Computational Topology for Approximations of Knots
CComputational Topology for Approximations ofKnots
J. Li and T. J. Peters and K. E. JordanAbstract.
The preservation of ambient isotopic equivalence underpiecewise linear (PL) approximation for smooth knots are prominentin molecular modeling and simulation. Sufficient conditions are givenregarding:(1) Hausdorff distance, and(2) a sum of total curvature and derivative.High degree B´ezier curves are often used as smooth representations,where computational efficiency is a practical concern. Subdivision canproduce PL approximations for a given B´ezier curve, fulfilling the abovetwo conditions. The primary contributions are:(i) a priori bounds on the number of subdivision iterations sufficientto achieve a PL approximation that is ambient isotopic to theoriginal B´ezier curve, and(ii) improved iteration bounds over those previously established. Introduction (a) Unknot VS. Knot (b) An intermediate step (c) Knot VS. Knot
Figure 1.
Ambient isotopic approximationFigure 1(a) demonstrates an example of topological difference, where a knot-ted B´ezier curve is defined by an unknotted control polygon [12]. Subdivisionis then used to generate new control polygons. Figure 1(b) shows the controlpolygon after one subdivision, where the topological difference remains. Fig-ure 1(c) shows the control polygon after two subdivisions, where the controlpolygon obtains the same topology as the underlying curve.The images are illustrative and a curve visualization tool [16] was used toexperimentally create these examples. Rigorous proofs of the topological dif-ference between the B´ezier curve and its initial control polygon were formu-lated [12, Section 2]. This serves as a cautionary note that graphics used to a r X i v : . [ c s . C G ] D ec approximate a curve may not have isotopic equivalence. Additional rigoroustopological analysis is important, as described here. Figure 1(b) and 1(c) arevisual examples that show successive subdivisions eventually produce topolog-ically correct PL approximations. The advantage of the bounds given here arediscussed in Remark 4.5.1.1. Topological background.
There is contemporary interest [1, 2, 5, 17, 20]to preserve topological characteristics such as homeomorphism and ambientisotopy between an initial geometric model and its approximation. Ambientisotopy is a continuous family of homeomorphisms H : X × [0 , → Y suchthat H ( X,
0) = X and H ( X,
1) = Y, for topological spaces X and Y [8]. It is particularly applicable for time varyingmodels, such as the writhing of molecules.A B´ezier curve is characterized by an indexed set of points, which formsa piecewise linear ( P L ) approximation of the curve, called a control polygon.The de Casteljau algorithm [7] is a subdivision algorithm associated to B´eziercurves which recursively generates control polygons more closely approximatingthe curve under Hausdorff distance [22, 23].An earlier algorithm [9] establishes an isotopic approximation over a broadclass of parametric geometry, but can not provide the number of subdivisioniterations for B´ezier curves. Other recent papers [3, 14] present algorithms tocompute isotopic PL approximation for 2 D algebraic curves. Computationaltechniques for establishing isotopy and homotopy have been established regard-ing algorithms for point-cloud by “distance-like functions” [4]. Ambient isotopyunder subdivision was previously established [20] for 3 D B´ezier curves of lowdegree (less than 4).Recent progress regarding isotopy under certain convergence criteria hasbeen made [6, 11, 13]. In particular, Denne and Sullivan proved that for home-omorphic curves, if their distance and angles between the first derivatives arewithin some given bounds, then these curves are ambient isotopic [6]. Thisresult has been applied to B´ezier curves [13]. Here we present an alternativeset of conditions for ambient isotopy that is explicitly constructed. It is usefulfor applications that require explicit maps between initial and terminal config-urations. Remark 1.2 will show that there is no need to test first derivatives.Instead, we test global conditions of distance and total curvature. It may alsobe useful when the conditions here are easier to be verified than those in thepreviously established method. Furthermore, the subdivision iteration boundestablished here is an improvement over the previous one (Remark 4.5).Moreover, this is alternative to a result regarding existence of ambient iso-topy for B´ezier curves [10]. The pure existence proof requires the convex hullsof sub-control polygons to be contained in a tubular neighborhood determinedby a pipe surface and may need more subdivision iterations and produce toomany
P L segments. The work here removes this convex hull constraint andproduces the isotopy using fewer subdivision iterations. omputational Topology for Approximations of Knots 3
A technique we will use is called pipe surface [15]. A pipe surface of radius r of a curve c ( t ), where t ∈ [0 ,
1] is given by p ( t, θ ) = c ( t ) + r [ cos ( θ ) n ( t ) + sin ( θ ) b ( t )] , where θ ∈ [0 , π ] and n ( t ) and b ( t ) are, respectively, the normal and bi-normalvectors at the point c ( t ), as given by the Frenet-Serret trihedron. Remark 1.1.
