Department of Computer Science, POSTECH, Pohang , Korea. (x 0 (t); y 0 (t)) 6= (0; 0) and N(t) is well dened on the

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1 Comparing Oset Curve Approximation Methos Gershon er +, In-Kwon, an Myung-Soo Kim + Department of Computer Science, Technion, IIT, Haifa 32000, Israel Department of Computer Science, POSTECH, Pohang , Korea Oset curves have iverse engineering applications, which have consequently motivate extensive research concerning various oset techniques. Oset research in the early 980s focuse on approximation techniques to solve immeiate application problems in practice. This tren continue until 988, when chek [, 2] applie non-linear optimization techniques to the oset approximation problem. Since then, it has become quite icult to improve the state-of-the-art of oset approximation. Oset research in the 990s has been more theoretical. The founational work of Farouki an Ne [3] clarie the funamental iculty of exact oset computation. Farouki an Sakkalis [4] suggeste the Pythagorean Hoograph curves which allow simple rational representation of their exact oset curves. Although many useful plane curves such as conics o not belong to this class, the Pythagorean Hoograph curves may have much potential in practice, especially when they are use for oset approximation. In a recent paper [5] on oset curve approximation, the authors suggeste a new approach base on approximating the oset circle, instea of approximating the oset curve itself. To emonstrate the eectiveness of this approach, we have mae extensive comparisons with previous methos. To our surprise, the simple metho of ler an Hanson [6] outperforms all the other methos for osetting (piecewise) quaratic curves, even though its performance is not as goo for high egree curves. The experimental results have reveale other interesting facts, too. If these etails ha been reporte several years ago, we believe, oset approximation research might have evelope somewhat ierently. This paper is intene to ll in an important gap in the literature. Qualitative as well as quantitative comparisons are conucte employing a whole variety of contemporary oset approximation methos for freeform curves in the plane. The eciency of the oset approximation is measure in terms of the number of control points generate while the approximations are mae within a prescribe tolerance. Oset of Planar Curves Given a regular parametric curve, C(t) (x(t); y(t)), in the plane, its oset curve C (t) by a constant raius, is ene by: C (t) C(t) + N(t); () where N(t) is the unit normal vector of C(t): N(t) (y0 (t);?x 0 (t)) px 0 (t) 2 + y 0 (t) 2 : (2) The regularity conition of C(t) guarantees that (x 0 (t); y 0 (t)) 6 (0; 0) an N(t) is well ene on the curve C(t). Equation (2) has a square root term in the enominator. Therefore, even if the given curve C(t) is a polynomial curve, its oset is not a polynomial or rational curve, in general. This funamental eciency has motivate the evelopment of various polynomial an rational approximation techniques of C (t). While the oset to a polynomial or rational parametric curve must be approximate, it is somewhat counter-intuitive that a close cousin of the oset, the evolute, is inee always representable as a rational curve (see Siebar on Evolute). Most oset approximation techniques are base on an iterative process of tting an approximation curve, measuring the accuracy, an subiviing the problem into smaller problems if the approximation error is larger than the tolerance. This ivie an conquer approach exploits the subivision property of the base curve C(t). Henceforth, we assume C(t) is represente by a Bezier or NURBS curve. Traitionally, the oset approximation error has been measure only at nite sample points along C(t), computing i kc(t i )?C a(t i)k?. er an Cohen [7] propose a symbolic metho which computes the global error between square istances: (t) kc(t)? C a (t)k 2? 2 : (3) The error function (t) is obtaine by symbolically computing the ierence an inner prouct of Bezier or NURBS curves (see Siebar on Symbolic Computation an Reference [8]). Therefore, it can be represente as a Bezier or NURBS scalar function. As a scalar el, the largest coecient of (t) globally bouns the maximal possible error ue to the convex hull property of Bezier or NURBS formulation. In this article, we exploit the error functional (t) of Equation (3) to measure all the oset approximation errors. This provies not only a global boun for each metho, but also an equal basis for the comparison of ierent methos.

