Guaranteeing the homotopy type of a set defined by non-linear inequalities

NICOLAS DELANOUE

LISA, University of Angers, 62 av. Notre Dame du Lac, Angers, 49000, France, e-mail: delanoue@istia.univ-angers.fr

LUC JAULIN

E3I2, ENSIETA, 2 rue Franc¸ois Verny, Brest Cedex 09, 29806, France, e-mail: luc.jaulin@ensieta.fr

and

BERTRAND COTTENCEAU

LISA, University of Angers, 62 av. Notre Dame du Lac, Angers, 49000, France, e-mail: bertrand.cottenceau@istia.univ-angers.fr

(Received: 24 June 2006; accepted: 24 March 2007)

Abstract.This paper provides an effective method to create an abstract simplicial complex homotopy equivalent to a given set_Sdescribed by non-linear inequalities (polynomial or not). To our knowledge, no other numerical algorithm is able to deal with this type of problem. The proposed approach divides Sinto subsets that have been proven to be contractible using interval arithmetic. The method is close to ˇCech cohomology and uses the nerve theorem. Some examples illustrate the principle of the approach. This algorithm has been implemented.

1. Introduction

Topological properties of a set S are significant information and have a lot of applications in different areas like robotics, computer-aided design…. There exist different approaches for computing topological properties of sets.

• In the case of semi-algebraic sets, G. E. Collins [2] proposes a finite partition into semi-algebraic subsets homeomorphic to open boxes. This algorithm is called cylindrical algebraic decomposition. Then, from this cell decomposition, a triangulation homeomorphic to the semi-algebraic set is created. Basu, Pollack, and Roy [1] propose a method which computes more efficiently the number of connected components and the first Betti number of semi-algebraic sets.

• Stander and Hart [7] combine interval analysis [13], [14] and Morse theory [12]

to compute the topology of an implicit surface defined by:

{(x,y,z)∈R³|ƒ(x,y,z) = 0, withƒ ∈C^∞(R³,R)}. (1.1)

A semi-algebraic set is a set defined by a quantifier-free Boolean formula with atomsP < 0, P= 0,P_∈P, wherePis a finite collection of polynomials.

Figure 1. v1is a star for this subset of_R²whereasv2is not.

In our article, sets are defined by a quantifier-free Boolean formula with atoms ƒ≤0,ƒ ∈F, whereF is a finite collection ofC¹(Rⁿ,R). The algorithm divides a set, denoted byS, with a finite cover{Si}i∈Isuch that:

∀J⊂I,

j∈J

Sj is contractible or empty. (1.2)

With this cover, an abstract simplicial complex homotopy equivalent toS is gen- erated. Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces (pathconnectedness, fun- damental group, homotopy groups, homology groups,_…). Combining these results with the Kenzo [15] program, the homology groups of a given setS (and the Betti numbers) can be computed.

The paper is organized as follows: Section 2 presents a sufficient condition to prove algorithmically that a set is contractible. Next, from a cover satisfying (1.2) created by a subdivision process, an algorithm which provides a simplicial complex homotopy equivalent to S is described in Section 3. To finish, some examples illustrate the principle of the approach. These illustrations have been computed by a new solver we developed in C++ called HIA(Homotopy via Interval type Analysis [9]).

2. Sufficient Condition of Contractibility

In this section, we recall definitions about star-shaped sets, homotopy equivalence between maps and topological sets. We describe basic relations between these notions. At the end, a sufficient condition of contractibility is given.

DEFINITION 2.1. A pointvis astarfor a subsetXof an Euclidean set ifXcontains all the line segments connecting any of its points andv.

DEFINITION 2.2. Ifvis a star for subsetXof an Euclidean set, one says thatXis star-shapedorv-star-shapedif one wants to clarify thatvis a star.

Figure 2. The two continuous maps_ƒ,g:X→Yare homotopic (_ƒg).

Figure 3. These two sets are homotopy equivalent (AB).

PROPOSITION 2.1.Let X and Y be two v-star-shaped sets, then X ∩Y and X∪Y are also v-star-shaped.

DEFINITION 2.3Homotopic maps. Two continuous mapsƒ,g:X→Yarehomo- topic(orƒishomotopictog) if there exists a continuous mapF :X_×[0,1]→Y, such that: F(x,0) = ƒ(x) andF(x,1) = g(x), ∀x ∈X. The map F is said to be a homotopy, and we write_ƒ gfor “_ƒ is homotopic tog.” Figure 2 illustrates this notion.

