Knot Experiments for Molecular Movies - Semantic Scholar
Knot Experiments for Molecular Movies J. Li, T. J. Peters, K. Marinelli∗, E. Kovalev†, K. E. Jordan‡ July 5, 2014
Abstract Computer graphics experiments led to examples where the graphical approximant is ambient isotopic to a spline, but a slight perturbation of the approximant destroys the isotopic equivalence. These serve as cautionary counterexamples for dynamic visualization of molecular simulations. These visual experiments facilitated the creation of a new operation on splines and formal proofs of its convergence.
Keywords: Knot, isotopy, computer animation, molecular simulation. 2000 MSC: 57Q37, 57Q55, 57M25, 68R10, 92E10.
1
Introduction
The backbone of a molecule can be represented as a spline curve. Computer rendering relies on a piecewise linear (PL) approximation, so that the graphics could mis-represent the embedding. In Figure 1, the spline is the trivial knot, yet the PL representation is the non-trivial knot 41 . These issues, shown here statically, become of significant concern for dynamic visualization.
Figure 1: Initial curves
∗
Department of Mathematics, University of Connecticut, Storrs, CT 06269. Department of Computer Science and Engineering, University of Connecticut, Storrs, CT 06269. ‡ IBM T.J. Watson Research Center, Cambridge, MA, U.S.A. †
1
The primary example presented in Section 4 lends caution regarding appropriate approximations for dynamic visualization [10, 14]. Here, the knot type of the PL curves remain unchanged under perturbation but the accompanying B´ezier curves are of different knot types. This example complements one previously published [5], where the perturbed PL structure becomes self-intersecting while the associated B´ezier curve remains simple. This example also complements [3], where sufficient conditions were given for perturbations of the control points to maintain isotopic equivalence of the splines, but no specific counterexamples were presented if the sufficient conditions were violated. The underlying intuition, as illustrated in Figure 1 is that often the initial control points are far away from the spline. So, it is not surprising that a spline and its control polygon could have different embeddings. To create the desired example, some subtlety was required. Having initial isotopy between the spline and its control polygon suggested a condition of them being close in Hausdorff distance. While it is well known that subdivision causes the control polygon to converge to the spline, this operation also could lead to changes in the knot type of the control polygon at each iteration, vastly complicating the analysis. A desirable alternative would be a method that maintained the knot type of the control polygon, while the spline converged to it. A newly introduced spline operation, called collinear insertion, is presented that meets those objectives. Computer graphics experiments were conducted to foster further intuition about this new operation, leading to formal proofs of its convergence properties and presentation of the example sought. It is shown that even small vertex perturbations can lead to incorrect knotting in the graphics display. Appreciating this subtlety is important to assess the pragmatic trade-offs of maintaining one PL approximation and perturbing it over multiple frames versus the more performance intensive approach of re-approximating the B´ezier curve for each frame. To create the primary example presented in Section 4, visual analysis of computer graphics was a fundamental, iterative tool. The resultant strategy was to move a single control point sufficiently far to change the knot type of the corresponding spline, but not moving it so far as to change the embedding of the defining PL control polygon, so that these results complement those already cited [5].
2
2
Related Work
The term ‘molecular movies’ includes “. . . molecular animations . . . ” [18]. The work [10] established sufficient conditions for preservation of knot type during dynamic visualization of ongoing molecular simulations, possibly requiring a new approximation for each iteration of the perturbed curve. These approximation algorithms are more intensive than just continuing to perturb the initial PL approximation, as is examined here. The preservation of topological characteristics in geometric modeling and graphics is of contemporary interest [1, 6, 9, 11, 13, 15]. The models here are 1-manifolds, represented by splines, specifically by B´ezier curves (Please see Section 3). Splines are defined by finitely many control points which become input to a standard PL approximation algorithm known as subdivision [21]. Sufficient conditions for a homeomorphism between a B´ezier curve and its control polygon have been studied [20], while topological differences have also been shown [4, 14, 21]. Motivated by applications for dynamic visualization in a high performance computing environment, the preservation of topological integrity during perturbations has been investigated [2, 10]. For example, when visualizing a molecule that is twisting and writhing under local chemical and kinetic changes, it is crucial that topological artifacts are not introduced by the visual approximations [10]. In particular, a perturbation on a control point of a B´ezier curve changes both the control polygon and the B´ezier curve. The paper [2] gives an upper bound on the perturbation of vertices of polyhedra to retain ambient isotopic equivalence. The paper [10, Proposition 5.2] provides a sufficient condition to ensure that both a control polygon and its B´ezier curve remain ambient isotopic during the perturbation.