The paper [15] provides the computation of the radius r onlyfor rational spline curves. However, the method of computing r is similar forother compact, regular, C , and simple curves, that is, taking the minimum of1 /κ max , d min , and r end , where κ max is the maximum of the curvatures, d min isthe minimum separation distance, and r end is the maximal radius around theend points that does not yield self-intersections.Pipe surfaces have been studied since the 19th century [19], but the pre-sentation here follows a contemporary source [15]. These authors perform athorough analysis and description of the end conditions of open spline curves.The junction points of a B´ezier curve are merely a special case of that analysis.We shall state the conditions. We assume throughout this paper that thespace curves are parametric, compact, simple (non-self-intersecting) and reg-ular (The first derivatives never vanish). Given two curves, P L and smoothrespectively (Usually, the
P L curve is an approximation of the smooth curve.),suppose that they are divided into sub-curves. Let L ( t ) : [0 , → R and C ( t ) : [0 , → R be the corresponding P L and smooth sub-curves. We re-quire that L (0) = C (0) and L (1) = C (1). In particular, for a B´ezier curve,subdivision produces sub-control polygons and the corresponding smooth sub-curves such that each pair of end points between the P L and smooth sub-curvesare connected.There exists a nonsingular pipe surface of radius r for C [15]. Denote thedisc of radius r centered at C ( t ) and normal to C as D r ( t ). Let a pipe section to be Γ = (cid:83) t ∈ [0 , D r ( t ). Denote the interior as int(Γ), and the boundary as ∂ Γ. Note that the boundary ∂ Γ consists of the nonsingular pipe surface andthe end discs D r (0) and D r (1). Define θ ( t ) : [0 , → [0 , π ] by θ ( t ) = η ( C (cid:48) ( t ) , L (cid:48) ( t )) , where the function η ( · , · ) denotes the angle between two vectors [13].1.2. Our two conditions.
The two primary conditions for this paper are nowstated.
Conditions 1 and for ambient isotopy are:(1) L \ { L (0) , L (1) } ⊂ int(Γ); and(2) T κ ( L ) + max t ∈ [0 , θ ( t ) < π ,where Γ is the pipe section of C and T κ ( L ) denotes the total curvature of L , i.e. the sum of exterior angles [13]. Conditions 1 and will guarantee ambient isotopy between not only thesub-curves L and C , but also the whole curves, which is more important. Remark 1.2.
We shall show later that, for a B´ezier curve, the number of sub-divisions for
Condition 2 is at most one more than that for a weaker condition T κ ( L ) < π (Lemma 4.3 in Section 4.2). This allows us to easily remove theburden of testifying the derivatives in order to find θ ( t ).2. Construction of Homeomorphisms
Constructing the ambient isotopy here relies upon explicitly constructinga homeomorphism. The explicit construction provides more algorithmic effi-ciency than only showing the existence of these equivalence relations.
Lemma 2.1.
Suppose L is a sub-control polygon and C is the correspondingB´ezier sub-curve. Then Conditions 1 and By the convergence in Hausdorff distance under subdivision, sufficientlymany subdivision iterations will produce a control polygon that fits inside anonsingular pipe surface. Furthermore, by the Angular Convergence [13, Theo-rem 4.1] and the lemma [13, Lemma 5.3], possibly more subdivisions will ensurethat each sub-control polygon lies in the corresponding nonsingular pipe sec-tion, which is the
Condition 1 . Denote the number of subdivision iterations toachieve this by ι .By the Angular Convergence, T κ ( L ) converges to 0 under subdivision. Be-cause the discrete derivative of the control polygon converges to the derivativeof the B´ezier curve [21] under subdivision, θ ( t ) converges to 0 for each t ∈ [0 , Condition 2 will be achieved by sufficiently many subdivision iterations, say ι . (The Details to find ι and ι are in Section 4.2.) (cid:3) Remark 2.2.
To obtain some intuition for these conditions, restrict our at-tention to a B´ezier curve. Consider L to be a sub-control polygon and C tobe the corresponding sub-curve. Condition 1 will ensure that L lies inside anonsingular pipe section, while Condition 2 will ensure a local homeomorphismbetween L and C . In particular, Conditions 1 and will be sufficient for usto establish the one-to-one correspondence using normal discs of C . Conditions 1 and are assumed in the rest of the section. Define a function ˜ L ( t ) : [0 , → L by letting˜ L ( t ) = D r ( t ) ∩ L, (2.1)where D r ( t ) is the normal disc of C at t .Define a map h : C → L for each p ∈ C by setting h ( p ) = ˜ L ( C − ( p )) . (2.2)We shall show that h is a homeomorphism. The subtlety here is to demon-strate the one-to-one correspondence by showing each normal disc of C inter-sects L at a single point (which will be the main goal of the following), andintersects C at a single point (which will be easy), under the assumption of Conditions 1 and . omputational Topology for Approximations of Knots 5 Figure 2.
Each normal disc intersects L at a single point2.1. Outline of the proof.