2 Evolute The evolute of C(t) is ene as: E(t) C(t) + N(t) (t) ; where (t) is the curvature of C(t) (x(t); y(t)): (t) x0 (t)y 00 (t)? x 00 (t)y 0 (t) (x 0 (t) 2 + y 0 (t) 2 ) 32 : That is, E(t) is a variable raius oset with oset raius (t) (t). Figure shows two examples of evolute curves. Quite surprisingly, E(t) is a rational curve, provie C(t) is a rational or polynomial curve: E(t) C(t) + N(t) (t) C(t) + C(t) + (y 0 (t);?x 0 (t)) (x 0 (t) 2 +y 0 (t) 2 ) 2 x 0 (t)y 00 (t)?x 00 (t)y 0 (t) (x 0 (t) 2 +y 0 (t) 2 ) 32 x 0 (t) 2 + y 0 (t) 2 x 0 (t)y 00 (t)? x 00 (t)y 0 (t) (y0 (t);?x 0 (t)): (a) C(t) 20 E(t) (b) E(t) C(t) In contrast to the oset computation in Equation (), there is no square root term in the representation of E(t). In Figure, the curves an their evolutes are both represente as B-spline curves. Figure : In (a), a B-spline curve, C(t), in light curve, is shown along with its evolute, E(t), in bol curve. In (b), the evolute E(t) for a cubic polynomial approximation of a circle, C(t), is shown. E(t) is scale up by a factor of 20. Qualitative Comparisons Control Polygon Base Methos Let C(t) be a B-spline curve with k control points of orer n an ene over a knot sequence ft i g ; 0 i < n + k. The i-th noe parameter value i of C(t) is ene as: i P i+n? ji+ t j ; (7) n? for 0 i < k. Hence, a noe parameter value is an average of n? consecutive knots in. Each control point, P i, of C(t), is associate with one noe, i. C( i ) is typically close to P i ; however, it is not the closest point of C(t) to P i, in general. b [9] translate each control point, P i, by N( i ), whereas ler an Hanson [6] translate each ege of the control polygon into the ege normal irection by a istance. Unfortunately, b [9] always uner-estimates the oset; i.e., (t) 0, for all t. (For the proof an relate issues, see Siebar on Uner an Over-Estimation.) ler an Hanson [6] o not uner-estimate the oset curve. In aition to computing the exact linear an circular oset curves, their metho outperforms all the other methos for the case of osetting (piecewise) quaratic curves. However, for osetting high egree curves, this simple metho has a similar performance to that of b [9]. uillart [0] solve the uner-estimating problem. The istance between P i an C( i ), an the curvature ( i ) of C(t) at i are taken into account. Numerical approximation is also taken to compute the closest point of C(t) to the control point P i, while using C( i ) as an initial solution. With all these enhancements, uillart [0] was able to oset the linear an circular segments exactly. er an Cohen [] took a ierent approach that exactly computes the osets of linear an circular elements. Using the values of (t) (in Equation (3)) at t 0 ; : : : ; n, the error in the neighborhoo of each control point, P i, is estimate an use to ajust the translational istance applie to P i. This perturbation base approach is an iterative metho that converges to the exact circular oset segment. For general curves, when the result of b [9] is use as an initial solution, the perturbation process typically reuces the oset approximation error of [9] by an orer of magnitue. In principle, this metho can be applie to any oset approximation metho that prouces piecewise polynomial curves. Most traitional techniques subivie C(t) at the mile of the parametric omain; however, a better caniate is the parameter of the location with the maximum error. Since (t) represents the exact 2

3 Symbolic Computation In this article, we have employe (t) (in Equation (3)) an m(t) (in Equation ()) to estimate the oset approximation error. Furthermore, we also nee to compute the composition of U(s(t)) (in Equation (0)). The symbolic computation of these equations involves the ierence, prouct, an sum of (piecewise) scalar polynomial or rational curves. Let C (t) P m i0 PiBi; (t) an C2(t) P n j0 PjBj;(t) be two (piecewise) polynomial regular parametric curves, in the Bezier or NURBS representations. The computation of C (t) C 2(t) can be accomplishe by elevating both C (t) an C 2(t) to a common function space. The orer of the common function space is equal to the maximal orer of C (t) an C 2(t). If either