DEFINITION 2.4Homotopy equivalence between sets. Two spaces X and Y are homotopy-equivalent(orof the same homotopy type) if there exist continuous maps ƒ:X→Yandg:Y →X, such that:_ƒ◦g1_Xandg◦ƒ1_Y, where 1_Xand 1_Yare the identity maps ofXandYrespectively. In this case,ƒis ahomotopy equivalence andgis thehomotopy inverseofƒ. We writeX Yfor “Xis homotopy-equivalent toY.” Figure 3 illustrates this notion.

DEFINITION 2.5. A topological spaceXwhich is homotopy-equivalent to a point, iscontractible.

Remark 2.1.A star-shaped set is contractible as illustrated in Figure 4.

The next result is a sufficient condition to prove that a set defined by only one inequality is star-shaped. This sufficient condition can be checked using interval analysis [5].

PROPOSITION 2.2. Letƒbe a C¹ function fromRⁿ toR, D be a convex set and S ={x∈D⊂Rⁿ|ƒ(x)≤0}. If there exists v inS such that

{x∈D|ƒ(x) = 0 and ∇ƒ(x)⋅(x−v)≤0}=∅ (2.1)

Figure 4. A star-shaped set is contractible.

then_S is star-shaped.

Proof. See [3].

COROLLARY 2.1.LetSbe a set defined by a quantifier-free Boolean formula with atomsƒ≤0,ƒ ∈{ƒi}i∈I, where{ƒi}i∈I is a finite collection ofC¹(D⊂Rⁿ,R). If D is convex and if there exists v∈D such that

∀i∈I, {x∈D|ƒi(x) = 0 and ∇ƒi(x)⋅(x−v)≤0}=∅ (2.2) thenS is contractible.

Proof. Combine Proposition 2.1, Remark 2.1, and Proposition 2.2.

3. Discretization

Section 2 shows that proving the contractibility of a set often amounts to checking that a set defined by non-linear inequalities is empty. From a set S, which can be an infinite collection of elements, a single point set{v}is created holding the homotopy type. Most of the subsets S ofRⁿ are not contractible. The algorithm, presented in this section, produces a set K(S) (a simplicial complex) which is homotopy equivalent toS. A simplicial complex can be finitely represented.

The section is organized as follows. First, we recall definitions and some properties related to simplicial complexes. In the next subsection, we present the idea of our approach and a proof that the two sets S and K(S) are homotopy equivalent. The last subsection presents explicitly our algorithm Homotopy via Interval Analysisand examples.

3.1. SIMPLICIALCOMPLEXES

This subsection is concerned with building up spaces from certain elementary spaces calledsimplices. A simplex is a generalization tondimensions of a triangle or a tetrahedron. These simplices are fitted together in such a way that two simplices meet (if at all) in a common face.

Figure 5. Two subsets ofR^2.

DEFINITION 3.1. Asimplicial complex Kis a finite set of simplices, all contained in_Rⁿ. Furthermore:

1. If_σnis a simplex ofKand_τpis a face of_σn, then_τpis inK.

2. If_σnand_σpare simplices ofK, then either_σn∩σpis empty, or it is a common face of_σnand_σp.

For example (Figure 5),K2is a simplicial complex whereas K1is not. A sim- plicial complexKis not a topological space, this is only a set whose elements are simplices. The set of points that lie in at least one of the simplices ofK, topologized as a subspace ofRⁿ, is a topological space, called the polyhedron of K, written

|K|. Simplicial complexes are sets of simplices lying in one particular Euclidean spaceRⁿ. To free ourselves of this restriction, let us define the notion ofabstract simplicial complex.

DEFINITION 3.2. An abstract simplicial complex K is a finite set of elements a⁰,a¹, … called abstract vertices, together with a collection of subsets (aⁱ⁰,aⁱ¹, …,aⁱⁿ), …calledabstract simplices, with the property that any subset of a simplex is itself a simplex; i.e.

σ ∈K, σ^′ ⊂σ⇒σ^′ ∈K. (3.1)

Thedimensionof an abstract simplicial complex is the maximum of the dimension of its simplices.

EXAMPLE 3.1. The set K =

∅,{a⁰},{a¹},{a²},{a³},{a⁴},{a⁰_,a¹},

…{a¹,a²},{a⁰,a²},{a³,a⁴},{a⁰,a¹,a²} (3.2) is an abstract simplicial complex. Its dimension is the dimension of{a⁰,a¹,a²}, i.e. 2.