For an arbitrary t ∈ [0 , D r ( t ). Following is the sketch of proving that D r ( t )intersects L at a single point. (See Figure 2.)(1) The essential initial steps are to select a non-vertex point of L , denotedas w , a plane, denoted as Ω , and an angle, denoted as θ ( t ) : (a) Define w and Ω: Pick a line segment of L whose slope is equal to L (cid:48) ( t ), denoted as (cid:126) . Choose an interior point of (cid:126) , denoted as w . Let Ω be the plane that contains w and is parallel to D r ( t ).(We use w to define two sub-curves of L , a ‘left’ sub-curve whichterminates at w , denoted as L l , and a ‘right’ sub-curve whichbegins at w , denoted as L r .)(b) Consider η ( C (cid:48) ( t ) , L (cid:48) ( t )) = θ ( t ). Since Ω is parallel to D r ( t ),a normal vector of Ω, denoted by (cid:126)n Ω has the same direction as C (cid:48) ( t ) and η ( (cid:126)n Ω , (cid:126) ) = η ( C (cid:48) ( t ) , L (cid:48) ( t )) = θ ( t ). Figure 3.
Similar angles θ ( t ) Remark:
Since η ( (cid:126)n Ω , (cid:126) ) = θ ( t ), Condition 2 implies that T κ ( L ) + η ( (cid:126)n Ω , (cid:126) ) = T κ ( L ) + θ ( t ) < π . Since w is an interior point of (cid:126) , the an-gle determined by Ω and L l , and the angle determined by Ω and L r , havethe same measure θ ( t ), as shown in Figure 3. So we obtain the similar in-equalities T κ ( L l ) + θ ( t ) < π and T κ ( L r ) + θ ( t ) < π , which will be crucial. (2) Prove, by Condition 2 , that Ω ∩ L r = w . Similarly, show that Ω ∩ L l = w . So Ω ∩ L = w . (Lemma 2.4)(3) Prove that any plane parallel to Ω intersects L at no more than a singlepoint. (Lemma 2.5)(4) Since D r ( t ) (cid:107) Ω, it will follow that D r ( t ) intersects L no more thana single point. Show, using Condition 1 , that D r ( t ) must intersect L ,and hence D r ( t ) ∩ L is a single point. (Lemma 2.6)2.2. Preliminary lemmas for homeomorphisms.
In order to work withtotal curvatures of
P L curves, an extension of the spherical triangle inequality[24], given in Lemma 2.3, will be useful, similar to previous usage by Milnor[18].
Figure 4.
Spherical triangle (cid:52)
ABC
Spherical triangle inequalities:
Consider Figure 4, and the three angles ∠ AOB , ∠ BOC , and ∠ AOC , formed by three unit vectors −→ OA , −−→ OB , and −−→ OC .(Note the common end point O . When we consider angles between vectors thatdo not share such a common end point, we move the vectors to form a commonend point.) Denote the arc length of the curve from A to B as (cid:96) ( (cid:100) AB ), andsimilarly for that from B to C as (cid:96) ( (cid:100) BC ) and that from A to C as (cid:96) ( (cid:100) AC ). Thetriangle inequality, (cid:96) ( (cid:100) AB ) ≤ (cid:96) ( (cid:100) BC ) + (cid:96) ( (cid:100) AC ), of the spherical triangle (cid:52) ABC provides that ∠ AOB ≤ ∠ BOC + ∠ AOC. (2.3) omputational Topology for Approximations of Knots 7
Lemma 2.3.
Suppose that (cid:126)v , (cid:126)v , . . . , (cid:126)v m , where m ∈ { , , . . . } , are nonzerovectors, then η ( (cid:126)v , (cid:126)v m ) ≤ η ( (cid:126)v , (cid:126)v ) + η ( (cid:126)v , (cid:126)v ) , + . . . , + η ( (cid:126)v m − , (cid:126)v m ) . (2.4) Proof.
The proof follows easily from Inequality 2.3. (cid:3)
Now, we adopt the notation shown in Figure 2 and formalize the proofoutlined in Section 2.1. We assume that the sub-curve on the right hand sideof Ω in Figure 3 is L r , and the other one is L l , where we denote the set ofordered vertices of L r as { v , v , . . . , v n } , with v = w .We have θ ( t ) ≤ max t ∈ [0 , θ ( t ). It is trivially true that T κ ( L r ) ≤ T κ ( L ), sothat with Condition 2: T κ ( L ) + max t ∈ [0 , θ ( t ) < π , we have T κ ( L r ) + θ ( t ) ≤ T κ ( L ) + max t ∈ [0 , θ ( t ) < π . (2.5)The statement and proof of Lemma 2.4 depend upon the point w chosen inStep 1 of the Outline presented in Section 2.1. There, the point w was definedas an interior point of a line segment (cid:126) of L , so that w is precluded from beinga vertex of the original PL curve L . Lemma 2.4.