3
Collinear Insertion
A fundamental intuition about spline curves is that the curve tends towards the control points. It follows that adding more collinear control points should draw the curve closer to those edges. Computer graphics experiments corroborated that intuition, as shown in Figure 2. Four points create the initial PL Figure 2: Collinear Insertions 3
curve shown. Using Hausdorff distance, the smooth curve that is farthest away is just the B´ezier curve created from those 4 initial points. The 3 additionally shown smooth curves are obtained from successive collinear insertions. A salient feature is that the PL embedding remains fixed under collinear insertion, facilitating further experiments and proofs. Here, this technique is called collinear insertion 1 , as formalized, below. Collinear insertion is similar to degree elevation [21] in that both methods produce higher degree B´ezier curves. In contrast to Figure 2, degree elevation changes the control polygon but preserves the B´ezier curve. Definition 3.1 [21] The examples here are parameterized B´ ezier curve, 3 denoted as B(t), of degree n with control points Pm ∈ R is defined by B(t) =
n X n m=0
m
tm (1 − t)n−m Pm , t ∈ [0, 1],
The P L curve determined by consecutive linear segments connecting the points {P0 , P1 , . . . , Pn } is called the control polygon. If P0 = Pn , then the polygon and the curve are closed. Otherwise, they are open. The operation of collinear insertion is easily adapted to other spline curves. The method of collinear insertion for a spline curve is now formalized. 1. For each line segment, add its midpoint2 as a new control point. 2. The process can be repeated k times, to obtain 2k ∗ n control points and a B´ezier curve of degree 2k ∗ n, while leaving the embedding of the control polygon fixed. Figure 3(a) shows a control polygon and its B´ezier curve, created by collinear insertion, as a modification of Figure 1. The PL curve more closely approximates the B´ezier curve and this example is subjected to further formal analyses.
4
The Defining Data for the Example
The 7 control points3 listed below are used. 1
This terminology is used in other fields [17], but with a very different meaning that should not cause confusion within this mathematical context. 2 For ease of exposition, the collinear points inserted are all midpoints. 3 At least four decimal digits are provided here, but higher accuracy is possible.
4
(1.3076, −3.3320, −2.5072), (−1.3841185, 4.6825505, 0.913541), (−3.2983075, −4.0566825, 2.686189), (−0.1232995, 2.768254, −2.463584), (3.9079915, −4.533357, 1.2263705), (−3.935983, −0.438272, −0.983365), (3.218174, 4.296123, 2.1124595).
Denote the ordered vertices of P1 (the original control points) as v0 , v1 , . . . , v6 . After 4 iterations of collinear insertion there are 112 control points for the B´ezier curve of degree 112 shown in Figure 3(a). The linear perturbation of one of of the original vertices is described below, as depicted in Figure 3(b). Denote the control polygon and the B´ezier curve in Figure 3(a) by P1 and B1 and it will be shown that both are ambient isotopic to 41 .
(a) Initial polygon and curve
(b) Perturbed image
Figure 3: Perturb v0 to v00 We fix v1 and v6 while translating v0 to (1.9817, −1.7646, −4.5897), denoted by v00 , as shown in Figure 3(b). Denote the resultant control polygon and B´ezier curve as P2 and B2 . The PL knots remain ambient isotopic, but B1 is changed from the knot 41 to an unknot B2 , as explained in Section 5.2).
