.

20
Example
  • (0,1/2 ? 0) ? (1/2,1 ? 0) ? (0,1 ?
    0,1)
  • ?(0,1/2 , 0) ? ? (1/2,1 , 0) ? ? (0,1 ,
    0,1)
  • ?(0,1 , 0)
  • ? 0,1 ? 0

21
The Derivative
22
Examples
23
The Derivative Operator
  • (0,1 ? IR) ? (0,1 ? IR) is monotone
    but not continuous. Note that the classical
    operator is not continuous either.
  • (a?b) ?x . ?
  • is not linear! For f x ? x
    -1,1 ? IR
    g x ? x -1,1 ? IR
  • (fg) (0) ? (0) (0)

24
Domain of Ties, or Indefinite Integrals
  • Recall ? (0,1 ? IR) ? (P(0,1 ? IR), ? )
  • Domain of ties ( T0,1 , ? )
  • Theorem. ( T0,1 , ? ) is a continuous Scott
    domain.

25
The Fundamental Theorem of Calculus
  • Define (T0,1 , ?) ? (0,1 ? IR)
  • ? ? ? f ?
    ?

26
Fundamental Theorem of Calculus
  • For f, g ? C10,1, let f g if f g r, for
    some r ? R.
  • We have

27
F.T. of Calculus Isomorphic version
  • For f , g ? 0,1 ? IR, let f g if f
    g a.e.
  • We then have

28
A Domain for C1 Functions
  • What pairs ( f, g) ? (0,1 ? IR)2 approximate a
    differentiable function?

29
Function and Derivative Consistency
  • Define the consistency relation Cons ? (0,1 ?
    IR) ? (0,1 ? IR) with (f,g) ? Cons if
    (?f) ? (? g) ? ?
  • In fact, if (f,g) ? Cons, there are least and
    greatest functions h with the above properties in
    each connected component of dom(g) which
    intersects dom(f) .

30
Consistency for basis elements
  • (?i ai?bi, ?j cj?dj) ? Cons is a finitary
    property
  • We will define L(f,g), G(f,g) in general and
    show that
  • (f,g) ? Cons iff L(f,g) ?G(f,g).
  • Cons is decidable on the basis.
  • Up(f,g) (fg , g) where fg t ?
    L(f,g)(t) , G(f,g)(t)

31
Function and Derivative Information
32
Updating
33
Consistency Test and Updating for (f,g)
  • Let O be a connected component of dom(g) with
    O ? dom(f) ? ?. For x , y ? O
    define
  • Define L(f,g)(x) supy?O?dom(f)(f (y)
    d(x,y)) and G(f,g)(x)
    infy?O?dom(f)(f (y) d(x,y))
  • Theorem. (f, g) ? Con iff ?x ? O. L (f, g) (x) ?
    G (f, g) (x).

34
Updating Linear step Functions
  • For (f, g) (?1?i?n ai?bi , ?1?j?m cj?dj)
    with f linear g standard,
  • the rational endpoints of ai and cj induce a
    partition y0 lt y1 lt y2 lt lt yk of
    the connected component O of dom(g).
  • Hence L(f,g) is the max of k2 linear maps.
  • Similarly for G(f,g)(x).

35
Updating Algorithm
36
Updating Algorithm (left to right)
f
1
1
37
Updating Algorithm (left to right)
38
Updating Algorithm (right to left)
39
Updating Algorithm (right to left)
40
Updating Algorithm (similarly for upper one)
41
Output of the Updating Algorithm
42
The Domain of C1 Functions
  • Lemma. Cons ? (0,1 ? IR)2 is Scott closed.
  • Theorem. D1 0,1 (f,g) ? (0,1?IR)2 (f,g)
    ? Cons is a continuous Scott domain, which can
    be given an effective structure.
  • Theorem. ??? C10,1 ? C00,1 ? (0,1 ? IR)2
    restricts
    to give a topological embedding
  • D1c ? D1
    (with C1 norm) (with Scott topology)

43
Higher Interval Derivative
44
Higher Derivative and Indefinite Integral
  • For f 0,1 ? IR we define 0,1 ?
    IR by
  • Then ?f ? ?2(a,b) a?b
  • ? (0,1 ? IR) ? (P(0,1 ? IR), ? )
  • ? a?b ? (a,b)
  • ? ?i ?I ai ? bi ?i?I ? (ai,bi)
  • ? is well-defined and Scott continuous.