The plane Ω intersects L only at the single point w .Proof. Here we prove Ω ∩ L r = w . A similar argument will show Ω ∩ L l = w .The oriented initial line segment of L r is −−→ wv which lies on (cid:126) . So η ( (cid:126)n Ω , −−→ wv ) = η ( (cid:126)n Ω , (cid:126) ) = θ ( t ) < π . For a proof by contradiction, assume that Ω intersects L r at some point u other than w . The possibility that −−→ wv ⊂ Ω is precluded by θ ( t ) < π/
2, sothe plane Ω intersects −−→ wv only at w . So u / ∈ −−→ wv .Denote the sub-curve of L r from w to u as L ( wu ). Then, since u / ∈ −−→ wv , theunion, L ( wu ) ∪ −→ uw , forms a closed P L curve, as Figure 5 shows. By Fenchel’stheorem we have T κ ( L ( wu ) ∪ −→ uw ) ≥ π. (2.6)Denote the exterior angle of the P L curve L ( wu ) ∪ −→ uw at w as α (Figure 5),that is, α = η ( −→ uw, −−→ wv ) . By Inequality 2.3, α = η ( −→ uw, −−→ wv ) ≤ η ( −→ uw, (cid:126)n Ω ) + η ( (cid:126)n Ω , −−→ wv ) . Since −→ uw ⊂ Ω, we have that η ( −→ uw, (cid:126)n Ω ) = π . Note also that η ( (cid:126)n Ω , −−→ wv ) = θ ( t ).So α ≤ π θ ( t ) . Figure 5.
The intersection u generates a closed P L curveDenote the exterior angle of the
P L curve L ( wu ) ∪ −→ uw at u as α . By thedefinition of exterior angles, we have α ≤ π , so that T κ ( L ( wu ) ∪ −→ uw ) = α + T κ ( L ( wu )) + α ≤ π θ ( t ) + T κ ( L ( wu )) + π. It follows from Inequality 2.6 that π θ ( t ) + T κ ( L ( wu )) + π ≥ π, so T κ ( L ( wu )) + θ ( t ) ≥ π . (2.7)By L ( wu ) ⊂ L r , we have T κ ( L r ) + θ ( t ) ≥ T κ ( L ( wu )) + θ ( t ) ≥ π . But this contradicts Inequality 2.5. (cid:3)
Lemma 2.5.
Any plane parallel to Ω intersects L at no more than a singlepoint.Proof. Suppose ˜Ω is a plane parallel to Ω. If ˜Ω ∩ L = ∅ , then we are done,so we assume that ˜Ω ∩ L (cid:54) = ∅ . If ˜Ω = Ω, then Lemma 2.4 applies, so we alsoassume that ˜Ω (cid:54) = Ω, implying that w / ∈ ˜Ω.Consider two closed half-spaces H l and H r such that H l ∪ H r = R and H l ∩ H r = Ω. Since Ω ∩ L l = Ω ∩ L r = w and L = L l ∪ L r is simple, we canassume that L l ⊂ H l and L r ⊂ H r .Suppose without loss of generality that ˜Ω ⊂ H r , as shown in Figure 6. Thensince L l ⊂ H l and H l ∩ H r = Ω (cid:54) = ˜Ω, we have ˜Ω ∩ L l = ∅ . Since we assumed˜Ω ∩ L (cid:54) = ∅ , it follows that ˜Ω ∩ L r (cid:54) = ∅ . Now, it suffices to show that ˜Ω ∩ L r isa single point. omputational Topology for Approximations of Knots 9 Since L r is compact and oriented, let ˜ w denote the first point of L r , at which˜Ω intersects L r . Since ˜Ω (cid:107) Ω and ˜Ω (cid:54) = Ω, we have ˜ w (cid:54) = w . We shall show that˜Ω ∩ L r = ˜ w . Figure 6.
