Bounding Interval Rational Bézier Curves with Interval Polynomial Bézier Curves

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1 Brigham Young University BYU ScholarsArchive All Faculty Publications Bounding Interval Rational Bézier Curves with Interval Polynomial Bézier Curves Thomas W. Sederberg Falai Chen See next page for additional authors Follow this and additional works at: Part of the Computer Sciences Commons Original Publication Citation Falai Chen, Thomas W. Sederberg and Wenping Lou, "Bounding interval rational Bézier curves with interval polynomial Bézier curves," Geometric modeling and processing, IEEE Computer Society, 3836,. BYU ScholarsArchive Citation Sederberg, Thomas W.; Chen, Falai; and Lou, Wenping, "Bounding Interval Rational Bézier Curves with Interval Polynomial Bézier Curves" (000). All Faculty Publications This Peer-Reviewed Article is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Faculty Publications by an authorized administrator of BYU ScholarsArchive. For more information, please contact

2 Authors Thomas W. Sederberg, Falai Chen, and Wenping Lou This peer-reviewed article is available at BYU ScholarsArchive:

3 Bounding Interval Rational Bézier Curves with Interval Polynomial Bézier Curves Falai Chen Department of Mathematics University of Science and Technology of China Hefei, Anhui, 3006, China Thomas W. Sederberg Computer Department Brigham Young University Provo, Utah 8460, USA Wenping Lou Department of Mathematics University of Science and Technology of China Hefei, Anhui, 3006, China Abstract In this paper, we put forward and study the problem of bounding an interval rational Bézier curve with an interval polynomial Bézier curve. We propose three different methods Hybrid Method, Perturbation Method and Linear Programming Method to solve this problem. Examples are illustrated to compare the three different methods. The empirical results show that the Perturbation Method and the Linear Programming Method produce much tighter bounds than the Hybrid Method, though they are computationally several times more expensive. Introduction In the communities of Approximation Theory and Computer Aided Geometric Design(CAGD), there is considerable interest in approximating functions (or curves/surfaces) with functions (or curves/surfaces) of simpler forms, for example, in approximating rational functions (or curves) with polynomial functions (or curves). Much literature has focused on these problems from different points of views, e.g., from pure approximation theory to applications in CAGD. However, as far as the authors are aware, almost all the related work is concerned with how accurate the approximation is and seldom consider the problem of how to transfer the approximation errors into subsequent applications. This problem can be important in some applications, e.g., in tolerance analysis in CAD and in numerical analysis. To solve this problem, Sederberg, et. al. [] introduced interval forms of curves and surfaces. Based on the new representations of curves and surfaces, several authors developed robust algorithms for geometric operations such as curve/curve intersections in CAD/CAM systems ([3] [8]). In this paper, we are interested in the following problem, a problem that has potential applications in CAD and numerical analysis: Given an interval rational function (or curve), bound it with an interval polynomial function (or curve) such that the bound is as tight as possible. We will develop three different methods Hybrid Method, Perturbation Method and Linear Programming Method to solve the problem in the following sections. Testing and comparisons are made between the three different approaches through illustration of examples. The empirical results show that the Hybrid Method generally produces much looser bounds than the other two methods, though it is computationally less expensive.. Interval Bézier Curves An interval [a; b] is the set of real numbers fxja 6 x 6 bg. Interval arithmetic operations are defined by [a; b] Λ [c; d] =fx Λ yjx [a; b] and y [c; d]g; () where Λ represents an arithmetic operation, Λ f+; ; ;=g. One can verify that [a; b] +[c; d] = [a + c; b + d] [a; b] [c; d] = [a d; b c] [a; b] [c; d] = [min(ac; ad; bc; bd); max(ac; ad; bc; bd)] [a; b]=[c; d] = [a; b] [=d; =c]; 0 6 [c; d]: Interval arithmetic is an important tool in numerical analysis, and it has many applications in other areas. For details, the reader is referred to Moore s book Interval Analysis []. ()

