More Accurate Representation of Conics by NURBS. One of the argument usually given to explain the popularity of NURBS is the fact that they

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1 More Accurate Representation of Conics by NURBS Carole Blanc Christophe Schlick LaBRI 1, 51 cours de la liberation, 405 Talence (FRANCE) 1 Introduction One of the argument usually given to explain the popularity of NURBS is the fact that they allow to dene free-form curves and surfaces (as almost every spline model) and provide also an exact representation of conic sections and thus of a large set of curves and surfaces that are intensively used in CAD : circular arcs, circles, cylinders, cones, spheres, surfaces of revolution, etc. Nevertheless, there is relatively few published work about the mathematical properties of the representation of conics by NURBS. To our knowledge, the most complete work has been done by Farin [2, ]. This paper does not pretend to ll this theoretical lack, it is simply intended to deal with the following problems : All the known representations by NURBS of curves and surfaces based on conics have only a C 1 continuity. Moreover, there does not exist any technique which would eventually allow to nd a parametrization with a higher level of continuity. The parametrization resulting from the representation of conics by NURBS can deviate signicantly from the ideal arc length parametrization. The only known solution to reduce this deviation is to increase the number of control points of the spline (using re- nement algorithms, for instance), but such a process converges quite slowly to the ideal parametrization. The solution that we propose in this paper uses an original reparametrization process, that we have called zigzag reparametrization, which is based on a particular family of rational polynomials. This techniques allows to get a higher order continuity as well as a more uniform parametrization. Its counterpart is that the degree of the NURBS is raised (in our case, from two to four). The remainder of the paper is divided into three main sections. Section 2 recalls the basis of the representation of conics by NURBS. Then, Section discusses about some weaknesses of this representation. Finally, Section 4 details the new reparametrization technique and shows how it can be used to correct the previous weaknesses. 2 Conics, Circles, Spheres and Splines 2.1 Conics The work of Moebius, at the beginning of the last century, has stated that a conic can be considered as the projection on IR 2 of a parabola dened in IR. Using that result, Forrest [5] has shown that a conic can also be expressed as a rational quadratic Bezier curve (and alternatively as a quadratic NURBS [11]). Such a curve is dened by three points P 0 ; P 1 ; P 2 and three coecients w 0 ; w 1 ; w 2 : 1 Laboratoire Bordelais de Recherche en Informatique (Universite Bordeaux I and Centre National de la Recherche Scientique). The present work is also granted by the Conseil Regional d'aquitaine. Technical Report, LaBRI,

2 8t 2 [0; 1] C(t) = w 0 B 2 0 (t) P 0 + w 1 B 2 1 (t) P 1 + w 2 B 2 2 (t) P 2 w 0 B 2 0 (t) + w 1 B 2 1 (t) + w 2 B 2 2 (t) (1) or by expliciting the Bernstein polynomials B 2 0(t), B 2 1(t) and B 2 2(t) : 8t 2 [0; 1] C(t) = w 0 (1? t) 2 P 0 + 2w 1 (1? t)t P 1 + w 2 t 2 P 2 w 0? 2t(w 0? w 1 ) + t 2 (w 0? 2w 1 + w) (2) Parameters w 0, w 1 and w 2 are linked by the conic invariant relationship w 0 w 2 =w 2 1 [5, 11, 8] which means that every set (w 0 ; w 1 ; w 2 ) providing the same value for the conic invariant provides also the same curve. Moreover, because P 0 and P 2 behave symmetrically, it is preferable to have w 0 = w 2. Finally, without loss of generality, one can assume that w 0 = w 2 = 1 and w 1 = w (this condition is sometimes called the normal form of the rational curve [2]) which leads to : 8t 2 [0; 1] C(t) = (1? t)2 P 0 + 2w (1? t)t P 1 + t 2 P 2 1? 2t(1? w) + 2t 2 (1? w) () The main advantage of the latter expression is that it enables a simple classication of the resulting curve: w < 1 : C(t) is an ellipse w = 1 : C(t) is a parabola w > 1 : C(t) is an hyperbola P 1 P 0 w<1 w=1 w>1 2.2 Circular arcs Figure 1: Three types of conics For geometric modelling, the circular arc is undoubtly the most useful particular case of conic sections. Such an arc is obtained when P 0 ; P 1 ; P 2 form an isosceles triangle and when w = cos (where is the angle between P 0 P 1 and P 0 P 2 ) [5, 11, 8]. For instance, a circular arc of length 2 starting from the trigonometric origin (see Figure 2) is obtained with the following control points : P 2 P 0 (1; 0) P 1 (1; tan ) P 2 (cos 2; sin 2) (4) which can be computed more eciently by : P 0 (1; 0) P 1 (1; T ) P 2 ( 1? T T 2 ; 2T ) where T = tan (5) 1 + T 2 Technical Report, LaBRI,