A parallel plane intersecting L Denote the sub-curve of L r from its initial point v to ˜ w as K , and thesub-curve from ˜ w to its end point v n as K , as shown in Figure 6. Since ˜ w isthe first intersection point of ˜Ω ∩ L r , but K ends in ˜ w , then it is clear that˜Ω ∩ K contains only ˜ w . Then in order to show ˜Ω ∩ L r = ˜ w , it suffices to showthat ˜Ω ∩ K = ˜ w .If ˜ w = v n , then it is the degenerate case: K = ˜ w , and we are done.Otherwise, there is a vertex v k for some k ∈ { , . . . , n } such that −−→ ˜ wv k is thenon-degenerate initial segment of K , where ˜ w (cid:54) = v k . Now we shall establishthe inequality: T κ ( K ) + η ( (cid:126)n Ω , −−→ ˜ wv k ) < π , to guarantee a single point of intersection, similar to arguments previouslygiven in Lemma 2.4. To this end, we use Inequality 2.3 to note that η ( (cid:126)n Ω , −−→ ˜ wv k ) ≤ η ( (cid:126)n Ω , −−→ v v ) + η ( −−→ v v , −−→ ˜ wv k )(2.8) = θ ( t ) + η ( −−→ v v , −−→ ˜ wv k ) . (2.9)The proof will be completed if we can show that T κ ( K ) + θ ( t ) + η ( −−→ v v , −−→ ˜ wv k ) < π . (2.10) Case1:
The intersection ˜ w is not a vertex, that is, ˜ w (cid:54) = v k − . Then˜ w is an interior point of −−−−→ v k − v k , and hence T κ ( K ) = η ( −−→ v v , −−→ v v ) + . . . + η ( −−−−−−→ v k − v k − , −−−−→ v k − ˜ w ), and η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) = 0. By Lemma 2.3, η ( −−→ v v , −−→ ˜ wv k ) ≤ η ( −−→ v v , −−→ v v ) + . . . + η ( −−−−−−→ v k − v k − , −−−−→ v k − ˜ w ) + η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) = T κ ( K ) . So T κ ( K ) + θ ( t ) + η ( −−→ v v , −−→ ˜ wv k ) ≤ T κ ( K ) + θ ( t ) + T κ ( K ) . We also have T κ ( L r ) = T κ ( K ) + η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) + T κ ( K ) = T κ ( K ) + T κ ( K ) , (since η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) = 0), so that T κ ( K ) + θ ( t ) + η ( −−→ v v , −−→ ˜ wv k ) ≤ T κ ( L r ) + θ ( t ) , which is less than π , by Inequality 2.5. Case2:
The intersection ˜ w is a vertex, that is, ˜ w = v k − , then T κ ( K ) = η ( −−→ v v , −−→ v v ) + . . . + η ( −−−−−−→ v k − v k − , −−−−→ v k − ˜ w ). By Lemma 2.3, η ( −−→ v v , −−→ ˜ wv k ) ≤ η ( −−→ v v , −−→ v v ) + . . . + η ( −−−−−−→ v k − v k − , −−−−→ v k − ˜ w ) + η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) ≤ T κ ( K ) + η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) . So T κ ( K ) + θ ( t ) + η ( −−→ v v , −−→ ˜ wv k ) ≤ T κ ( K ) + θ ( t ) + T κ ( K ) + η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) . But by the definition of the total curvature for a
P L curve, T κ ( K ) + T κ ( K ) + η ( −−−−→ v k − ˜ w, −−→ ˜ wv k ) = T κ ( L r ) . So T κ ( K ) + θ ( t ) + η ( −−→ v v , −−→ ˜ wv k ) ≤ T κ ( L r ) + θ ( t ) , which is less than π , by Inequality 2.5.So Inequality 2.10 holds, which is an inequality analogous to Inequality 2.5.If in the proof of Lemma 2.4, we change Ω to ˜Ω, L r to K and θ ( t ) to η ( (cid:126)n Ω , −−→ ˜ wv k ), then a similar proof of Lemma 2.4 will show that ˜Ω ∩ K = ˜ w .This completes the proof. (cid:3) Lemma 2.6.
For an arbitrary t ∈ [0 , , the disc D r ( t ) intersects C at aunique point, and also intersects L at a unique point.Proof. First, we have C ( t ) ∈ D r ( t ) ∩ C . If there is an additional point, say C ( t ) ∈ D r ( t ) ∩ C where t (cid:54) = t , then we have that C ( t ) (cid:54) = C ( t ) because C is simple, and hence D ( t ) (cid:54) = D ( t ). Since also C ( t ) ∈ D r ( t ), we have that C ( t ) ∈ D r ( t ) ∩ D r ( t ). But this contradicts the non-self-intersection of Γ. So D r ( t ) ∩ C must be a unique point.Now, we show that D r ( t ) ∩ L (cid:54) = ∅ . If t = 0 or t = 1, then since L (0) ∈ D r (0) and L (1) ∈ D r (1), we have that D r ( t ) ∩ L (cid:54) = ∅ .Otherwise if t ∈ (0 , D r ( t ) ∩ L = ∅ . Since L ⊂ Γ by
Condition 1 , the contrary assumption implies that L ⊂ Γ \ D r ( t ). Because C is an open curve, we have that D r (0) (cid:54) = D r (1). So omputational Topology for Approximations of Knots 11 Γ \ D r ( t ) consists of two disconnected components, but this implies that L isdisconnected, which is a contradiction. So D r ( t ) ∩ L (cid:54) = ∅ . (2.11)Since D r ( t ) (cid:107) Ω (as discussed in Section 2.1), Lemma 2.5 implies that theplane containing D r ( t ) intersects L at no more than a single point, which, ofcourse, further implies that D r ( t ) intersects L at no more than a single point.This plus Inequality 2.11 shows that D r ( t ) ∩ L is a single point.If D r ( t ) ∩ L = D r ( t ) ∩ L for some t (cid:54) = t , then D r ( t ) and D r ( t ) intersect,which contradicts the non-self-intersection of Γ. So there is an one-to-onecorrespondence between the parameter t and the point D r ( t ) ∩ L for t ∈ [0 , (cid:3) Lemma 2.7.