(2)
(2)
2
(2)
2
(2)
45
Domains of C 2 functions
  • Theorem. Cons (f0,f1,f2) is decidable on basis
    elements.
  • (The present algorithm to check seems to be
    NP-hard.)
  • D2 (f0,f1,f2) ? (I0,1?IR)3 Cons
    (f0,f1,f2)
  • Theorem. ????? restricts to give a topological
    embedding D2c ? D2

46
Domains of C k functions
  • The decidability of Cons on basis elements for k
    ? 3 is an open question.
  • Dk (fi)0?i?k ? (0,1?IR)k1 Cons
    (fi)0?i?k

47
Part (II) Domain-theoretic Solution of
Differential Equations
  • Develop proper data types for ordinary
    differential equations.
  • Solve initial value problem up to any given
    precision.

48
Picards Theorem
49
Picards Solution Reformulated
  • Apv (f,g) ? (f , ?t. v(t,f(t)))

50
A domain-theoretic Picards theorem
  • To obtain Picards theorem with domain theory, we
    have to make sure that derivative updating
    preserves consistency.
  • (f , g) is strongly consistent, (f , g) ?S-Cons,
    if
  • ? h ? g we have (f ,
    h) ? Cons
  • Q(f,g)(x) supy?O?Dom(f) (f (y) d(x,y))
  • R(f,g)x) infy?O?Dom(f) (f (y)
    d(x,y))
  • Theorem. If f , f , g, g 0,1 ? R are
    bounded and g, g are continuous a.e. (e.g. for
    polynomial step functions f and g), then (f,g) is
    strongly consistent iff for any connected
    component O of dom(g) with O ? dom(f) ? ? ,
    we have ?x ? O. Q(f,g)(x),
    R(f,g)(x) ? f (x) , f (x)
  • Thus, on basis elements strong consistency is
    decidable.

51
A domain-theoretic Picards theorem
  • Consider any initial value f ? 0,1 ? IR with
  • (f, ?t. v (t , f(t) )
    ) ? S-Cons
  • Then the continuous map Up ? Apv has a least
    fixed point above (f, ?t.v (t , f(t))) given by
  • (fs, gs) ?n ?0 (Up ? Apv )n
    (f, ?t.v (t , f(t) ) )

52
The Classical Initial Value Problem
  • Suppose v Ih for a continuous h -1,1 ? R
    ? R which satisfies the Lipschitz property around
    (t0,x0) (0,0).
  • Then h is bounded by M say in a compact rectangle
    K around the origin. We can choose positive a ? 1
    such that -a,a ?-Ma,Ma ? K.
  • Put f ?n ?0 fn where fn -a/2n,a /2n ?
    -Ma/2n , Ma/ 2n
  • Then (f , -a,a ? -M ,M ) ? S-Cons, hence

    (f, ?t. v(t , f(t) ) ) ? S-Cons since

    (-a,a ? -M ,M ) ?
    ?t. v (t , f(t) )
  • Theorem. The domain-theoretic solution

    (fs, gs) ?n ?0 (Up ? Apv )n (f, ?t. v (t ,
    f(t) ))
    gives the unique classical solution through (0,0).

53
Computation of the solution for a given precision
? gt0
  • We express f and v as lubs of step functions
  • f ?n ?0 fn
    v ?n ?0 vn
  • Putting Pv Up ? Apv the solution is obtained
    as
  • For all n ?0 we have un- ? un1- ? un1 ?
    un with un - un- ? 0
  • Compute the piecewise linear maps un- , un until
  • the first n ?0 with un - un- ? ?

54
Example
v is approximated by a sequence of step
functions, v0, v1, v ?i vi
.
t
The initial condition is approximated by
rectangles ai?bi (1/2,9/8) ?i ai?bi,
v
t
55
Solution
At stage n we find un - and un
.
56
Solution
At stage n we find un - and un
.
57
Solution
At stage n we find un - and un
.
un - and un tend to the exact solution f t ?
t2/2 1
58
Computing with polynomial step functions
59
Part III A Domain-Theoretic Model of Geometry
  • To develop a Computable model for Geometry and
    Solid Modelling, so that
  • the model is mathematically sound, realistic
  • the basic building blocks are computable
  • it bridges theory and practice.

60
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!

61
The Intersection of two lines
  • With floating point arithmetic, find the point P
    of the intersection L1 ? L2. Then
    min_dist(P, L1) gt 0, min_dist(P,
    L2) gt 0.

62
The Convex Hull Algorithm
With floating point we can get
63
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (i) AC,
or
64
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (i) AC,
or (ii) just AB, or
65
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (i) AC,
or (ii) just AB, or (iii) just BC, or
66
The Convex Hull Algorithm
A, B C nearly collinear
With floating point we can get (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.
67
A Fundamental Problem in Topology and Geometry
  • Subset A ? X topological space. Membership
    predicate ?A X ? tt, ff
  • 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.