C (t) or C 2(t) is a B-spline curve, the common function space is ene by consiering both knot vectors an an preserving the lowest egree of continuity at each knot. Once the common function space is etermine, both C (t) an C 2(t) are elevate to this space via egree raising an renement. (See [8] for more etails as well as the extension to rationals.) The computation of C (t)c 2(t) is somewhat more involve. Here, we consier only the case of Bezier polynomial curves. (See [8] for the more general cases of piecewise polynomials an rationals.) The i-th Bernstein Bezier basis function of egree k is ene by: B k i (t) k i t i (? t) k?i : (4) The prouct of two Bernstein Bezier basis functions is: Bi k (t)bj(t) l k t i (? t) k?i l t j (? t) l?j i j k l t i+j (? t) k+l?i?j i j Therefore, we have: C (t)c 2(t)? k i? k+l i+j? l j mx i0 mx Bi+j k+l (t): (5) i0 j0 mx P ib m i (t) i0 j0 m+n X k0 j0 P jb n j (t) P ip jb m i (t)b n j (t)? m? n i j P ip j? m+n i+j B m+n i+j (t) Q kb m+n k (t); (6) ( m+n i+j ) where Q k accumulates all the combinatorial terms ( m i )( n P ip j j), for k i + j. Hence, C(t)C2(t) is represente as a Bezier polynomial curve of egree m + n. square error function, one can n the parameter location of the maximal error an subivie C(t) there. Alternatively, instea of subiviing C(t), one can insert new knots into C(t) at the parameter locations with error larger than the allowe tolerance. er an Cohen [7] took this renement approach. Interpolation Methos ss [2] use a cubic Hermite curve to approximate the oset curve. The cubic Hermite curve is etermine by interpolating the position an velocity of the exact oset curve at both enpoints. The numerical approximation proceure of ss [2] is quite unstable when the oset curve becomes almost at. Therefore, instea of using the original algorithm [2], we compute the rst erivative of the oset curve base on the following simple close form equation (see also [3]): C 0 (t) ( + (t))c 0 (t); (8) where (t) is the curvature of C(t). chek [] suggeste a least squares solution for the etermination of C 0 (t) at the curve enpoints. That is, at each enpoint of C (t), the irection of C 0 (t) is maintaine to be parallel to C0 (t); however, instea of using Equation (8), their lengths are etermine so that the cubic Hermite curve best ts C (t) in the least squares sense. For computational eciency, only nite samples of C (t) are use in the optimization. chek an Wissel [2] use a general non-linear optimization technique to approximate a high egree spline curve with low egree spline curves. They applie the same technique to approximate an exact oset curve with low egree spline curves. The least squares base methos [, 2] are expecte to perform better than other methos. However, there still remains a question about whether the least squares solution is optimal when searching for the smallest number of (say cubic) curve segments to approximate an exact oset curve. The answer is negative, in general. In the special case of osetting quaratic curves, the simple metho of ler an Hanson [6] performs much better than the least squares methos [, 2]. It is important to question how this unexpecte result coul be obtaine. The answer might be quite useful in improving the accuracy of oset approximation. The least squares solution optimizes the integrate summation of the least squares errors in the approximation. Therefore, even if a small portion of the approximation curve has a large error, as long as the rest of the curve tightly approximates the exact curve, the overall least squares error can be very small. That is, the least squares solution provies an optimal solution with respect to an L 2 norm. When this L 2 optimal solution is further evaluate with respect to the L norm (of Equation (3)), the optimal- 3