5
Topological Comparisons
The following three Subsections show that • the two PL curves are ambient isotopic to the knot 41 , • the initial B´ezier curve is a nontrivial knot, • the perturbed B´ezier curve is the trivial knot. 5
5.1
Simple PL Curves that Are Ambient Isotopic Knots
Figure 4 shows a regular projection [16] of P1 . The conceptual overview is that the simplicity of P1 a necessary condition for P1 to be a knot, as can be shown by elementary calculations on each pair of line segments. Some efficiency is gained by bounding box tests or oriented line segments [8]. To ensure correctness, the calculations were performed independently by two of the co-authors4 . Figure 4 depicts 6 crossings, with 3 consecutive overcrossings. Two of these can be reFigure 4: Regular Projection moved by a Reidemeister move of Type 2b [16] in order to establish the knot type of 41 [16]. The simplicity of P2 could directly be proven by the same type of explicit calculations for the simplicity of P1 . However, the simplicity of P2 will also follow from showing that no self-intersections are introduced by the translation of v0 to v00 . This argument is completed in the rest of this paragraph, showing, more generally that the knot type 41 is preserved for all the instances between v0 and v00 . The previous pairwise line segment calculations left only three pairs for more detailed consideration. Two of those pairs arise from the movement of v0 v1 relative to intersect the unchanged segments v2 v3 and v6 v0 to intersect the unchanged v3 v4 . Since the movement to v00 is monotonically decreasing in z, both these are eliminated by examination. Consider the last pair of v0 v1 versus v5 v6 . Elementary geometric calculations find the point q that is the intersection of the plane formed by the three points v0 , v1 , v00 with the infinite line containing the two points v5 , v6 . Let T denote the triangle formed by the vertices v0 , v1 , v00 . Consider the projections of q and T onto the xy-plane, denoted by qxy and Txy , respectively. An easy sign test [22], shows that qxy lies in the open half plane that is disjoint from Txy , implying that q ∈ / T , which suffices to exclude this last case. 4
Some undergraduate students also verified the calculations.
6
5.2
Nontrivial Knot Occurrence
For the nontrivial knottedness of B1 , we start from a projection obtained by taking each z-coordinate to be zero, as shown in Figure 5(a). Working in MatLab, we used the simplex search method [12] implemented by the function ‘fminsearch’ to find pairs of parameters where the projected curve is self-intersecting. We then evaluated the curve by Horner’s method to find 4 crossings: (−0.13, 0.93, −0.69), (−0.13, 0.93, −1.29) & (−0.44, 1.82, −0.29), (−0.44, 1.82, 0.52) (−1.87, −1.0, 0.43), (−1.87, −1.0, −0.34) & (1.88, −1.06, −0.52), (1.88, −1.06, −1.13).
Visual inspection of the diagram knot diagram in Figure 5(a) leads to identification of the knot 41 . Additional corroboration is provided by noting the corresponding Alexander polynomial [16]to be 1 − 3t + t2 , where the annotations within Figure 5(a) provide ordered labels for the crossings and regions for calculation of the Alexander polynomial.
(a) Annotations for Alexander Polynomial
(b) Perturbed Unknot
Figure 5: Original and Perturbed B´ezier Curves
5.3
Resultant Trivial Knot
For the unknottedness of B2 , consider the projection shown in Figure 5(b), obtained by taking each z-coordinate to be zero, where the orientation is noted by the cursor and the numbers label the crossings consistently with those of Figure 5(a). Denote the projected B´ezier curve by C2d(t). Define a function f nS(t1 , t2 ) : [0, 1] × [0, 1] → R by f nS(t1 , t2 ) = ||C2d(t1 ) − C2d(t2 )||. 7
Again, the MatLab function ‘fminsearch’, is used to find the roots of f nS(t1 , t2 ), with t1 6= t2 , for self-intersections, where f nS(t1 , t2 ) is a polynomial of degree 112. The self-intersections are listed below and are labeled as ‘1, 2, 3’ and ‘4’ in Figure 5(b): (0.83, 0.44, −2.71), (0.83, 0.44, −1.21) & (−0.04, 2.09, −1.28), (−0.04, 2.09, 0.68); (−1.87, −1.0, 0.43), (−1.87, −1.0, −0.34) & (2.05, −1.41, −0.35), (2.05, −1.41, −4.19).