3 P 1 P Circles and Spheres P φ Figure 2: Circular arc The control triangle P 0 P 1 P 2 given by Equation 5 allows only to dene a circular arc sweeping less than 180 degrees 2 (T becomes innite for 2 = ). Therefore, three circular arcs are needed at least to obtain a full circle. These arcs are pieced together (the last point of each arc becomes the rst point of the next one) giving a circle dened by n arcs and 2n + 1 control points. The resulting curve can be considered either as a piecewise rational Bezier or as a NURBS. It should be noted that arcs of dierent lengthes may be pieced together, but the concatenation of similar arcs provides a better parametrization. Among all the possible representations, the following ones appear particularly interesting. P 1 P 1 P 2 P 2 P 2 P 1 P 0 P 0 P 0 Figure : Three dierent representations of a circle 2..1 Triangle-based circle The rst exact representation of a circle by a NURBS has been proposed by Versprille [11]. In that model, the circle is composed of three arcs (each of length 2=) and dened by seven 2 Several authors have proposed representations for circular arcs sweeping over 180 degrees by using innite control points or negative weights [7, 8]. Such constructions are less interesting in a CAD orientation, because the convex hull property is lost and the resulting parametrization is usually bad. But note that the reparametrization process proposed in the present paper may eventually be employed to improve the quality of the parametrization. Technical Report, LaBRI, 1995

4 control points forming an equilateral triangle (see Figure, Left) : p P 0 (1; 0) P 1 (1; ) P2 (?1 p 2 ; 2 ) P (?2; 0) P 4 (?1 2 ;? p 2 ) p P 5(1;? ) P6 (1; 0) where even-indexed (resp. odd-indexed) points have their weights set to 1 (resp. 1=2) Square-based circle The main drawback of the triangle-based circle is that the size of the control polygon is relatively large compared to the circle. An improvement, proposed by Tiller [10], is to dene the circle by 4 arcs (each of length =2) using nine control points regularly placed on a square (see Figure, Middle) : P 0 (1; 0) P 1 (1; 1) P 2 (0; 1) P (?1; 1) P 4 (?1; 0) P 5 (?1;?1) P 6 (0;?1) P 7 (1;?1) P 8 (1; 0) where even-indexed (resp. odd-indexed) points have their weights set to 1 (resp. p 2=2). 2.. Hexagon-based circle The square-based circle is by far the most commonly used in modelling softwares, mainly because of its simplicity. Nevertheless, there is another simple representation which provides a much tighter convex hull as well as a better parametrization. In this model, the circle is composed of six arcs (each of length =) and dened by 1 control points forming an hexagon (see Figure, Right) : P 0 (1; 0) P 1 (1; p ) P 2( 1 p 2 ; 2 ) P (0; 2 p 2 ) P 4(?1 2 ; p p 2 ) P 5(?1; ) P 12(1; 0) where even-indexed (resp. odd-indexed) points have their weights set to 1 (resp. p =2) Sphere Starting from a NURBS-based representation of the circle, a NURBS-based representation of the sphere can be easily obtained by computing a rational tensor product of a circle (providing the parallels) by a half-circle (providing the meridians). The most popular representation, proposed by Tiller [10], is based on the circumscribing cube (dened with 45 control points) but one can also imagine a tighter control lattice having a kind of hexahedrical shape (dened with 91 control points) as shown on Figure 4. Figure 4: Two dierent representations of a sphere In fact, any regular polygon can be used to dene the circle, but the square and the hexagon are certainly the representations that provide the best tradeo between cost, accuracy and ease of use. Technical Report, LaBRI,