4 Uner an Over-Estimation The oset approximation of b [9] is formally ene as follows: C a (t) i0 i0 (P i + N( i)) B i(t) P ib i(t) + C(t) + V (t): i0 N( i)b i(t) where kn( i)k P, for i 0; : : : ; n. The vector el n curve V (t) N(i)Bi(t) has all its control points i0 N( i) on the unit circle S. By the convex hull property, we have kv (t)k an D(t) (b) kc(t)? C a (t)k kv (t)k kv (t)k : C(t) (a) C a (t) If N( i) 6 N( j), for some 0 i; j n, we have min kv (t)k <, an this results in an error in the oset approximation. Hence, this metho always unerestimates the exact oset. Figure 2(a) shows a quartic Bezier curve C(t) an its oset approximation C a (t). In Figure 2(b), the ierence vector el D(t) C(t)? C a (t) is completely containe in a isk of raius. All the control points of D(t) are on the circumference of the isk. Uner-estimation of osets may lea to unesirable results. For example, in NC machining, the unerestimation leas to gouging. Assume the unerestimation of the oset is boune from below by: min min(kv (t)k): When we translate control point P i in the irection of N( i), by a istance min, the resulting curve completely over-estimates the exact oset (see Figure 3): kc(t)? C a (t)k V (t) min kv (t)k V (t) : One can reuce the relative error in the oset approximation by alternating the uner an over-estimations. This can be one by ajusting the oset istance at each control point appropriately. Figure 4 shows an example of this approach. We use the same quartic Bezier curve as in Figures 2 an 3. The quartic Bezier oset approximation curve interpolates the exact oset at ve iscrete locations, corresponing to the noe values, i 4 ; 0 i 4. Figure 2: In (a), an oset approximation C a (t) compute by translating the control points of the original curve C(t) (ashe lines) by an amount equal to the oset istance will always uner-estimate the real oset. In (b), D(t) C(t)? C a (t) is foun to be fully containe in a circle of the oset raius size,. (b) D(t) C a (t) C a (t) C(t) D(t) C(t) (b) (a) Figure 3: In (a), an oset approximation C a (t) of a quartic Bezier curve C(t) (ashe lines) is compute by forcing C a (t) to over-estimate the error. D(t) (t) is shown in (b). Compare with Figure 2. C(t)?C a (a) Figure 4: In (a), an oset approximation C a (t) of a quartic Bezier curve C(t) (ashe lines) is compute by enforcing C a (t) to interpolate at ve locations on the exact oset curve compute at the noe values on C a(t). D(t) C(t)? Ca (t) is shown in (b). Compare with Figures 2 an 3. 4

5 ity is no longer guarantee. This is an important observation which suggests possible improvements over the nearly optimal solutions [, 2]. Pham [3] suggeste a simple B-spline interpolation metho to approximate the oset curve. Finite sample points are generate on the exact oset curve, an they are interpolate by a piecewise cubic B-spline curve. It is also interesting to note that this simple metho also performs pretty well. In many examples, its performance is only slightly worse than an sometimes even better than the local least squares methos [, 2]. Circle Approximation Methos Assume the base curve C(t), t 0 t t, is a polynomial curve with no inection point, an a unit circular arc U(s), s 0 s s, is parameterize so that: C 0 (t 0 ) k U 0 (s 0 ) an C 0 (t ) k U 0 (s ): If one can compute a reparameterization s(t) so that: C 0 (t) k U 0 (s(t)); the oset curve is then computable as: C (t) C(t) + U(s(t)): (9) The oset curve is not a polynomial or rational curve; therefore, we have to approximate U(s) an/or s(t) by a polynomial or rational. et al. [5] approximate the unit circle U(s) with piecewise quaratic polynomial curve segments Q j (s), j 0; : : : ; n. The Hoograph curve Q 0 j (s) is piecewise linear; therefore, the parallel constraint: C 0 (t) k Q 0 (s(t)) provies the reparameterization of s(t) as a rational polynomial of egree?, where is the egree of C(t). For a polynomial curve C(t) of egree, the resulting oset approximation (compute as in Equation (9)) is a rational curve of egree 3? 2. (For a rational curve C(t) of egree, the oset approximation curve is of egree 5? 