Note that the orientation of B2 is shown by arrows in Figure 5(b). By comparing the z-coordinates of points listed above, we find that ‘1’ and ‘2’ are under crossings, while ‘3’ and ‘4’ are over crossings. It follows that B2 is simple. The resultant diagram, after Reidemeister moves, has only one crossing, so that B2 is unknotted.
6
Convergence Theorem
For a spline curve, b, the well-known technique of subdivision [21] generates a sequence `i , i ∈ N of PL approximations of b such that the sequence `i convergences, under the Hausdorff distance, exponentially to b. The technique of collinear insertion could be considered a dual to subdivision, in that there is a given PL curve ` and a sequence of spline curves bi is generated. The visual experiments suggested the following conjecture, which is now proved. The sequence bi converges in distance to `. We show that collinear insertion causes the distance between the control polygon and the associated B´ezier curve to converge to zero, as the number of iterations becomes unbounded. A previously published theorem on splines [19, Theorem 4.1] is central to our proof, below. Let N denote the natural numbers {1, 2, 3, . . .}. For k ∈ N, we consider the central binomial coefficient [7], denoted as 2k . k Lemma 6.1 For k ∈ N,
2k k
4k ≤√ . 2k + 1
8
Proof: The proof is by induction, the relation is informally attributed to P. Erdos [7] . For k ∈ N , let
2k k
N1 (2k) =
2k . 22k+2
Lemma 6.1 For k ∈ N , k N1 (2k) < √ . 2 2k + 1 Proof: N1 (2k) =
2k k
2k 22k+2
Abstract Computer graphics experiments led to examples where the graphical approximant is ambient isotopic to a spline, but a slight perturbation of the approximant destroys the isotopic equivalence. These serve as cautionary counterexamples for dynamic visualization of molecular simulations. These visual experiments facilitated the creation of a new operation on splines and formal proofs of its convergence.
Keywords: Knot, isotopy, computer animation, molecular simulation. 2000 MSC: 57Q37, 57Q55, 57M25, 68R10, 92E10.
1
Introduction
The backbone of a molecule can be represented as a spline curve. Computer rendering relies on a piecewise linear (PL) approximation, so that the graphics could mis-represent the embedding. In Figure 1, the spline is the trivial knot, yet the PL representation is the non-trivial knot 41 . These issues, shown here statically, become of significant concern for dynamic visualization.