4 An interval polynomial is a polynomial whose coefficients are intervals: [p](t) := n [a k ;b k ]B n k (t); 0 6 t 6 ; (3) where B n k (t) = n k tk ( t) n k, k =0; ;::: ;nare Bernstein bases. An interval polynomial can be also expressed in the form where [R i ] and w i > 0, i =0; ;::: ;nare interval control points and weights respectively. An interval (polynomial or rational) Bézier curve defines a region (a slender tube) in the plane which consists of all the Bézier curves whose control points satisfy P i [P i ] for i =0; ;::: ;n. Figure shows a sample cubic interval polynomial Bézier curve. An interval polynomial (or ratio- [p](t) =[p min (t);p max (t)]; 0 6 t 6 ; (4) where and p min (t) = p max (t) = n n a k B n k (t) b k B n k (t) p min (t) and p max (t) are called lower bound (denoted by lb([p](t)) )andupper bound (denoted by ub([p](t)) )of [p](t) respectively. The width of an interval polynomial can be defined by W ([p](t)) = jjp max (t) p min (t)jj; (5) where the norm jj:jj is the standard norm, such as jj:jj, jj:jj and jj:jj. An interval rational function is defined [r](t) := [a k;b k ]! k B n k (t)! kb n k (t) ; 0 6 t 6 ; (6) where! k, k =0; ;::: are weights (It is also possible to define the weights to be intervals. Since the algorithms in the following sections are similar for both cases, we assume the weights are real numbers in this paper). It is similar to define the upper bound, the lower bound and the width of an interval rational function. An interval polynomial Bézier curve is a polynomial Bézier curve whose control points are vector-valued intervals (i.e., rectangular regions in a plane): [P](t) = n [P i ]Bi n (t); (7) where [P i ] = [a i ;b i ] [c i ;d i ] = ([a i ;b i ]; [c i ;d i ]), i = 0; ;::: ;nare the interval control points. Similarly, an interval rational Bézier curve is defined [R](t) = [R i]! i B n i (t) w ib n i (t) ; (8) Figure. A sample interval Bézier curve nal) Bézier curve can be viewed as a vector-valued interval polynomial (or rational function). For example, the interval polynomial Bézier curve (7) can be rewritten in the form [P](t) =([x](t); [y](t)); (9) where [x](t) and [y](t) are interval polynomials: [x](t) = [y](t) = n n [a k ;b k ]Bk n (t) (0) [c k ;d k ]Bk n (t) () With this form, the problem of bounding interval rational Bézier curves with interval polynomial Bézier curves can be conveniently converted to the problem of bounding interval rational functions with interval polynomials. 3. Bounding Interval Rational Bézier Curves with Interval Polynomial Bézier Curves Before dealing with the problem of bounding interval rational Bézier curves with interval polynomial Bézier curves, we solve the problem of bounding an interval rational function with an interval polynomial. Problem Given an interval rational function [r](t) := find an interval polynomial [p](t) := [r k]! k B P k n (t) n w kbk n ; () (t) m [p k ]Bk n (t) (3)

5 such that [p](t) bounds [r](t) [r](t) ρ [p](t); for 0 6 t 6 ; (4) and that the width of [p](t) is as tight as possible. [p](t) is called an interval polynomial bound of [r](t). Since the process of finding an upper bound is similar to that of finding a lower bound, we can solve the following problem instead Problem Given a rational function of degree n R(t) := find a polynomial of degree m such that and P (t) := R k! k B P k n (t) n! kbk n ; (5) (t) m P k Bk n (t) (6) P (t) > R(t); 0 6 t 6 (7) jjp (t) R(t)jj (8) is minimized. In the following, we will propose three different methods Hybrid Method, Perturbation Method and Linear Programming Method to solve the above problem. 3.. Hybrid Method In [], Sederberg, et al proposed a method called the hybrid curve method to approximate rational functions with polynomials. Indeed, this method also provides a way to compute an upper polynomial bound of a rational function. The main idea of the Hybrid Method is as follows. Any rational function R(t) can be expressed as a polynomial, one of whose coefficients is a rational function: r k! k Bk n R(t) = P (t) n! kbk n (t) m ;k6=l P k B m k (t) +M (t)bm l (t); (9) where M (t) is a rational function of degree n: M (t) = M k! k B n k (t)! kb n k (t) ; (0) and l =[m=] is an integer. The coefficients of P i, i =0; ;::: ;m;i 6= l and M i, i =0; ;::: ;ncan be computed as follows: For i =0; ;::: ;l, P 0 = R 0 ; P = R 0 + n!(r P0) m! 0 ;. P i = R 0 + min(n;i) P j= For i = m + n;::: ;n+ l +, ( n j)( m i j)! j (Rj P i j ) ( m i )! 0 : P m = R n ; P m = R n + n!n (Rn Pm) m! n ; P i n = R n + and M i =. min(i+l;n) j=max(0;i+l m) n P j=max(0;i m) n j m ( n j)( i j)! m j (Rj Pi j ) ( i n)! m ; n m l i+l j!j (R j P i+l j ) () () n i!i ; i =0; ;::: ;n: (3) Theorem Let M max = max 06k6nM k and M min = min 06k6nM k. Then polynomial P (t) = m ;k6=l is an upper bound of R(t), and P k B m k (t) +M maxb m l (t) (4) jjp (t) R(t)jj 6 c(mmax M min ); (5) where c is a constant: l m l m l m l c = : (6) l m m Proof: Since from (9), one has R(t) 6 M (t) 6 Mmax; 0 6 t 6 ; m ;k6=l P k B m k (t) +M maxb m l i.e., P(t) is a polynomial upper bound of R(t). On the other hand, (t) =P (t); jjp (t) R(t)jj 6 (Mmax M min )max 06t6B m l (t): Since Bl m (t) attains maximum value at t = l m in [0; ], (5) follows immediately. The Hybrid Method generally provides a loose bound for a rational function, and it has the disadvantage that there is a restricted condition under which the polynomial converges to the rational function when m!. For the specific convergence condition, the reader is referred to [0].