4.) For a quaratic polynomial curve C(t), this technique also provies a simple metho to represent the exact oset curve C (t) as a rational curve of egree six. Assume that the exact circle, Q(s), 0 s, is represente by a rational quaratic curve. Then, the parallel constraint: C 0 (t(s)) k Q 0 (s) provies the reparameterization of t(s) as a rational polynomial of egree two. Therefore, the exact oset curve C (t) is a rational curve of egree six. Even with the high egree of six, the exact oset capability suggests this metho as the metho of choice for osetting (piecewise) quaratic polynomial curves, especially for high precision oset approximation. However, this exact oset capability oes not exten to rational quaratic curves. (There are some rational quaratic curves which have no exact rational parametrization of their oset curves.) One can attempt to globally approximate s(t) by maximizing the constraint energy: max s(t) Z t t 0 hc 0 (t); U 0 (s(t))i kc 0 (t)k ku 0 t: (0) (s(t))k This approach was taken in et al. [4], in which the composition of U(s(t)) (U s)(t) is carrie out symbolically [8] (see also Siebar on Symbolic Computation). The oset approximation in [5] epens on the metho use for the piecewise quaratic approximation to the circle. The error in the oset approximation stems only from the quaratic polynomial approximation of the circular arc, scale by the oset raius. et al. [5] use ve ierent circle approximation methos. Two of the ve methos generate G -continuous circle approximations with quaratic Bezier curve segments. In the rst metho, the unit circle U(s) is totally containe in the close convex region boune by the quaratic curve segments. The corresponing oset curve approximation completely over-estimates the exact oset curve. In the secon metho, the quaratic curve segments pass through both the interior an exterior of the unit circle U(s). Therefore, the oset approximation curve both over an uner-estimates the exact oset curve, while the approximation error is reuce by half from the over-estimating rst metho. We use this secon metho, referre to as in the next section, for comparison with other methos. In contrast, et al. [4] approximate the reparametrization s(t), while representing the circle U(s) exactly by a rational quaratic curve. In this metho, the error stems only from the inaccurate reparameterization function s(t), which results in a mismatch in the parallel constraint of C 0 (t) k U 0 (s(t)). To the authors' knowlege, this is the only oset approximation metho for which the use of (t) is completely ineective in the global error boun. The term (t) is always equal to zero. et al. [4] measure the angular eviation of U(s(t)) from the exact oset irection N(t) by using the following error function: m (t) hc(t)? Ca (t); C0 (t)i 2 2 kc 0 (t)k 2 : () The error is equal to zero if orthogonality is preserve. Otherwise, it is equal to cos 2, where is the angle between U(s(t)) an C 0 (t). Quantitative Comparisons We consier how eciently each metho approximates the oset curve, given a prescribe tolerance. Several examples of Bezier an B-spline curves are given, both in polynomial an rational forms. All 5

6 the methos (compare in this article) are implemente using the IRIT [5] soli moeling system that has been evelope at the Technion, Israel, with some of the oset approximation methos implemente at POSTECH, Korea. Methos uner Comparison We quantitatively compare the following methos: : The simple metho of b [9] in which the control points are translate by the oset istance. This metho always creates unerestimate osets. (See Siebar on Uner an Over-Estimation.) : An aaptive oset renement approach that was suggeste in er an Cohen [7]. Instea of subiviing the base curve, whenever the error is too large, the oset curve is rene to yiel a better approximation (by using more control points). The error analysis of (t) is exploite to n better caniate locations for re- nement. This metho also uner-estimates the oset curves. : The enhancement suggeste by uillart [0] that allows the exact oset representation of linear as well as circular segments. : The metho of ler an Hanson [6] in which the eges of the control polygon, rather than the control points, are translate. ss: The metho of ss [2] that ts a cubic Bezier curve to each oset curve segment to interpolate the bounary points an velocities of the exact oset curve. Pham: The metho of Pham [3] that interpolates a sequence of nite sample points on the exact oset curve by a non-uniform piecewise cubic B-spline curve. (The original metho of Pham [3] uses a uniform