The enumeration of all elements of an abstract simplicial complexKseems to be useless, since{a⁰,a¹,a²}∈Kimplies that{a⁰},{a¹},{a²},{a⁰,a¹},{a¹,a²}, {a⁰,a²}are also elements ofK.

The dimension of an abstract simplex is the number of vertices minus 1.

Figure 6. A realization ofK.

NOTATION 3.1. With V a finite collection of elements (abstract vertices) V = {a⁰,a¹, …,aⁿ}and 2^V the power set ofV, a simplicial complex is a subset of 2^V satisfying (3.1). Letting{σ1, …,σm}be included in 2^V(not necessarily an abstract simplicial complex), we denote by_σ1+· · ·+_σmthe following abstract simplicial complex:

σ1+· · ·+_σm:=

i=m i= 1

2^σi.

Thanks to this notation, the abstract simplicial complexKdefined in Example 3.1 is written:K={a⁰,a¹,a²}+{a³,a⁴}, and if no ambiguity can arise, we will write K=a⁰a¹a²+a³a⁴.

DEFINITION 3.3. A realization of an abstract simplicial complexKis a simplicial complexKhavingKas an abstraction.

Remark 3.1. IfK₁ and K₂ are two realizations of an abstract simplicial complex K, then|K₁|and|K₂|are homeomorphic [11]. Hence, topological properties of an abstract simplicial complexKare well defined.

DEFINITION 3.4. LetKbe an abstract simplicial complex, and{x}a vertex. We denote byC(x_,K) the following set:

C(x,K) =K∪

s_∈K

{{x} ∪s}.

C(x,K) is an abstract simplicial complex called theconeofK fromx. With Nota- tion 3.1, a cone can be interpreted as a product noted by∗; withK =_σ1+· · ·+_σm one has:

C(x,K) =x∗(_σ1+· · ·+_σm) =x_σ1+· · ·+x_σm.

The reader can check thatσ1+· · ·+σmis the smallest, with inclusion defined on 2^2V as order relation, abstract simplicial complex that containsσ1, …,σm, as simplices.

Figure 7. A realization ofC(x,K).

EXAMPLE 3.2. ForKdefined in Example 3.1,C(x,K) is equal to C(x,K) = K∪

{x},{x,a⁰},{x,a¹},{x,a²},{x,a³},{x,a⁴},

…{x,a⁰,a¹},{x,a¹,a²},{x,a²,a³}, {x,a⁴,a⁵},{x,a⁰,a¹,a²}.

(3.3)

With Notation 3.1:

C(x,K) = x∗{a⁰a¹a²+a³a⁴}

= xa⁰a¹a²+xa³a⁴. Figure 7 shows a realization ofC(x,K).

PROPOSITION 3.1.LetKbe an abstract simplicial complex, andC(x,K)a cone ofK, thenC(x,K) {x}.

Proof. LetKbe a realization ofK, by construction|K|is|{x}|-star-shaped.

3.2. HOMOTOPYTYPEISPRESERVED

To guarantee a simplicial complex has the same homotopy type as setS, the main idea of our approach is to create an abstract simplicial complex from a finite cover {Si}i∈I of S. This cover has to be such that each Si is contractible and compact, moreover, the intersection of elements of any sub collection of{Si}i∈I

has to be contractible (or empty). To simplify our presentation, let us introduce the Notation 3.2.

NOTATION 3.2. With Ian index set andJa subset ofI, let us denote bySJ the set

j∈J

Sj. With this notation:S3∩S4 ∩S9is denoted byS_{3,4,9}.

DEFINITION 3.5. Let S be a topological subset of Rⁿ, {Si}i∈I is a compact contractible coverofSif:

• Iis finite.

• ∀i∈I, Siis compact.

Figure 8. A compact contractible cover{Sⁱ}i_∈Iof_S.

• ∀J⊂I, SJ is contractible or empty.

Figure 8 is an example of compact contractible cover.

DEFINITION 3.6. Let {Si}i_∈I be a compact contractible cover of a set S. We denote byJ those index sets J such thatSJ is non-empty. An abstract simplicial complexK(S) is said to beadaptedto{Si}i_∈Iif it is the smallest simplicial complex satisfying the following conditions:

• ∀J_∈J, an abstract vertex (a^J) is inK(S).