6 3.. Perturbation Method The Hybrid Method presented in the last subsection generally produces quite a loose bound. In this subsection, we will propose an improved algorithm to solve Problem. Given rational function (5), we perturb R(t) with another rational function ffl(t) such that R(t) +ffl(t) is a polynomial where and R(t) +ffl(t) =P (t) :=! 0 k = m+n m m P m+n ffl k! 0 k ffl(t) = Bm+n k (t) P m+n!0 k Bm+n k (t) i+j=k From (7), we have = n m! i k i m+n R k! k Bk n (t) + P i B m i (t) n j=0 P k Bk m (t); (7) (8) m + n k : (9) m i ffl k! 0 kb m+n k! j Bj n (t): (30) Writing both sides of the above equation in Bernstein form and comparing the coefficients, one get ffl k = min(k;m) i=max(k n;0) (P i R k i )ff k;i ; (3) k = 0; ;::: ;m + n, where for max(k n; 0) 6 i 6 min(k; m),! k m+n k k i i ff k;i = P j+l=k m i! l k j m+n k m j ; (3) otherwise, ff k;i =0. Now we wish to make the perturbation ffl(t) is as small as possible. To this end, we minimize P m+n ffl k.let L(P 0 ;::: ;P m ):= := min(k;m) i=max(k n;0) m+n ffl k (P i R k i )ff k;i A (33) From l min(k;m) (P i R k i )ff k;i ff k;l i=max(k n;0) =0; l =0; ;::: ;m; (34) we obtain a system of linear equations for P i, i = 0; ;::: ;m: 0 m min(i+n;l+n) 0 m min(i+n;l+n) k=max(i;l) ff k;i ff k;l A Pi ff k;i ff k;l A Rk i ; (35) l =0; ;::: ;m:by solving the above equations and substituting P i into (3), the values of ffl i are obtained. Now a polynomial upper bound of R(t) can be computed as follows. Theorem Let fflmax = max 06i6m+nffl i and ffl min = min 06i6m+nffl i. Then polynomial P (t) ffl min is an upper bound of R(t), and jjp (t) ffl min R(t)jj 6 fflmax ffl min : (36) Proof: From (7), we have R(t) 6 P (t) ffl min ; which means P (t) ffl min is an upper bound of R(t). To get (36), one need only notice that P (t) R(t) = ffl(t) Linear Programming Method In this approach, we take jjp (t) R(t)jj = m + m R 0 R(t)dtis a con- as the minimization target, where c = stant. To satisfy condition (7), one must have m! k B n k (t) > n n P k Bk m (t) for any t [0; ]: Or equivalently, m+n i+j=k > m+n i+j=k P i! j k i R j! j k i m + n k m i P k c (37) R k! k Bk n (t); (38) m + n k m i B m+n k (t) B m+n k (t):