B-spline curve; however, we have moie the metho.) Whenever the oset approximation error is larger than the prescribe tolerance, more sample oset points are use for a better t. : The global least squares approximation that ts a uniform piecewise cubic B-spline curve to the oset curve. Whenever the oset approximation error is larger than the prescribe tolerance, more control points are use for a better t. : The least squares metho of chek [, 2] that ts a cubic Bezier curve to each oset curve segment. Whenever the error is larger than the tolerance, the base curve is subivie into two subsegments an the oset approximation is repeate recursively. : The approach suggeste by et al. [5] that approximates the curve of the convolution between C(t) an the oset circle U(s) of raius. Traitionally, the oset approximation error has been measure only at nite sample points of C(t) an C a (t). As previously mentione, we aopt the symbolic approach of error estimation [7]. Therefore, we can provie an L global upper boun on the oset approximation error for each of the methos uner comparison. The global error boun is erive by symbolically computing the error function (t) (in Equation (3)). Because of the convex hull property of the Bezier or NURBS representation of the scalar function (t), we can easily etermine its upper boun as the maximum coecient of the Bezier or NURBS basis functions. Comparison Results an Remarks Figures 5{6 show the results of osetting (piecewise) quaratic curves. We compare the number of control points with respect to the accuracy of oset approximation. In these examples, the metho of ler an Hanson [6] outperforms all the other methos even if the base curve has sharp corners with high curvature (Figure 6). This surprising result has never been reporte in the literature. In fact, we have assume that the least squares methos provie near optimal solutions to the oset approximation problem. However, the superior performance of ler an Hanson [6] tells us that this is not true, in general. At this moment, we have no clear explanation of the unerlying geometric properties of this unusual phenomenon. Nevertheless, it is not icult to point out at least two possible sources of the nonoptimality in the current least squares methos: As iscusse above, the least squares methos provie the optimal solutions in an L 2 norm, which may be quite ierent from the optimal solutions in an L norm. The least squares optimization proceure solves an over-constraine problem, the solution of which epens on the istribution of nite sample points on the oset curve. In some egenerate cases, the least squares solution may have large variation epening on the istribution of ata points. Further investigations are require to eliminate these limitations, an this may avance the state-of-theart of oset curve approximation. Figures 7, 9, an 0 show other examples of osetting (piecewise) cubic B-spline curves. Throughout the conucte tests, we have observe the following consistent results: The uner-estimating oset approximation metho,, is oing quite poorly. The aaptive oset renement approach,, is better than, especially when high precision is esire. In the case of osetting (piecewise) quaratic curve segments, the simple metho of ler an Hanson [6] outperforms all the other methos, especially when high precision is require. 6

7 Offset Error 0? ? ? ? ? ? ? 0?2 0?3 0?4 0? ? ? ? ? ? ? Offset Error 0? 0?2 0?3 0?4 Figure 5: An oset approximation (light curve) for a quaratic polynomial B-spline curve with eight control points. 