Figure 1: Initial curves
∗
Department of Mathematics, University of Connecticut, Storrs, CT 06269. Department of Computer Science and Engineering, University of Connecticut, Storrs, CT 06269. ‡ IBM T.J. Watson Research Center, Cambridge, MA, U.S.A. †
1
The primary example presented in Section 4 lends caution regarding appropriate approximations for dynamic visualization [10, 14]. Here, the knot type of the PL curves remain unchanged under perturbation but the accompanying B´ezier curves are of different knot types. This example complements one previously published [5], where the perturbed PL structure becomes self-intersecting while the associated B´ezier curve remains simple. This example also complements [3], where sufficient conditions were given for perturbations of the control points to maintain isotopic equivalence of the splines, but no specific counterexamples were presented if the sufficient conditions were violated. The underlying intuition, as illustrated in Figure 1 is that often the initial control points are far away from the spline. So, it is not surprising that a spline and its control polygon could have different embeddings. To create the desired example, some subtlety was required. Having initial isotopy between the spline and its control polygon suggested a condition of them being close in Hausdorff distance. While it is well known that subdivision causes the control polygon to converge to the spline, this operation also could lead to changes in the knot type of the control polygon at each iteration, vastly complicating the analysis. A desirable alternative would be a method that maintained the knot type of the control polygon, while the spline converged to it. A newly introduced spline operation, called collinear insertion, is presented that meets those objectives. Computer graphics experiments were conducted to foster further intuition about this new operation, leading to formal proofs of its convergence properties and presentation of the example sought. It is shown that even small vertex perturbations can lead to incorrect knotting in the graphics display. Appreciating this subtlety is important to assess the pragmatic trade-offs of maintaining one PL approximation and perturbing it over multiple frames versus the more performance intensive approach of re-approximating the B´ezier curve for each frame. To create the primary example presented in Section 4, visual analysis of computer graphics was a fundamental, iterative tool. The resultant strategy was to move a single control point sufficiently far to change the knot type of the corresponding spline, but not moving it so far as to change the embedding of the defining PL control polygon, so that these results complement those already cited [5].
2
2
Related Work
The term ‘molecular movies’ includes “. . . molecular animations . . . ” [18]. The work [10] established sufficient conditions for preservation of knot type during dynamic visualization of ongoing molecular simulations, possibly requiring a new approximation for each iteration of the perturbed curve. These approximation algorithms are more intensive than just continuing to perturb the initial PL approximation, as is examined here. The preservation of topological characteristics in geometric modeling and graphics is of contemporary interest [1, 6, 9, 11, 13, 15]. The models here are 1-manifolds, represented by splines, specifically by B´ezier curves (Please see Section 3). Splines are defined by finitely many control points which become input to a standard PL approximation algorithm known as subdivision [21]. Sufficient conditions for a homeomorphism between a B´ezier curve and its control polygon have been studied [20], while topological differences have also been shown [4, 14, 21]. Motivated by applications for dynamic visualization in a high performance computing environment, the preservation of topological integrity during perturbations has been investigated [2, 10]. For example, when visualizing a molecule that is twisting and writhing under local chemical and kinetic changes, it is crucial that topological artifacts are not introduced by the visual approximations [10]. In particular, a perturbation on a control point of a B´ezier curve changes both the control polygon and the B´ezier curve. The paper [2] gives an upper bound on the perturbation of vertices of polyhedra to retain ambient isotopic equivalence. The paper [10, Proposition 5.2] provides a sufficient condition to ensure that both a control polygon and its B´ezier curve remain ambient isotopic during the perturbation.
3
Collinear Insertion
A fundamental intuition about spline curves is that the curve tends towards the control points. It follows that adding more collinear control points should draw the curve closer to those edges. Computer graphics experiments corroborated that intuition, as shown in Figure 2. Four points create the initial PL Figure 2: Collinear Insertions 3
curve shown. Using Hausdorff distance, the smooth curve that is farthest away is just the B´ezier curve created from those 4 initial points. The 3 additionally shown smooth curves are obtained from successive collinear insertions. A salient feature is that the PL embedding remains fixed under collinear insertion, facilitating further experiments and proofs. Here, this technique is called collinear insertion 1 , as formalized, below. Collinear insertion is similar to degree elevation [21] in that both methods produce higher degree B´ezier curves. In contrast to Figure 2, degree elevation changes the control polygon but preserves the B´ezier curve. Definition 3.1 [21] The examples here are parameterized B´ ezier curve, 3 denoted as B(t), of degree n with control points Pm ∈ R is defined by B(t) =
n X n m=0
m
tm (1 − t)n−m Pm , t ∈ [0, 1],
The P L curve determined by consecutive linear segments connecting the points {P0 , P1 , . . . , Pn } is called the control polygon. If P0 = Pn , then the polygon and the curve are closed. Otherwise, they are open. The operation of collinear insertion is easily adapted to other spline curves. The method of collinear insertion for a spline curve is now formalized. 1. For each line segment, add its midpoint2 as a new control point. 2. The process can be repeated k times, to obtain 2k ∗ n control points and a B´ezier curve of degree 2k ∗ n, while leaving the embedding of the control polygon fixed. Figure 3(a) shows a control polygon and its B´ezier curve, created by collinear insertion, as a modification of Figure 1. The PL curve more closely approximates the B´ezier curve and this example is subjected to further formal analyses.