7 Thus a sufficient condition for (7) to hold is d k := i+j=k (P i R j )! j k i m + n k m i > 0; (39) k = 0; ;::: ;m+ n. An immediate consequence of the above discussion is the following Theorem 3 Let P i, i =0; ;::: ;mbe the solutions of the following linear programming problem ρ Min P m P i s.t. d k > 0; k =0; ;::: ;m+ n: (40) Then polynomial P (t) defined in (6) is an upper bound of R(t), and where! 00 k =,, :::,m+n. jjp (t) R(t)jj 6 max 06k6m+n min(k;m) i=max(k n;0)! k i k i d k! 00 k ; (4) m + n k ; (4) m i Proof: We need only prove the second part of the theorem. To estimate jjp (t) R(t)jj, we write P (t) R(t) in a rational Bernstein polynomial: P (t) R(t) = P m+n ( m+n k (t) m P m+n!0 k Bm+n k (t) ) d kb m+n where w 0 k is defined as in (9). Now (4) follows directly from the above equation. Remarks: The Hybrid Method automatically solves the constrained upper bound problem, i.e., in addition to satisfying (5 8), the following constraints must be held: 8 < : P (i) (0) = R (i) (0) P (i) () = P (i) () i =0; ;::: ;μ; (43) where μ 6 [ m ]. The Perturbation Method and the Linear Programming Method can also be adapted to solve the constrained upper bound problem. For the Perturbation Method, just set ffl k =0, k =0; ;::: ;μ;m μ;::: ;m, and solve for the corresponding P k from (3). The remaining P k can be obtained from (35) with l = μ + ;::: ;m μ. For the Linear Programming Method, P k, k =0; ;::: ;μ;m μ;::: ;m can be solved by setting the corresponding d k = 0, and the remaining P k are obtained by solving the linear programming problem (40). ; 3.4. Bounding Interval Rational Bézier Curves with Interval Polynomial Bézier Curves In the previous subsections, we derived three different methods to bound an interval rational function with an interval polynomial. These three methods lead directly to algorithms of bounding an interval rational Bézier curve with an interval polynomial Bézier curve. The main results are based on the following fact. Theorem 4 Given an interval rational Bézier curve of degree n If and [R](t) =([x](t); [y](t)): (44) [μx](t) = [μy](t) = m m [μx i ]B m i [μy i ]B m i (t) (45) (t) (46) are degree m interval polynomial bounds of [x](t) and [y](t) respectively, then interval polynomial Bézier curve [P](t) = ([μx](t); [μy](t)) = m ([μx i ]; [μy i ])B m i (t) (47) bounds interval rational Bézier curve [R](t), i.e., [R](t) ρ [P](t). Proof: Straightforward. To measure how tight a bound is, we define the bounding error as follows e([x](t); [μx](t)) = max(jjub([x](t)) ub([μx](t))jj ; jjlb([x](t)) lb([μx](t))jj ) e([y](t); [μy](t)) = max(jjub([y](t)) ub([μy](t))jj ; jjlb([y](t)) lb([μy](t))jj ) e([r](t); [P](t)) = max(e([x](t); [μx](t)); e([y](t); [μy](t))) (48) It is a little hard to compute the bounding errors by the above definitions. Fortunately, Theorems, and 3 give a good estimation for the bounding error by each of the three methods. For a given interval rational Bézier curve [R](t) and tolerance ffl > 0, we use one of the three methods to find an interval polynomial Bézier curve [P](t) to bound [R](t). If the bounding error is larger than ffl, we subdivide [R](t) at parameter value t == and then bound each segment

8 with an interval polynomial Bézier curve respectively. This process is continued until the bounding error is less than ffl for each segment. At last, we find a piecewise interval polynomial Bézier curve which bound the original interval rational Bézier curve [R](t). 4. Examples and Comparison In this section, we will compare the bounding errors and the computational costs between the three methods Hybrid Method (HM), Perturbation Method (PM) and Linear Programming Method (LPM) for bounding an interval rational Bézier curve with an interval polynomial Bézier curve through illustration of examples. We implemented these examples on a 66 MHZ Pentium PC. Example Let [r](t) be a cubic interval rational function with the following interval coefficients and weights: [a 0 ;b 0 ]=[0; =4];! 0 =; [a ;b ]=[; 6=5];! =; [a ;b ]=[ ; 3=4];! =; [a 3 ;b 3 ]=[; 7=6];! 3 = We use degree m =4; 5 and 6 interval polynomials to bound [r](t) respectively. The bounding errors (B.E.) and the computational time (C.T.) in CUP seconds for the three methods are list in Table.. Figure. Figure.0 show the corresponding figures. Table. Comparison of the bounding errors and the computational costs (I) m HM PM LPM B.E. C.T. B.E. C.T. B.E. C.T. m = m = m = Figure 3. Hybrid Method m = Figure 4. Hybrid Method m = Figure 5. Perturbation Method m = Figure. Hybrid Method m =4 Figure 6. Perturbation Method m =5