0? In the case of osetting (piecewise) cubic curve segments, the least squares methos, an, perform much better than all the other methos, especially when high precision is require. In many examples, the local cubic B-spline interpolation metho, Pham, has similar { an sometimes even better { performance to. However, its performance eteriorates when the base curve has a raius of curvature similar to the oset raius. The only geometrical metho that approaches the eciency of the least squares methos is followe not so closely by. For the case of osetting (piecewise) cubic curves, the global least squares metho,, outperforms all the other methos, while it is closely followe by the local least squares metho,, an also by the local cubic B-spline interpolation metho, Pham. Many practical situations require the prouction of local optimal solutions base only on the local ata Figure 6: An oset approximation (light curve) for a quaratic polynomial with sharp corners. that is available. For example, for ata storage saving, we can store only the subivision locations of the curve, instea of all the control points that are generate. We then use the local methos to generate the control points (on the y) by consiering only local ata. In this case, an Pham are the methos of choice. As iscusse in the above observation, the performance of Pham is closely relate to the raius of curvature of the base curve. When the raius of curvature is similar to the oset raius, the sample oset points are clustere together. The B-spline interpolation of these clustere points generates unulation, which is the main source of large approximation error. In this case, it is better to use a smaller number of ata points for the interpolation. Figures {2 exemplify this phenomenon by comparing the relative performances of ierent oset approximation methos. Given a xe base curve, by increasing the oset raius graually, we can observe that Pham's 7

8 0? ? ? ? ? ? ? ? ? ? ? ? Offset Error 0? 0?2 0?3 0?4 0? Figure 7: An oset approximation (light curve) for a polynomial cubic Bezier curve. Offset Error 0? 0?2 0?3 0?4 0? Figure 8: An oset approximation (light curve) for a rational quaratic B-spline circle with nine control points. metho has the worst relative performance near the oset istance which starts to evelop cusps in the oset curve. For a better visualization of the relative performance, only four methos are shown in the graphs of Figures {2. There is another source of unulation we have to consier in Pham's metho. That is, the mismatch in spees between the two curves, i.e., the base curve an the oset curve, also cause eterioration in the quality of the oset approximation. For the implementation of Pham, we use a non-uniform cubic B- spline curve, in which the ata points of the oset inherit the knot values of the base curve points. When the oset ata points are clustere, their knot values are much sparser compare with the oset curve length. This unnatural assignment of knot values generates unulation. Therefore, for a better oset approximation, it is also important to rearrange the knot values of the oset ata points. The superior performance (in the quaratic case) of the simple metho,, suggests the possibility of improvement over the current least squares methos. This improvement may be achieve by resolving the limitations of least squares methos as is- cusse above. The limitation resulting from the L 2 norm seems more serious. To resolve this problem, we nee to evelop an ecient algorithm to compute an optimize the L norm of the oset approximation error; that is, the maximum of the error function (t) (in Equation (3)): max t nkc(t)? C a (t)k2? 2 o ; (2) or a more precise geometric istance measure base on the following Hausor metric: n o max max min kc(s)? C a t s (t)k 2? 2 ; (3) max s min t n kc(s)? C a (t)k2? 2 o where s is assume to be a local perturbation of the parameter t. Note that the metho of b [9] essentially moels the L norm of Equation (2) in terms of the maximum an minimum magnitues of the istance curve, D(t) C(t)? C a (t), in Figures 2{4. Let's consier a variant of b [9] which uses the least 8

9 0? ? ? ? ? ? Offset Error 0? 0?2 0?3 0?4 0?5 0? ? ? ? ? ? Offset Error 0? 0?2 0? Figure 9: An oset approximation (light curve) for a cubic polynomial B-spline curve with 3 control points. 