5 Discussion.1 Study of the continuity As the circle is obtained by concatenation of several circular arcs, the question of continuity at the junction points (knots, in the B-spline terminology) arises naturally. To study this problem, let us take the case where two arcs, respectively dened by the control triangles P 0 P 1 P 2 and P 2 P P 4 (see Figure 5), are connected together. When the arcs have the same length, we have necessarily w 0 = w 2 = w 4 = 1 and w 1 = w = w. Moreover, let us suppose that the parameter range is [0; 1] for the rst segment and [1; 2] for the second one. If we use the 1? and 1 + notation to express the left and right limit for the curve at the parameter value t = 1, Equation gives : C(1? ) = P 2 C 0 (1? ) = 2w (P 2?P 1 ) C 00 (1? ) = 2 (P 0?P 2 ) + 4w (1?w) (P 1?P 2 ) C(1 + ) = P 2 C 0 (1 + ) = 2w (P?P 2 ) C 00 (1 + ) = 2 (P 4?P 2 ) + 4w (1?w) (P?P 2 ) (6) P P 2 P 1 φ P 4 P 0 Figure 5: Junction of two circular arcs φ Obviously, there is a C 0 continuity at point P 2. Moreover, as the two arcs have the same length, triangles P 0 P 1 P 2 and P 2 P P 4 are similar, which means in particular : P 2? P 1 = P? P 2 (7) Therefore the junction has also a C 1 continuity. The similarity of triangles P 0 P 1 P 2 and P 2 P P 4 provides also (see Figure 5) : P 4? P 2 = P 0? P 2 + 2(1 + cos 2) (P 2? P 1 ) = P 0? P 2 + 4w 2 (P 2? P 1 ) (8) The substitution of Equation 8 in the expression of C 00 (1 + ) shows that the only possibilities to get C 00 (1? ) = C 00 (1 + ) are w = 0 (a 180 o arc, which is forbidden) or w = 1 (a 0 o arc, which is without interest). This leads to the following conclusion [8, 2] : There is no possibility to dene a circle with a C 2 continuity when using a quadratic NURBS (or a quadratic piecewise rational Bezier)..2 Study of the parametrization Let us take again the circular arc of length 2 starting from the trigonometric origin, described on Figure 2. It is well-known that the best parametrization of a curve is the arc length parametrization (also called canonical parametrization or constant-speed parametrization) because the variation of this parameter represents the exact distance when following the curve Technical Report, LaBRI,