The map ˜ L ( t ) given by Equation 2.1 is well defined, one-to-oneand onto.Proof. It is well defined by Lemma 2.6. Suppose ˜ L ( t ) = ˜ L ( t ), then D r ( t ) ∩ L = D r ( t ) ∩ L which is not empty by Lemma 2.6. So D r ( t ) ∩ D r ( t ) (cid:54) = ∅ .Since Γ is nonsingular, it follows that D r ( t ) = D r ( t ). Since C is simple, if D r ( t ) = D r ( t ), then t = t . Thus ˜ L is one-to-one. Since L ⊂ Γ, each pointof L is contained in some disc D r ( t ). So ˜ L is onto. (cid:3) Lemma 2.8.
The map ˜ L ( t ) given by Equation 2.1 is continuous.Proof. Let Γ t t be the portion of Γ corresponding to [ t , t ], that isΓ t t = (cid:91) t ∈ [ t ,t ] D r ( t ) . Suppose that s ∈ [0 ,
1] is an arbitrary parameter. Then by Lemma 2.7, thereis a unique point q ∈ L such that q = ˜ L ( s ) = D r ( s ) ∩ L . We shall prove thecontinuity of ˜ L ( t ) at s by the definition, that is, for ∀ (cid:15) >
0, there exists a δ > | t − s | < δ implies || ˜ L ( t ) − ˜ L ( s ) || < (cid:15) .Note that D r ( s ) divides Γ into Γ s and Γ s . Since C is an open curve, itfollows that D r (0) (cid:54) = D r (1), and that Γ s and Γ s intersect at only D r ( s ).By Lemma 2.6, D r ( s ) ∩ L is a single point, so L is divided by D r ( s ) into twosub-curves, denoted as K and K , that is K ⊂ Γ s and K ⊂ Γ s , as shownin Figure 7. Case1:
The parameter s is such that s (cid:54) = 0 and s (cid:54) = 1. Consider Γ s first.Since K is oriented, we can let v be the first vertex of K that is nearest (indistance along K ) to q . For any 0 < (cid:15) < || qv || , let q (cid:48) ∈ qv such that || qq (cid:48) || = (cid:15) .By Lemma 2.7, q (cid:48) = ˜ L ( τ ) = D r ( τ ) ∩ L for some τ ∈ ( s, qq (cid:48) ∩ intΓ sτ (cid:54) = ∅ . To verify this, observe qq (cid:48) ⊂ qv ⊂ K ⊂ Γ s and Γ s = Γ sτ ∪ Γ τ , so qq (cid:48) ⊂ Γ sτ ∪ Γ τ . If qq (cid:48) ∩ intΓ sτ = ∅ , then thesegment qq (cid:48) is contained in D r ( s ) ∪ Γ τ which is disconnected. This implies qq (cid:48) is disconnected, which is a contradiction. Figure 7. If | s − τ | < δ , then || q − q (cid:48) || < (cid:15) Secondly, note that the subset Γ sτ of a nonsingular pipe section is connected(since C is C ), and qq (cid:48) is a line segment jointing the end discs of Γ sτ , and hasintersections with interior of Γ sτ . This geometry implies that qq (cid:48) ⊂ Γ sτ . (2.12)Let δ = τ − s . For an arbitrary t ∈ ( s, s + δ ) = ( s, τ ), Inclusion 2.12 impliesthat ˜ L ( t ) = D r ( t ) ∩ qq (cid:48) . Since neither ˜ L ( t ) (cid:54) = q or ˜ L ( t ) (cid:54) = q (cid:48) , it follows that˜ L ( t ) ∈ int( qq (cid:48) ). So || ˜ L ( t ) − ˜ L ( s ) || < || qq (cid:48) || = (cid:15). This shows the right-continuity. We similarly consider the Γ s and obtain theleft-continuity. Case2:
The parameter s is such that s = 0 or s = 1. We similarly obtainthe right-continuity if s = 0, or the left-continuity if s = 1. (cid:3) Theorem 2.9. If L and C satisfy Conditions 1 and
2, then the map h definedby Equation 2.2 is a homeomorphism.Proof. By Lemma 2.7, ˜ L ( t ) is one-to-one and onto. By Lemma 2.8, ˜ L ( t ) iscontinuous. Since ˜ L is defined on a compact domain, it is a homeomorphism.Note that C is simple and open, so C ( t ) is one-to-one, and it is obviouslyonto. The map C ( t ) is also continuous and defined on a compact domain,so C ( t ) a homeomorphism. Since h is a composition of C − and ˜ L , h is ahomeomorphism. (cid:3) Remark 2.10.