0?4 0? squares technique to optimize the oset istance at each control point so that the istance curve D(t) is a best t to the oset circle of raius. This metho measures the oset error in the L sense of Equation (2). (Note that the approximation of D(t) to an oset circle still has the limitation of L 2 norm.) Although we have not provie all the etails of in this article, the metho of et al. [5] actually measures the oset approximation error uner the L norm of Equation (3), which is more precise than the L norm of Equation (2). We expect that future oset approximation techniques (while incorporating these L norms into their optimization proceures) may provie more accurate results than the current least squares methos. Conclusion We have compare several contemporary oset approximation techniques for freeform curves in the plane. In general, the least squares methos perform very well. However, for the case of osetting quaratic curves, the simple metho of ler an Hanson [6] is the metho of choice. Therefore, the least squares methos nee further improvement to prouce near-optimal solutions in all cases. Some of the current methos [5, 7, 9] have geometric repre- Figure 0: An oset approximation (light curve) for a cubic polynomial perioic B-spline curve with four control points. sentations of the oset approximation error (in certain L norms), whereas none of the current least squares methos have such geometric interpretation of their respective error bouns. We also pointe out two limitations of the current least squares methos: (i) the L 2 norm employe in these methos an (ii) the epenency on the nite sample points use in the optimization. The B-spline interpolation metho also nees further investigation to eliminate the curve unulation resulting from the curve spee mismatch between the base curve an the oset curve. In this respect, there are still many ways to improve the current state-of-the-art of oset curve approximation. We hope that the experimental results reporte in this article an the relate remarks will serve as useful guielines for future research. References [] chek, J., (988), \Spline Approximation of Oset Curves," Computer Aie Geometric Design, Vol. 5, pp. 33{40. 9

10 Offset Distance Offset Distance Figure 2: Pham's metho has the worst performance for the oset istances aroun 0.4. Base curve is a cubic B-spline curve. Tolerance of 0.00 is use for oset approximation error Figure : Pham's metho has the worst relative performance for the oset istances between 0.4 an 0.6. Base curve is a cubic B-spline curve. Tolerance of is use for oset approximation error. [2] chek, J., an Wissel, N., (988), \Optimal Approximate Conversion of Spline Curves an Spline Approximation of Oset Curves," Computer-Aie Design, Vol. 20, No. 8, pp. 475{483. [3] Farouki, R., an Ne, C., (990), \Analytic Properties of Plane Oset Curves" & \Algebraic Properties of Plane Oset Curves," Computer Aie Geometric Design, Vol. 7, pp. 83{99 & pp. 0{27. [4] Farouki, R., an Sakkalis, T., (990), \Pythagorean Hoograph," IBM J. Res. Develop., 34, pp [5], I.-K., Kim, M.-S., an er, G., (995), \Planar Curve Oset Base on Circle Approximation," to appear in Computer-Aie Design. [6] ler, W., an Hanson, E., (984), \Osets of Two Dimensional Proles," IEEE Computer Graphics & Application, Vol. 4, pp. 36{46. [7] er, G., an Cohen, E., (99), \Error Boune Variable Distance Oset Operator for Free Form Curves an Surfaces," International Journal of Computational Geometry an Applications, Vol., No., pp. 67{78. [8] er, G., (992). \Free Form Surface Analysis using a Hybri of Symbolic an Numeric Computation," Ph.D. thesis, University of Utah, Computer Science Department. [9] b, B., (984), Design of Sculpture Surfaces Using The B-spline Representation, Ph.D. thesis, University of Utah, Computer Science Department. [0] uillart, S., (987), \Computing Oset of B-spline Curves," Computer-Aie Design, Vol. 9, No. 6, pp. 305{309. 0

11 [] er, G., an Cohen, E., (992), \Oset Approximation Improvement by Control Points Perturbation." Mathematical Methos in Computer Aie Geometric Design II, Tom Lyche an Larry L. Schumaker, Acaemic Press, pp [2] ss, R., (983), \An Oset Spline Approximation for Plane Cubic Splines," Computer- Aie Design, Vol. 5, No. 4, pp. 297{299. [3] Pham, B., (988), \Oset Approximation of Uniform B-splines," Computer-Aie Design, Vol. 20, No. 8, pp. 47{474. [4], I.-K., Kim, M.-S., an er, G., (995), \Planar Curve Oset by Curve Tangent Matching," Technical Report, CS-CG-TR , Dept. of Computer Science, POSTECH, November, 995. [5] IRIT soli moeller (995), IRIT 5.0 User's Manual, Technion.