• ∀J_∈J, an abstract simplicial complexKJ defined by KJ =a^J∗

J′ ∈J |SJ′ ⊂SJ

KJ′

.

is an abstract simplicial subcomplex ofK(S).K(S) is usually called the nerve of the cover{Si}i∈I.

THEOREM 3.1.If{Si}i∈Iis a compact contractible cover of_{S ⊂R}ⁿandK(S)an abstract simplicial complex adapted to {Si}i∈I then K(S) and S are of the same homotopy type.

Proof. K(_S) is the nerve of the cover{Si}i∈I. From the nerve theorem [4],K(_S)

andSare of the same homotopy type.

3.3. A NEWALGORITHM: HOMOTOPYTYPE VIAINTERVALANALYSIS

This subsection presents a new algorithm, called HIA (Homotopy via Interval Analysis). This algorithm is often able to create a compact contractible cover of a set defined by non-linear inequalities. In a second pass, it creates an abstract simplicial complex adapted to this cover. The cover{Si}i∈I is created thanks to a tiling. Therefore, before introducing the main step of this algorithm, definitions and notations are introduced.

Figure 9. An abstract simplicial complex adapted to{Sⁱ}ⁱ∈I.

Figure 10. _SandK(S) are homotopy equivalent.

Figure 11. p1andp2are respectively examples of boxes of_R²and_R³.

DEFINITION 3.7. A box is a Cartesian product of compact real intervals.

Figure 11 shows a box ofR²and one ofR³.

DEFINITION 3.8. A tilingP is a finite collection of boxes{p_i}i_∈Isatisfying:

i=j_⇒m(p_{i ∩}p_j) = 0,

wheremis the classical Lebesgue Measure.

NOTATION 3.3. LetSbe a part ofRⁿand{p_i}i∈Ia tiling such thatS ⊂

i∈Ip_i. Let us denote by{Si}i_∈Ithe cover ofSwhereSi =S ∩pi,i∈I. So the bisection ofSi

is defined via the bisection of the boxp_i.

Figure 12. Example of a tilingPwith 9 boxes.P={p1, …,p9}.

The main idea of this algorithm is a subdivision process. From a cover{Si}i∈I

created thanks to a tiling{pi}i_∈I(see Notation 3.3), a sub-algorithm checks if:

∀J _⊂I_,

j_∈J

Sj is contractible or empty (Sj:=S ∩p_j).

If it is not satisfied, then eachSiresponsible for this failure is bisected. The following algorithm uses:

• The tilingP_∗is such that:∀{p_j}j∈J ⊂P_∗,

j∈JSjis contractible or empty.

• The tilingP_Δ£, nothing is known about its boxes.

At the end of the algorithm, using P_∗ = {p_i}i∈I, the sub-algorithm Adapted Triangulationcreates an abstract simplicial complexK(S) adapted to the cover{Si :=S ∩p_i}i∈I.

ALGORITHM 1 [HIA—Homotopy type via Interval Analysis].

Require:Sa subset ofRⁿ,X₀a box ofRⁿwhich containsS. Ensure:A triangulationK(S) which is homotopy equivalent toS.

1: Initialization:P_∗:=∅,P_Δ :={X₀} 2: while P_Δ=∅do

3: Pop from the last element ofP_Δinto the boxp 4: if

∀{p_j}j∈J ⊂P_∗∪{p},

j_∈J

Sj is proven contractible or empty

then

5: Push{p}intoP_∗; 6: else

7: Bisect(p); then Push back the two resulting boxes intoP_Δ; 8: end if

an adapted triangulationK(S) from a cover{Si}i∈IofS. The idea is to add a cone toK(S) for eachJ inJ, and to do this, in such away, that any two cones can be attached by a cone previously created.

ALGORITHM 2 [Adapted Triangulation].

Require:A setSand a cover{Si}i∈IofS.

Ensure:An adapted triangulationK(S) of{S_i}i∈I. {Initialization:}

1: K(S)←∅,J ←∅.

{Remove useless indices, i.e. CreateJ:}

2: for allJ⊂Ido: ifSJ is contractiblethenJ ←J ∪{J}end for {Add cones:}

3: K(S)←

i∈I

Cone({i})

Above,Cone(J) withJ∈J is defined recursively by:

Cone(J) =a^J∗

⎛

⎜⎜

⎝

J′ ∈J |S_{J′ ⊂}SJ

Cone(J^′)

⎞

⎟⎟

⎠

with convention thatCone(∅) =∅anda^J∗∅=a^J.