A very natural way to define a homeomorphism between simplecurves C and L would be by f ( p ) = L ( C − ( p )). An easy method to extend f to a homotopy is the straight-line homotopy. However, we were not able toestablish that a straight-line homotopy based upon f would also be an isotopy,where it would be necessary to show that each pair of line segments generated omputational Topology for Approximations of Knots 13 is disjoint. Our definition of h in Equation 2.2 was strategically chosen so thatthis isotopy criterion is easily established, since the normal discs are alreadypairwise disjoint.3. Construction of Ambient Isotopies
Note that L and C fit inside a nonsingular pipe section Γ of C . For a similarproblem, an explicit construction has appeared [17, Section 4.4] [9]. The proofof Lemma 3.3, below, is a simpler version of a previous proof [9, Corollary 4].The construction here relies upon some basic properties of convex sets, whichare repeated here. For clarity, the complete proof of Lemma 3.3 is given here. (a) Rays outward (b) Variant of a push Figure 8.
Convex subsetThe Images in Figures 8(a) and 8(b) were created by L. E. Miller and areused, here, with permission.
Lemma 3.1. [9, Lemma 6]
Let A be a compact convex subset of R with non-empty interior. For each point p ∈ int ( A ) and b ∈ ∂A , the ray going from p to b only intersects ∂A at b (See Figure 8(a).) Lemma 3.2. [9, Lemma 7]
Let A be a compact convex subset of R with non-empty interior and fix p ∈ int ( A ) . For each boundary point b ∈ ∂A , denote by [ p, b ] the line segment from p to b . Then A = (cid:83) b ∈ ∂A [ p, b ] . Lemma 3.3.
There is an ambient isotropy between L and C with compactsupport of Γ , leaving ∂ Γ fixed.Proof. We consider each normal disc D r ( t ) for t ∈ [0 , p = D r ( t ) ∩ C and q = h ( p ) with h defined by Equation 2.2, then define a map F p,q : D r ( t ) → D r ( t ) such that it sends each line segment [ p, b ] for b ∈ ∂D r ( t ), linearly onto theline segment [ q, b ] as Figure 8(b) shows. The two previous lemmas (Lemma 3.1and Lemma 3.2), will yield that F p,q is a homeomorphism, leaving ∂D r ( t ) fixed[17, Lemma 4.4.6].In order to extend F p,q to an ambient isotopy, define H : D r ( t ) × [0 , → D r ( t ) [17, Corollary 4.4.7] by H ( v, s ) = (cid:26) (1 − s ) p + sq if v = pF p, (1 − s ) p + sq ( v ) if v (cid:54) = p, where F p, (1 − s ) p + sq is a map on D r ( t ) analogous to F p,q , sending each linesegment [ p, b ] for b ∈ ∂D r ( t ), linearly onto the line segment [(1 − s ) p + sq, b ].It is a routine [17, Corollary 4.4.7] to verify that H ( v, s ) is well defined onthe compact set D r ( t ), continuous, one-to-one and onto, leaving ∂D r ( t ) fixed.Now, we naturally define an ambient isotopy T t : R × [0 , → R on the planecontaining D r ( t ) by T t ( v, s ) = (cid:26) H ( v, s ) if v ∈ D r ( t ) v otherwise . We then define T : R × [0 , → R by T ( v, s ) = (cid:26) T t ( v, s ) if v ∈ D r ( t ) v otherwise . The fact that the normal discs D r ( t ) are disjoint ensures that T is an ambientisotopy [17, Corollary 4.4.8], with compact support of Γ, leaving ∂ Γ fixed. (cid:3) Ambient Isotopy for B´ezier Curves
Now we apply this result to a simple, regular, composite, C B´ezier curve B and the control polygon P .4.1. Ambient isotopy.
There exists a nonsingular pipe surface [15] of radius r for B , denoted as S B ( r ). Denote the nonsingular pipe section determinedby S B ( r ) as Γ B . Also, for each sub-control polygon of B , there exists a cor-responding nonsingular pipe sections. Denote the nonsingular pipe sectioncorresponding to the k th control polygon as Γ k . Theorem 4.1.
Each sub-control polygon P k of a B´ezier curve B will eventuallysatisfy Conditions 1 and B and P with compact support of Γ B , leaving ∂ Γ B fixed.Proof. By Lemma 2.1,
Conditions 1 and can be achieved by subdivisions.Now consider each sub-control polygon P k satisfying Conditions 1 and , andthe corresponding B´ezier sub-curves B k . Use Lemma 3.3 to define an ambientisotopy Ψ k : R × [0 , → R between B k and P k , for each k ∈ { , , . . . , i } .Define Ψ : R × [0 , → R by the compositionΨ = Ψ ◦ Ψ ◦ . . . ◦ Ψ i . Note that Ψ k fixes the complement of int(Γ k ), and int(Γ k ) ∩ int(Γ k (cid:48) ) = ∅ for all k (cid:54) = k (cid:48) . So the composition Ψ is well defined. Since each Ψ k is anambient isotopy, the composition Ψ is an ambient isotopy between B and P with compact support of Γ B , leaving ∂ Γ B fixed. (cid:3) omputational Topology for Approximations of Knots 15 Sufficient subdivision iterations.