6 from one point to another. In our example, this canonical parametrization is given by the trigonometric parametrization : 8 2 [0; 2] C() = (cos ; sin ) By combining Equation and Equation 5, we obtain the rational Bezier parametrization of our arc : 8t 2 [0; 1] C(t) =! 2 1? 2(1? w)t? 2(1? w)t2 1? 2(1? w)t + 2(1? w)t 2 ; 2T wt? 2T w(1? w)t 1? 2(1? w)t + 2(1? w)t 2 The quality of this parametrization can be evaluated by comparison with the canonical parametrization. In the ideal case, the two parametrizations are related by a linear function 4 : (9) 8t 2 [0; 1] (t) = 2t In the other cases, a quantitative evaluation of the parametrization is obtained by computing the distance between the actual function (t) and the ideal function 2t. Using an Euclidian norm, this distance (t) (sometimes called chordal deviation) is given by : (t) = s ((t)? 2t) (10) Consequently, to evaluate our parametrization, we have to nd the actual expression of (t). For that, we involve another classical parametrization of the circular arc, the half-tangent parametrization : 8s 2 [0; 1] C(s) = 1!? T 2 s T 2 s 2 ; 2T s 1 + T 2 s 2 where T = tan (11) The main advantage of this parametrization is that it provides a simple relationship between the arc length and the parameter s : 8s 2 [0; 1] (s) = 2 arctan T s Therefore, if we are able to nd s(t), we will get immediatly (t). Fortunately, a fundamental result of projective geometry, developped by Moebius, states that two parametrizations of the same curve by a rational polynomial of the same degree (Equation 9 and Equation 11) are related by a rational linear function 5 [10, 2] : 9(a; b; c; d) 2 IR 4 = s(t) = a + bt c + dt (12) In our case, parameters t and s are equal at the boundaries of the range [0; 1]. This enables a more precise expression : 4 This is not the case here because it would mean that the cosine function could be expressed as a rational polynomial. 5 This result has been extended very recently [6] to piecewise rational polynomial curves (e.g. NURBS). Technical Report, LaBRI,

7 (1 + p)t 9p 2 IR = s(t) = 1 + pt (1) The substitution of Equation 1 in Equation 11 provides p = w? 1. Therefore : s(t) = wt 1? t + wt (14) and nally : (t) = 2 arctan t p 1? w 2 1? t + wt (15) Figure 6: (t) and 0 (t) for = 0:25; 0:5; 0:75; 1; 1:25; 1:5 (in radians). Figure 7: (t) for for = 0:25; 0:5; 0:75; 1; 1:25; 1:5 (in radians). The plot of this function, for several values for, is shown on the left part of Figure 6. One can see that the funtion becomes less \straight" (in fact, it oscillates around the ideal straight line) when increases. This phenomenon is more apparent on the right part of the gure which shows the deviations of 0 (t) from the ideal horizontal lines. But the best quantitative information is given by the chordal deviation (t) on Figure 7 which leads to the following conclusion [4] : Except for very small angles, the representation of a circular arc (or more generally, a conic) by a quadratic NURBS (or a quadratic piecewise rational Bezier) involves a parametrization that deviates from the arc length in a non-negligible way. Technical Report, LaBRI,

8 4 Zigzag Reparametrization The previous discussion has shown that the usual representation of circles and spheres by NURBS curves or surfaces contains two main weaknesses : rst, the parametrization can deviate signicantly from the arc length, second, the continuity of the parametrization is C 1 at best. In fact, these unsatisfactory points have the same origin : as the degree of the curve is only quadratic there are not enough degrees of freedom to correct one (or both) faults. A natural idea is therefore to raise the degree in order to get more parameters to manipulate. Unfortunately, the classical process to raise the degree of a Bezier or a B-spline without modifying its shape [1, 7, 2] does not work at all in this particular case. Indeed, neither the level of continuity nor the deviation of the parametrization are changed. For that reason, we propose here an original method for raising the degree (restricted to rational or piecewise rational curves) which allows to control precisely the resulting parametrization. 4.1 Principle The general principle of the method is borrowed from the reparametrization scheme dened by Moebius. As we have seen above (see Equation 12), the reparametrization of a rational curve by a linear rational polynomial does not modify its shape and its degree. A corollary of this result is that a reparametrization by a quadratic rational polynomial doubles the degree of the initial curve, a cubic rational polynomial triples the degree and so on. Therefore, the conic dened by Equation for instance, can be represented by a quartic rational curve when we replace t by S(t) where S is a quadratic rational polynomial. In the most general formulation for S(t), there are 6 degrees of freedom : S(t) = a + bt + ct2 d + et + f t 2 but several characteristics appear desirable for this reparametrization. For instance, it is important for the two parametrizations to have the same domain of variation (S(0) = 0 and S(1) = 1). Moreover, due to the isoscelism of the control triangle, the symmetry of the parametrization should be conserved (S(1?t) = 1?S(t) or in other words S 0 (1?t) = S 0 (t)). All these constraints imply that there is nally only one degree of freedom. If we call this parameter p and dene it as the value of derivative at the boundaries (S 0 (0) = S 0 (1) = p), we obtain : S(t) = pt + (1? p)t 2 1? 2(1? p)t + 2(1? p)t 2 (16) The shape of this function (see Figure 8) has inspired the name of our reparametrization technique : zigzag parametrization. Notice that p has to be positive (or eventually null) in order to get an increasing function in the range [0; 1]. Technical Report, LaBRI,