4
The Defining Data for the Example
The 7 control points3 listed below are used. 1
This terminology is used in other fields [17], but with a very different meaning that should not cause confusion within this mathematical context. 2 For ease of exposition, the collinear points inserted are all midpoints. 3 At least four decimal digits are provided here, but higher accuracy is possible.
4
(1.3076, −3.3320, −2.5072), (−1.3841185, 4.6825505, 0.913541), (−3.2983075, −4.0566825, 2.686189), (−0.1232995, 2.768254, −2.463584), (3.9079915, −4.533357, 1.2263705), (−3.935983, −0.438272, −0.983365), (3.218174, 4.296123, 2.1124595).
Denote the ordered vertices of P1 (the original control points) as v0 , v1 , . . . , v6 . After 4 iterations of collinear insertion there are 112 control points for the B´ezier curve of degree 112 shown in Figure 3(a). The linear perturbation of one of of the original vertices is described below, as depicted in Figure 3(b). Denote the control polygon and the B´ezier curve in Figure 3(a) by P1 and B1 and it will be shown that both are ambient isotopic to 41 .
(a) Initial polygon and curve
(b) Perturbed image
Figure 3: Perturb v0 to v00 We fix v1 and v6 while translating v0 to (1.9817, −1.7646, −4.5897), denoted by v00 , as shown in Figure 3(b). Denote the resultant control polygon and B´ezier curve as P2 and B2 . The PL knots remain ambient isotopic, but B1 is changed from the knot 41 to an unknot B2 , as explained in Section 5.2).
5
Topological Comparisons
The following three Subsections show that • the two PL curves are ambient isotopic to the knot 41 , • the initial B´ezier curve is a nontrivial knot, • the perturbed B´ezier curve is the trivial knot. 5
5.1
Simple PL Curves that Are Ambient Isotopic Knots
Figure 4 shows a regular projection [16] of P1 . The conceptual overview is that the simplicity of P1 a necessary condition for P1 to be a knot, as can be shown by elementary calculations on each pair of line segments. Some efficiency is gained by bounding box tests or oriented line segments [8]. To ensure correctness, the calculations were performed independently by two of the co-authors4 . Figure 4 depicts 6 crossings, with 3 consecutive overcrossings. Two of these can be reFigure 4: Regular Projection moved by a Reidemeister move of Type 2b [16] in order to establish the knot type of 41 [16]. The simplicity of P2 could directly be proven by the same type of explicit calculations for the simplicity of P1 . However, the simplicity of P2 will also follow from showing that no self-intersections are introduced by the translation of v0 to v00 . This argument is completed in the rest of this paragraph, showing, more generally that the knot type 41 is preserved for all the instances between v0 and v00 . The previous pairwise line segment calculations left only three pairs for more detailed consideration. Two of those pairs arise from the movement of v0 v1 relative to intersect the unchanged segments v2 v3 and v6 v0 to intersect the unchanged v3 v4 . Since the movement to v00 is monotonically decreasing in z, both these are eliminated by examination. Consider the last pair of v0 v1 versus v5 v6 . Elementary geometric calculations find the point q that is the intersection of the plane formed by the three points v0 , v1 , v00 with the infinite line containing the two points v5 , v6 . Let T denote the triangle formed by the vertices v0 , v1 , v00 . Consider the projections of q and T onto the xy-plane, denoted by qxy and Txy , respectively. An easy sign test [22], shows that qxy lies in the open half plane that is disjoint from Txy , implying that q ∈ / T , which suffices to exclude this last case. 4
Some undergraduate students also verified the calculations.