68
Non-computability of the Membership Predicate
  • There is discontinuity at the boundary of the
    set.

False
True
69
Non-computable Operations in Classical CG SM
  • ?A Rn ? tt, ff not continuous means it is not
    computable, even for simple objects like
    A0,1n.
  • x ? A is not decidable even for simple objects
    for A 0,?) ? R, we just have the
    undecidability of x ? 0.
  • The Boolean operation ? is not continuous, hence
    noncomputable, wrt the natural notion of topology
    on subsets ? C(Rn) ? C(Rn) ? C(Rn), where
    C(Rn) is compact subsets with the Hausdorff
    metric.

70
Intersection of two 3D cubes
71
Intersection of two 3D cubes
72
Intersection of two 3D cubes
73
This is Really Ironical!
  • Topology and geometry have been developed to
    study continuous functions and transformations on
    spaces.
  • The membership predicate and the binary operation
    for ? are the fundamental building blocks of
    topology and geometry.
  • Yet, these fundamental functions are not
    continuous in classical topology and geometry.

74
Elements of a Computable Topology/Geometry
  • The membership predicate ?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.
  • ?A is now a continuous function.

75
Elements of a Computable Topology/Geometry
  • Note that ?A?B iff int Aint B int
    Acint 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 1tt is open its called the interior
    of the object. f 1ff is open its called
    the exterior of the object.

76
Partial Solid Objects
  • We have now introduced partial solid objects,
    since X \ (f 1tt ? f
    1ff)
    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 1tt ? g 1tt f 1ff
    ? g 1ff Therefore, f ? g means g has more
    information about an idealized real solid object.

77
The Geometric (Solid) Domain of X
  • The geometric (solid) domain S (X) of X is the
    poset (X ? tt, ff ?, ? )
  • S(X) is isomorphic to the poset SO(X) of pairs of
    disjoint open sets (O1,O2) ordered componentwise
    by inclusion

78
Properties of the Geometric (Solid) Domain
  • Theorem For a second countable locally compact
    Hausdorff space X (e.g. Rn), S(X) is bounded
    complete and ?continuous with (U1, U2) ltlt (V1,
    V2) iff the closures of U1 and U2 are compact
    subsets of V1 and V2 respectively.

79
Examples
  • A x?R2 ? x 1 ? 1, 2 represented in the
    model by Arep (int A, int Ac)
  • ( x ? x lt 1, R2 \ A ) is a classical (but
    non-regular) solid object.

80
Boolean operations and predicates
  • Theorem All these operations are Scott
    continuous and preserve classical solid objects.

81
Subset Inclusion
  • Subset inclusion is Scott continuous.

82
General Minkowski operator
  • For smoothing out sharp corners of objects.
  • SbRn (A, B) ? SRn Bc is bounded ?(Ø,Ø).
  • All real solids are represented in SbRn.
  • Define _?_ SRn ? SbRn ? SRn
    ((A,B) , (C,D)) ? (A ? C , (Bc ? Dc)c)
    where A ? C ac a? A, c? C
  • Theorem _?_ is Scott continuous.

83
An effectively given solid domain
  • The geometric domain SX can be given effective
    structure for any locally compact second
    countable Hausdorff space, e.g. Rn, Sn, Tn,
    0,1n.
  • Consider XRn. The set of pairs of disjoint open
    rational bounded polyhedra of the form K (L1 ,
    L2) , with L1 ? L2 ?, gives a basis for
    SX.
  • Let Kn (p1 ( K n ) , p2 ( K n) ) be an
    enumeration of this basis.
  • (A, B) is a computable partial solid object if
    there exists a total recursive function ßN?N
    such that ( K ß(n) ) n ?0 is an increasing
    chain with

(A , B) ( ?n p1 ( K ß(n) ) , ?n p2 (
K ß(n) ) )
84
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.

85
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.

86
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.
  • We will enrich the notion of domain-theoretic
    computability to include a quantitative measure
    of convergence.

87
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 rgt0 C ? Dr D ? Cr
    where Dr x ? y ? D.
    x-y ? r

88
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.