9 Figure 8: Zigzag reparametrization function S(t) for p = 0; 0:5; 1; 2; 6; Zigzag reparametrization of a conic If we apply our zigzag reparametrization on the Bezier curve dened by Equation, we obtain : C(t) = a2 P 0 + 2wab P 1 + b 2 P 2 a 2 + 2wab + b 2 (17) with a = 1? (2? p)t + (1? p)t 2 and b = pt + (1? p)t 2 The result is a quartic rational curve but it is not a rational Bezier since it is not written in terms of Bernstein polynomials anylonger. But, as the Bernstein polynomials form a basis of the polynomials space, we are insured that there exists a Bezier curve that is equivalent to Equation 17. To nd this quartic rational Bezier curve, we have to nd ve points (Q 0 ; Q 1 ; Q 2 ; Q ; Q 4 ) and ve weights (w 0 ; w 1 ; w 2 ; w ; w 4 ) obeying : C(t) = w 0(1?t) 4 Q 0 + 4w 1 t(1?t) Q 1 + 6w 2 t 2 (1?t) 2 Q 2 + 4w t (1?t) Q + w 4 t 4 Q 4 w 0 (1?t) 4 + 4w 1 t(1?t) + 6w 2 t 2 (1?t) 2 + 4w t (1?t) + w 4 t 4 (18) Obviously, Q 0 = P 0 and Q 4 = P 2 because of the interpolation of the boundaries. We also know that we can let w 0 = 1 and w 4 = 1 for symmetry reasons, without loss of generality. The remainding weights w 1, w 2 and w are obtained by comparing term by term, the denominators of Equation 18 and Equation 17. The remainding points Q 1, Q 2 and Q are obtained similarly by comparing the numerators. Finally, the zigzag reparametrization of Equation 17 provides a quartic rational Bezier (and therefore quartic NURBS) dened by : Q 0 = P 0 Q 1 = P 0 + wp w Q 2 = p2 P 0 + 2w(1 + p 2 )P 1 + p 2 P 2 2(p 2 + p 2 w + w) Q = P 2 + wp w Q 4 = P 2 w 0 = 1 w 1 = 1 + w p 2 w 2 = p2 + p 2 w + w w = 1 + w p w 4 = 1 (19) 2 Three important comments can be made about this formulation : Technical Report, LaBRI,