9 in Table Table. Comparison of the bounding errors and the computational costs (II) Figure 7. Perturbation Method m =6 HM PM LM m errors time errors time errors time m = m = m = Figure 8. Linear Programming Method m = Figure. Hybrid Method m =4 Figure 9. Linear Programming Method m = Figure 0. Linear Programming Method m =6 Figure. Hybrid Method m =5 Example In this example, we consider a case where the Hybrid Method diverges while the other two methods converges. The interval rational function [r](t) is defined by [a 0 ;b 0 ]=[ =8; =8];! 0 =; [a ;b ]=[7=8; ];! =3; [a ;b ]=[7=4; ];! =8; [a 3 ;b 3 ]=[7=8; ];! 3 =; [a 4 ;b 4 ]=[ =8; =8];! 4 = The degree 4, 5 and 6 interval polynomial bounds by the three methods are shown in Figure -Figure 9. The corresponding bounding errors and computational time are list Figure 3. Hybrid Method m =6

10 Figure 4. Perturbation Method m = Figure 9. Linear Programming Method m = Figure 5. Perturbation Method m =5.5 Example 3 In the last example, we use piecewise interval polynomial curves to bound an interval rational curve. Let the control points and weights of the interval rational curve be: [R 0 ] = ([50; 70]; [340; 355]);! 0 =; [R ] = ([90; 0]; [00; 0]);! =; [R ] = ([340; 360]; [330; 340]);! =; [R 3 ] = ([430; 455]; [0; 30]);! 3 = We recursively subdivide the interval rational curve at the parameter value t = and then bound each segment using an interval polynomial curve of degree m =4and 5 respectively. Table 3 and 4 list the corresponding bounding errors and the computational time for the three methods Figure 6. Perturbation Method m =6 Table 3. Comparison of the bounding errors and computational costs (III) (m =4) Figure 7. Linear Programming Method m = Figure 8. Linear Programming Method m =5 number of HM PM LM subdivisions errorstime errorstime errors time From the above examples, we can draw a conclusion that the bounding errors by the Perturbation Method and the Linear Programming Method are much smaller than those by the Hybrid Method, even though they are computationally several times more expensive. Consequently, if we use a piecewise interval polynomial curve to bound an interval rational curve such that the bounding error is less than some given tolerance, the Hybrid Method will generally need more segments than the other two methods. Furthermore, examples show that the Hybrid Method has a much more restrict convergence condition than the other two methods do.

11 Table 4. Comparison of the bounding errors and computational costs (III) (m =5) number of HM PM LM subdivisions errorstime errorstime errors time Conclusions In this paper, we put forward a new problem in CAD and interval analysis communities bounding an interval rational function (or Bézier curve) with an interval polynomial function (or Bézier curve). We proposed three different methods to solve the problem, and comparisons of bounding errors and computational costs are made between these three methods. The experimental results show that the Perturbation Method and the Linear Programming Method produce much tighter bounds than the Hybrid Method does, although they are computationally several times more expensive. Further research problems include: () for the Perturbation Method and the Linear Programming Method, explore whether the interval polynomial bounds converge to the original interval rational curve when m goes to infinity; and () generalizing the results in this paper to surface cases. Acknowledgement The first and third authors are supported by the National Natural Science Foundation of China (grants and ) and the 973 Project on Mathematical Mechanics (grant G ). The authors are sincerely thankful for the financial support of K.C. Wong Foundation. References [] R.E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, NJ, 966. [] T.W. Sederberg, and R.T. Farouki, Approximation by interval Bézier curves, IEEE Comput. Graph. Appl., 99, 5(), [3] C.Y. Hu, T. Maekawa, E.C. Sherbrooke, and N.M. Patrikalakis, Robust interval algorithm for curve intersections, Computer Aided Design, 996, 8(6/7), [4] C.Y. Hu, N.M. Patrikalakis, and. Ye, Robust interval solid modeling: Part I: representations, Computer Aided Design, 996, 8(0), [5] C.Y. Hu, N.M. Patrikalakis, and. Ye, Robust interval solid modeling: Part II: boundary evaluation, Computer Aided Design, 996, 8(0), [6] C.Y. Hu, T. Maekawa, N.M. Patrikalakis, and. Ye, Robust interval algorithm for surface intersections, Computer Aided Design, 997, 9(9), [7] S.T. Tuohy, T. Maekawa, G. Shen, and N.M. Patrikalakis, Approximation of measured data with interval B-splines, Computer Aided Design, 997, 9(), [8] F. Chen, and W. Lou, Degree Reduction of Interval Bézier curves, accepted by Computer Aided Design. [9] T.W. Sederberg, and M. Kakimoto, Approximating rational curves using polynomial curves, NURBS for Curve and Surface Design, G.Farin,eds.,SIAM, Philadelphia, 99, [0] G. Wang, T.W. Sederberg, and F. Chen, On the convergence of polynomial approximation of rational functions, Journal of Approximation Theory, 997, 89,