89
Boolean Intersection is not Hausdorff computable
is Hausdorff computable.
However Q?(0,1 ? 0) r,1 ? 0 ? R2 is
not Hausdorff computable.
90
Lebesgue Computability
  • (A , B) ? S k, kd is Lebesgue computable iff
    there exists an effective chain Kß(n) of basis
    elements with ß N?N a total recursive
    function such that
  • (A , B) ( ?n p1 ( K ß(n) ) , ?n
    p2 ( K ß(n) ) )
  • µ(A) - µ(p1 ( K ß(n) ) ) lt 1/2 n µ(B)
    - µ(p2 ( K ß(n) ) ) lt 1/2 n
  • A computable function is Lebesgue computable if
    it preserves Lebesgue computable sequences.
  • Theorem Boolean operations are Lebesgue
    computable.

91
Hausdorff and Lebesgue computability
  • Hausdorff computable ? Lebesgue
    computable Complement of a Cantor set with
    Lebesgue measure 1 r with r lim rn left
    computable but non-computable real.
  • start with
  • stage 1
  • stage 2
  • At stage n remove 2n open mid-intervals of length
    sn/2n.

92
Hausdorff and Lebesgue computability
  • Lebesgue computable ? Hausdorff computable
  • Let 0 lt rn ? Q with rn ? r, left
    computable, non-computable 0 lt r lt 1.

93
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.

94
Conclusion
  • Our model satisfies
  • A well-defined notion of computability
  • Reflects the observable properties of geometric
    objects
  • Is closed under basic operations
  • Captures regular and non-regular sets
  • Supports a methodology for designing robust
    algorithms

95
Data-types for Computational Geometry and Systems
of Linear Equations
  • The Convex Hull
  • Voronoi Diagram or the Post Office problem
  • Delaunay Triangulation
  • The Partial Circle through three partial points

96
The Outer Convex Hull Algorithm
97
The Inner Convex Hull Algorithm
98
The Convex Hull Algorithm
99
The Convex Hull Algorithm
100
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, with the Hausdorff metric.
  • Let (IR2, ? ) be the domain of rectangles in R2.
  • For x(T1,T2,,Tm)?(IR2)m, define
  • Cm (IR2)m ? SR2, Cm(x)
    (Im(x),Em(x)) with
  • Em(x)?(Hm (y))c y?(R2)m, yi?Ti, 1 ? i ?
    m
  • Im(x) ?(Hm (y))0 y?(R2)m, yi?Ti, 1 ? i ?
    m

101
The Convex Hull is Computable!
  • Proposition Em(x)(H4m((Ti1,Ti2,Ti3,Ti4))1?i?m)c
    Im(x)Int(?Hm((Tin))1?i?m)
    n1,2,3,4).
  • Theorem The map Cm (IR2)m ? SR2 is Scott
    continuous, Hausdorff and Lebesgue computable.
  • Complexity
  • Em(x) is O(m log m).
  • Im(x) is also O(m log m).
  • We have precisely the complexity of the
    classical convex hull algorithm in R2 and R3.

102
Voronoi Diagrams
  • We are given a finite number of points in the
    plane.
  • Divide the plane into components closest to
    these points.
  • The problem is equivalent to the Delaunay
    triangulation of the points
  • (1) Triangulate the set of given points so
    that the interior of the circumference circles do
    not contain any of the given points.

(2) Draw the perpendicular bisectors of
the edges of the triangles.
103
Voronoi Diagram Partial Circles
  • The centre of the circle through the three
    vertices of a triangle is the intersection of the
    perpendicular bisectors of the three edges of the
    triangle.
  • The partial circle of three partial points in the
    plane is obtained by considering the Partial
    Perpendicular Bisector of two partial points in
    the plane.

104
Partial Perpendicular Bisector of Two Partial
Points
105
PPBs for Three Partial Points
106
Partial Circles
Each partial circle is defined by its interior
and exterior. The exterior (interior) consists of
all those points of the plane which are outside
(inside) all circles passing through any three
points in the three rectangles.
The exterior is the union of the exteriors of the
three red circles.
The Interior is the intersection of the interiors
of the three blue circles.
107
Partial Circles
With more exact partial points, the boundaries of
the interior and exterior of the partial circle
get closer to each other.
108
Partial Circles
  • The limit of the area between the interior and
    exterior of the partial circle, and the Hausdorff
    distance between their boundaries, is zero.
  • We get a Scott continuous map C (IR2)3?SR2
  • We obtain a robust Voronoi algorithm.

109
Current and Further Work
  • Solving Differential Equations with Domains
  • Differential Calculus with Several Variables
  • Implicit and Inverse Function Theorems
  • Reconstruct Geometry and Smooth Mathematics with
    Domain Theory
  • Continuous processes, robotics,

110
THE END http//www.doc.ic.ac.uk/ae
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