Illustration.To illustrate Algorithm 2, let us use the cover given in Figure 13. In this case, the set:{J |J⊂I}has 2⁵elements since #I = 5. But only some of those are such that

j∈JSjis contractible. After step 2, one has:

J =

{1},{1,4},{1,3},{1,3,4},{2},{2,4},{2,5},{3},{3,4},{3,5}, {3,4,5},{4},{4,5},{5}.

To explain what is done at step 3, let us see howCone({1}) is computed, for example. To do it, the collection ofJ^′ ∈J such thatSJ′ ⊂S_{1} has to be known.

This collection is composed of S_{1}, S_{1_,4}, S_{1_,3}, andS_{1_,3_,4}. More generally, elements of{S_J, J ∈J } can be partially ordered by inclusion. Figure 13 shows how these elements are ordered whereSJ′ ⊂SJ is represented byJ^′←J.

Figure 13. Partial order relation on{SJ, J_∈J }.

Figure 14. Partial order relation on{S^J, J_∈J }with renamed elements.

To make clearer our explanation, let us renamea^{1} bya,a^{3} byb,a^{4} byc, etc. Figure 14 shows how thea^J are renamed.

After step 3, one obtains this abstract simplicial complex:

K(S) = a∗(ƒ ∗m+g∗m) +b∗

g∗m+h∗(m+n) +i∗n +c∗

ƒ ∗m+h∗(m+n) +j∗n+k +d∗(i∗n+j∗n+l) +e∗(k+l), K(S) = aƒm+agm+bgm+bhm+bhm+bin

+cƒm+chm+chn+cjn+ck+din+djn+dl+ek+el.

Geometrical illustration is given at Figure 21 in which a realization of the abstract simplicial complexKis provided.

EXAMPLE 3.3. Figures 15 and 16 are respectively examples of realizations of simplicial complexes generated byHIAfor the sets:

S1={(x,y)∈R²|x²+y²+xy−2≤0 and −x²−y²−xy+ 1≤0} and

S2={(x,y)∈R²|x²+y²−6≤0 and 0.2 cos(x−y)−sin(yx)−0.6≤0}.

HIA algorithm has been implemented in C++ / OpenGL and can be downloaded at http://www.istia.univ-angers.fr/_∼delanoue/.

Figure 15. Example of a set and a realization of the simplicial complex generated byHIA.

Figure 16. Example of a set and a realization of the simplicial complex generated byHIA.

Figure 17. A rope suspended between two pointsAandB.

EXAMPLE 3.4. This example illustrates why topological properties of sets can be really useful in robotics. To our knowledge, the set studied here is not a semi- algebraic set. Hence, to our knowledge no other algorithm is able to deal with this problem. Let us consider a rope, with lengthL suspended between two points A andB, which are endpoints of two arms (see Figure 17).

The independent parameters needed to specify an object configuration are the real numbers _α and _β. The robot has to satisfy some constraints. The rope must not touch the floor, and the lower point of the rope, denoted byC, must be outside the area delimited by the dotted lines. One would like to study the topological properties of the feasible configuration setS.

S =(_α,β)∈[0,π]×[_β⁻,β⁺] such that y_C≥0 and C∈ .

To apply our algorithm, first, it is needed that the setS could be described by a quantifier-free Boolean formula with atoms ƒ ≤ 0, ƒ ∈F where F is a finite

Figure 18. Variations of the functionz→ ƒ(_β0,z), wherez0 = arccos(^L_β).

collection ofC¹(Rⁿ,R). Secondly, to prove that a subsetSJofSis contractible, one also needs to be able to evaluate D_ƒ on a bounded box for all_{ƒ ∈} F. The next paragraphs show how these two problems can be solved.

From the Hamilton principle, it is possible to show that Cartesian coordinates ofCare given by:

xC = g₁(_α,β) = 2 cos_α+_β, yC = g2(_α,β) = 2 sin(_α)−acosh

β

a

+a,

whereais a positive real solution of (3.4) for a fixed real positive number_β: asinh

β

a

−L= 0. (3.4)

Letz= ^β_a, therefore (3.4) is equivalent to (3.5) where the unknown variable isz:

ƒ(_β,z) = sinh(z)−L

βz= 0. (3.5)

We have Dƒ(_β,z) =

Lz

β^2,cosh(z)− L β

. (3.6)

With_β =_β0, Figure 18 presents the function:

R^+∗z→ ƒ(_β0,z)∈R .