6
5.2
Nontrivial Knot Occurrence
For the nontrivial knottedness of B1 , we start from a projection obtained by taking each z-coordinate to be zero, as shown in Figure 5(a). Working in MatLab, we used the simplex search method [12] implemented by the function ‘fminsearch’ to find pairs of parameters where the projected curve is self-intersecting. We then evaluated the curve by Horner’s method to find 4 crossings: (−0.13, 0.93, −0.69), (−0.13, 0.93, −1.29) & (−0.44, 1.82, −0.29), (−0.44, 1.82, 0.52) (−1.87, −1.0, 0.43), (−1.87, −1.0, −0.34) & (1.88, −1.06, −0.52), (1.88, −1.06, −1.13).
Visual inspection of the diagram knot diagram in Figure 5(a) leads to identification of the knot 41 . Additional corroboration is provided by noting the corresponding Alexander polynomial [16]to be 1 − 3t + t2 , where the annotations within Figure 5(a) provide ordered labels for the crossings and regions for calculation of the Alexander polynomial.
(a) Annotations for Alexander Polynomial
(b) Perturbed Unknot
Figure 5: Original and Perturbed B´ezier Curves
5.3
Resultant Trivial Knot
For the unknottedness of B2 , consider the projection shown in Figure 5(b), obtained by taking each z-coordinate to be zero, where the orientation is noted by the cursor and the numbers label the crossings consistently with those of Figure 5(a). Denote the projected B´ezier curve by C2d(t). Define a function f nS(t1 , t2 ) : [0, 1] × [0, 1] → R by f nS(t1 , t2 ) = ||C2d(t1 ) − C2d(t2 )||. 7
Again, the MatLab function ‘fminsearch’, is used to find the roots of f nS(t1 , t2 ), with t1 6= t2 , for self-intersections, where f nS(t1 , t2 ) is a polynomial of degree 112. The self-intersections are listed below and are labeled as ‘1, 2, 3’ and ‘4’ in Figure 5(b): (0.83, 0.44, −2.71), (0.83, 0.44, −1.21) & (−0.04, 2.09, −1.28), (−0.04, 2.09, 0.68); (−1.87, −1.0, 0.43), (−1.87, −1.0, −0.34) & (2.05, −1.41, −0.35), (2.05, −1.41, −4.19).
Note that the orientation of B2 is shown by arrows in Figure 5(b). By comparing the z-coordinates of points listed above, we find that ‘1’ and ‘2’ are under crossings, while ‘3’ and ‘4’ are over crossings. It follows that B2 is simple. The resultant diagram, after Reidemeister moves, has only one crossing, so that B2 is unknotted.
6
Convergence Theorem
For a spline curve, b, the well-known technique of subdivision [21] generates a sequence `i , i ∈ N of PL approximations of b such that the sequence `i convergences, under the Hausdorff distance, exponentially to b. The technique of collinear insertion could be considered a dual to subdivision, in that there is a given PL curve ` and a sequence of spline curves bi is generated. The visual experiments suggested the following conjecture, which is now proved. The sequence bi converges in distance to `. We show that collinear insertion causes the distance between the control polygon and the associated B´ezier curve to converge to zero, as the number of iterations becomes unbounded. A previously published theorem on splines [19, Theorem 4.1] is central to our proof, below. Let N denote the natural numbers {1, 2, 3, . . .}. For k ∈ N, we consider the central binomial coefficient [7], denoted as 2k . k Lemma 6.1 For k ∈ N,
2k k
4k ≤√ . 2k + 1
8
Proof: The proof is by induction, the relation is informally attributed to P. Erdos [7] . For k ∈ N , let
2k k
N1 (2k) =
2k . 22k+2
Lemma 6.1 For k ∈ N , k N1 (2k) < √ . 2 2k + 1 Proof: N1 (2k) =
2k k
2k 22k+2