10 Only the position of point Q 2 depends on the value of parameter p. A nice consequence of this is that the zigzag reparametrization has a very simple geometric interpretation : changing the value of p consists in moving Q 2 along a straight line (more precisely, an half line because we force p 0). The simultaneous modication of w 1 ; w 2 and w keeps the curve inchanged (except its parametrization) despite the displacement of Q 2. When p = 1, we obtain exactly the points and weights that are provided by the classical degree raising technique mentionned above. In other words, it means that the zigzag reparametrization is a kind of generalization of this process, which provides an additional degree of freedom. The principle of the zigzag reparametrization can be extended to non-symmetric reparametrization by letting two degrees of freedom (p = S 0 (0) and q = S 0 (1)) for Equation 16. Such an extension can be used for the reparametrization of non-symmetric curves such as ellipses or parabolas. The principle of the zigzag reparametrization can also be extended above the degree 2. This extension is done by dening a family of rational polynomials where each member S k (t) of degree k fullls : S k (0) = 0 S k (1) = 1 8i = 0::k 9p i 2 IR + = di S (0) = p i dt i d i S dt i (1) = (?1)i?1 p i Therefore, each successive member of this zigzag polynomial family includes one additional parameter that enables more and more precision on the control of the parametrization. 4. A C 2 parametrization of the circle A rst application of the zigzag parametrization will be to obtain a circle with a C 2 parametrization. Starting again from the conguration of control points illustrated on Figure 5, we can check the continuity at point P 2 : C 0 (1? ) = 2wp (P 2?P 1 ) C 00 (1? ) = 2p 2 (P 0?P 2 ) + 4w (1+p?p 2?2wp 2 ) (P 1?P 2 ) C 0 (1 + ) = 2wp (P?P 2 ) C 00 (1 + ) = 2p 2 (P 4?P 2 ) + 4w (1+p?p 2?2wp 2 ) (P?P 2 ) (20) As previously, C 1 continuity is fullled thanks to Equation 7. To study the C 2 continuity, we have to substitute Equation 7 and Equation 8 in the expression of C 00 (1 + ) : C 00 (1 + ) = 2p 2 (P 0?P 2 ) + 4w (p 2?p?1) (P 1?P 2 ) Finally, the comparison of C 00 (1? ) and C 00 (1 + ) provides : p 2 (1 + w)? p? 1 = 0 (21) This quadratic equation has always two solutions ( = 4w+5 > 0) but only the positive solution is useful for our reparametrization : p = 1 + p 5 + 4w 2 (1 + w) (22) Technical Report, LaBRI,

11 Therefore, Equation 22 combined with Equation 19 enables a C 2 continuous junction for circular arcs of the same length, with the resulting curve being represented either by quartic rational Bezier or a quartic NURBS. As an illustration, Figure 9 represents the circle with a C 2 parametrization obtained by the zigzag reparametrization of the triangle-based circle, using for p the value provided by Equation 22. Q 4 Q Q 2 Q 1 Q 0 Figure 9: C 2 reparametrization of the triangle-based circle 4.4 A quasi-ideal parametrization of the circle A second application of the zigzag reparametrization will be to obtain a circle with a quasi-ideal parametrization, as close as possible to the trigonometric parametrization. As we have seen in Section, the parametrization error can be quantied by computing the chordal deviation (t) between the canonic parametrization and the actual one. The substitution of Equation 16 in Equation 10 provides the expression of the chordal deviation for a circular arc on which the zigzag reparametrization has been applied. The formulation of this new chordal deviation is function of parameter p : v u t (t) = 2 arctan p! 2 1? w2 (pt + (1? p)t 2 ) 1? 2t + (1 + w)(pt + (1? p)t 2 )? 2t (2) What we have here is a classical optimization problem : nd the optimal value of a parameter (in our case, the reparametrization factor p) which minimizes a given condition (in our case, the chordal deviation (t)). A rst possibility is to minimize the L 2 norm of (t) by employing a least-squares minimization method. The drawback of this technique (and all the other techniques of the same family) is that it provides only a numerical (and not an analytical) solution. Thus, for each new angle needed by the user, the whole least-squares process has to be restarted to nd the new parameter p. Another possibility is to minimize the L 1 norm of (t) which requires only to compute the maximal chordal deviation. This maximum has an analytical expression which can be computed by a symbolic calculation software such as Maple or Mathematica, but this expression is so complex 6 that it is useless in practice. For all that reasons, we propose here a third solution which is only a heuristic but, in fact, is very close to the optimum. If we study the variation of the maximal chordal deviation for 6 The formulation found by Maple requires no less than three printed pages. Technical Report, LaBRI,