From the Weierstrass intermediate value theorem, and sinceD2ƒ(_β0,z) > 0 if z∈]z₀; +

∞

[, there exists a uniquez^∗in ]z₀; +

∞

[ such thatƒ(_β0,z^∗) = 0. Therefore, the function denoted by_ϕ, which associates with a positive real number_β0a positive real numberz^∗ such thatƒ(_β0,z^∗) = 0, is well defined. Cartesian coordinates ofC are given by:

x_C = g₁(_α,β) = 2 cos_α+_β, yC = g2(_α,β) = 2 sin(_α)− ^β

ϕ(_β)cosh_ϕ(_β) + ^β ϕ(_β)^.

(3.7)

c_{1 ⇔ ƒ1}(_α,β) = −g₂(_α,β) ≤0 , c₂ ⇔

ƒ2(_α,β) = g₁(_α,β)−l ≤0, c₃ ⇔

ƒ3(_α,β) = −g₁(_α,β) +r ≤0, c_{4 ⇔ ƒ4}(_α,β) = g₂(_α,β)−b ≤0, c_{5 ⇔ ƒ5}(_α,β) = −g₂(_α,β) +t ≤0_, and = ]l_,r[_×]b_,t[.

It only remains to show that for alliin{1, …,5},ƒi is aC¹function, and how D_ƒican be computed. Functionsƒ2andƒ3are clearly differentiable sinceg₁is. One can obtainDƒ2andDƒ3fromDg₁:

D₁g₁(_α,β) = −2 sin(_α), D₂g₁(_α,β) = 1.

Functionsƒ1,ƒ4, andƒ5are differentiable ifg₂is. Andg₂is differentiable if_ϕis a differentiable function. Let us show that_ϕis differentiable. SinceD_2ƒ(_β,z^∗)>0, D₂ƒ(_β,z^∗) is an isomorphism ofR, from the implicit function theorem, one has that ϕis aC¹function and moreover:

ϕ^′(_β) = −D₂⁻¹ƒ

β,ϕ(_β)◦D1ƒ

β,ϕ(_β),

ϕ^′(_β) = −

L_ϕ(_β) β² cosh_ϕ(_β)− L

β

= −L_ϕ(_β) β²coshϕ(β)−βL^. One obtains:

D1g2(_α,β) = 2 cos(_α), D₂g₂(_α,β) = L

βcoshϕ(β)−L

1−cosh_ϕ(_β)

ϕ(β) + sinh_ϕ(_β)

+1−coshϕ(β) ϕ(_β) ^.

Numerical application: Figure 19 presents the feasible configuration set S with

= ]0.8,1.2[×]0.5,0.7[,L= 4, and (_α,β)∈[0,π]×[1/2,7/4]. It also provides

A program (Rope.exe) associated with this example can be found athttp://www.istia.

univ-angers.fr/_∼delanoue/.

Figure 19. The feasible configuration set_Sand a triangulation generated byHIA.

Figure 20. The circleS¹and its enlargementS¹_ε with_ε<1.

a realization of the simplicial complex generated byHIA. The simplicial complex K(S) can be collapsed, and one can affirm thatSis homotopy equivalent to a circle.

4. Discussion

The proposed method is efficient (e.g. time computing is less than 1 sec for Exam- ple 3.3) but does not always terminate because for some sets, it is not possible to decompose these ones with a finite collection of star-shaped sets. The circle S¹ = {(x,y) ∈ R², x² +y² −1 = 0} is an example that the HIA algorithm cannot handle. In general, the HIA algorithm is not able to deal with mani- folds. To expand our algorithm to manifolds, a algorithm could enlarge it without changing the homotopy type. For example, the circle S¹ could be enlarged into S_ε¹={(x,y)∈R², (x²+y²−1)²≤ε}as showed in Figure 20.

The presence of singularities has not been discussed. If there is a critical pointx₁ satisfyingƒ(x1) = 0,HIAalgorithm won’t terminate because the sufficient condition of Proposition 2.2 can not be checked using interval analysis. But on the other hand, thanks to Proposition 2.2, all critical points do not have to be found to guarantee the homotopy type ofM⁰=ƒ⁻¹(]−

∞

; 0]) [12].

Figure 21. Adapted Triangulationalgorithm step by step.

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