Now we consider sufficient numbersof subdivision iterations to achieve the ambient isotopy defined by Theorem 4.1,that is, we shall have a control polygon that satisfies
Conditions 1 and . Thenumber of subdivisions for Condition 1 is given in the paper [13, Lemma 6.3].To obtain the number of subdivisions for
Condition 2 , we consider the following,for which we let P (cid:48) ( t ) = l (cid:48) ( P, i )( t ) (the first derivative of the control polygon P ), and denote the angle between B (cid:48) ( t ) and P (cid:48) ( t ) as θ ( t ), for t ∈ [0 , Lemma 4.2. [13, Theorem 6.1]
For any < ν < π , there is an integer N ( ν ) such that each exterior angle of P will be less than ν after N ( ν ) subdivisions,where N ( ν ) = (cid:100) max { N , log( f ( ν )) }(cid:101) , (4.1) N = 12 log( N ∞ ( n − (cid:107) (cid:52) P (cid:48) (cid:107) σ ) , and f ( ν ) = 2 M (1 − cos( ν ))( σ − B (cid:48) dist ( N )) . Note that for a B´ezier curve of degree n , there are n − P k . To have T κ ( P k ) < π , it suffices to makeeach exterior angle smaller than π n − . By Lemma 4.2, this can be gained by N ( π n − ) subdivisions. Condition 2 is motivated by the weaker condition T κ ( P k ) < π . We couldn’tderive the same results by using this weaker condition instead, but our Condi-tion 2 requires at most one more subdivision, as shown below.
Lemma 4.3.
Condition 2 will be fulfilled by at most N ( π n − )+1 subdivisions.Proof. To prove T κ ( P k ) + max t ∈ [0 , θ ( t ) < π , it suffices to prove T κ ( P k ) < π t ∈ [0 , θ ( t ) < π . To have T κ ( P k ) < π , by Lemma 4.2, N ( π n − ) subdivisions will be sufficient.The definition given by Equation 4.1 implies that N ( π n −
1) ) ≤ N ( π n −
1) ) + 1 . On the other hand, by [13, Section 6.3], for all t ∈ [0 , − cos( θ ( t )) ≤ B (cid:48) dist ( i ) σ , where B (cid:48) dist ( i ) := 12 i N ∞ ( n − || ∆ P (cid:48) || . So to have max t ∈ [0 , θ ( t ) < π , it suffices to set2 B (cid:48) dist ( i ) σ < π , which implies i ≥
12 log( N ∞ ( n − || ∆ P (cid:48) || σ ) + 1 . Comparing it with Equation 4.1 , we find that it is at most one more than N ( ν ) for any 0 < ν < π . The conclusion follows. (cid:3) Let N (cid:63) = max { N ( π n −
1) ) + 1 , N (cid:48) ( r ) } , (4.2)where r is the radius of S r ( B ). Theorem 4.4.
Performing N (cid:63) or more subdivisions, where N (cid:63) is given byEquation 4.2, will produce an ambient isotopic P for B .Proof. According to [13, Lemma 6.3],
Condition 1 is satisfied after N (cid:48) ( r ) sub-divisions. By Lemma 4.3, Condition 2 is satisfied after N ( π n +1) ) + 1 subdivi-sions. Then Theorem 4.1 can be applied to draw the conclusion. (cid:3) Now we compare this result with the existing one [13].
Remark 4.5.
To obtain ambient isotopy, the previously established result [13]needs max { N ( π n − ) , N (cid:48) ( r ) } + 2 subdivision iterations [13, Remark 6.1]. Incontrast, Theorem 4.4 implies max { N ( π n − ) , N (cid:48) ( r ) } + 1 will be sufficient. Asubdivision doubles the number of line segments. Therefore, with only one lesssubdivision, the work here produces much less line segments, which may beuseful especially for applications with very complex shapes.5. Conclusions
We give two conditions regarding distance, and total curvature combinedwith derivative, to guarantee the same knot type. It can be directly appliedto B´ezier curves. This work is alternative to an existence result of requiringthe containment of convex hulls of sub-control polygons, and another resultusing conditions of distance and derivative. The approach here allows fewersubdivision iterations and less line segments by explicitly constructing ambientisotopies. Moreover, we showed that it is possible to verify the condition oftotal curvature only, other than total curvature combined with derivative, witha price of one additional subdivision. Testing the global property of totalcurvature may be easier than testing the local property of derivative in somepractical situations. It may find applications in computer graphics, computeranimation and scientific visualization. omputational Topology for Approximations of Knots 17
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