12 dierent values of, one can see that this maximum is always obtained for a value t that belongs in [0:19; 0:24]. If we take a value in that range (t = 1=5 seems to work well) we are insured that the chordal deviation at that point (1=5) will not be too far from the maximal deviation. Therefore, we simply have to force (1=5) = 0, that is : to get a quasi-optimal value for p : (1 + 4p) sin? tan(=5) = 0 (24) p? (1 + 4p) cos p = 4? 2 cos (=5) + cos(=5)?1 + 8 cos (=5)? 4 cos(=5) (25) Figure 10: (t) for for = 0:25; 0:5; 0:75; 1; 1:25; 1:5 (in radians). Figure 11: (t) and 0 (t) for = 0:25; 0:5; 0:75; 1; 1:25; 1:5 (in radians). Figure 10 shows the plot of the chordal deviation (t) obtained with the quasi-optimal value p. Notice that there is a factor 100 in the scale between Figure 7 and Figure 10; it means that the parametrization error has been reduced over 100 times. This result is conrmed on Figure 11 which shows the variations of (t) and 0 (t) for dierent values of when using the quasi-optimal parameter : the resulting curves are almost straight, no more oscillations are visible. Technical Report, LaBRI,

13 Q 4 Q Q 2 Q 1 Q 0 Figure 12: Quasi-optimal reparametrization of the triangle-based circle As an illustration, Figure 12 represents the circle with a quasi-optimal parametrization obtained by the zigzag reparametrization of the triangle-based circle, using the parameter p provided by Equation 25. The comparison of Figure 9 and Figure 12 shows that the C 2 continuous circle and the quasi-optimal circle are almost identical. In fact, the dierence between the reparametrizations is so small (p ' 1:21525 for the rst and p ' 1:20669 for the second) that the variation of the position of the control points is hardly visible. To conclude, the following table groups all the gures that are relative to the zigzag reparametrization of the triangle-based, square-based and hexagon-based circles. Notice that the chordal deviation can be reduced over 1000 times by the reparametrization. Notice also that the analytic L 1 minimization oers only an average of 5% additional precision compared to our heuristic minimization, for a computation cost that is much more expensive. This conrms the nice behaviour of the \(1=5) = 0" heuristic. Type of representation Triangle-based Square-based Hexagon-based Original parametrization Chordal deviation 1:68E?2 8:61E? :2E? C 2 reparametrization Continuity factor p 1: :1129 1:04750 Analytic reparametrization Optimal factor p 1: : :04706 Chordal deviation 6:11E?5 1:82E?5 :04E?6 Error reduction factor = Heuristic reparametrization Quasi-optimal factor bp 1: : : Chordal deviation b 6:8E?5 1:89E?5 :20E?6 Error reduction factor = b References [1] R. Bartels, J. Beatty, B. Barsky, An Introduction to Splines for Computer Graphics and Geometric Modeling, Morgan Kaufmann, [2] G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, [] G. Farin, From Conics to NURBS : a Tutorial and Survey, Computer Graphics & Applications, v12, n5, p78-86, Technical Report, LaBRI,

14 [4] R. Farouki, T. Sakkalis, Real Rational Curves are not Unit Speed, Computer Aided Geometric Design, v8, n2, p , [5] A. Forrest, Curves and Surfaces for Computer-Aided Design, PhD Thesis, University of Cambridge, [6] E. Lee, M. Lucian, Moebius Reparametrization of Rational B-Splines, Computer Aided Geometric Design, v8, p21-28, [7] L. Piegl, Innite Control Points : a Method for representing Surfaces of Revolution using Boundary Data, IEEE Computer Graphics & Applications, v7, n, p45-55, [8] L. Piegl, W. Tiller, A Menagerie of Rational B-Spline Circles, IEEE Computer Graphics & Applications, v9, n5, p48-56, [9] L. Piegl, On NURBS : a Survey, IEEE Computer Graphics & Applications, v11, n1, p55-71, [10] W. Tiller, Rational B-Splines for Curve and Surface Representation, IEEE Computer Graphics & Applications, v, n6, p61-69, 198. [11] K. Versprille, Computer Aided Design Applications of the Rational B-Spline Approximation Form, PhD Thesis, University of Syracuse, Technical Report, LaBRI,