Degree reduction of Béezier curves using constrained Chebyshev polynomials of the second kind

Abstract In this paper a constrained Chebyshev polynomial of the second kind with C1-continuity is proposed as an error function for degree reduction of Bézier curves with a C1-constraint at both endpoints. A sharp upper bound of the L∞ norm for a constrained Chebyshev polynomial of the second kind with C1-continuity can be obtained explicitly along with its coefficients, while those of the constrained Chebyshev polynomial which provides the best C1-constrained degree reduction are obtained numerically. The representations in closed form for the coefficients and the error bound are very useful to the users of Computer Graphics or CAD/CAM systems. Using the error bound in the closed form, a simple subdivision scheme for C1-constrained degree reduction within a given tolerance is presented. As an illustration, our method is applied to C1-constrained degree reduction of a plane Bézier curve, and the numerical result is compared visually to that of the best degree reduction method.


Introduction
Degree reduction of Bezier curves is one of the most important problems in CAGD (Computer Aided Geometric Design) or CAD/CAM. In general, degree reduction cannot be done exactly, leading to approximation problems. Much effort has been directed at dealing with these problems in the past twenty years. Most publications focus on particular aspects of the problems, such as best degree reduction [7,13], -constraints [1,2,10,14], the L p -norm [3,8,11,[16][17][18] and simple algorithms [12,19,[21][22][23].
It is well-known that the error function of the best degree reduction in the uniform (Loo) norm is the Chebyshev polynomial up to the leading coefficient. But, in many actual CAD/CAM systems, it is required [2] that the approximate curve is continuous to order k > 0 at each junction point of consecutive curve segments. In order to reduce the degree of Bezier curves with C°, C 1 or C 2 -constraints at both endpoints, constrained Chebyshev polynomials are necessary as the error functions, which are the best approximate monic polynomials to zero [13,14] for each case. In the C°constraint case, constrained Chebyshev polynomials can be expressed in terms of classical Chebyshev polynomials, but the other cases can be obtained numerically using the (modified) Remes algorithm [4,22]. Recently, Kim and Ahn [10] proposed a good C 1 -constrained degree reduction method using constrained Jacobi polynomials. The method gives the coefficients of the polynomials explicitly, and also presents the Loo norm of the polynomials in closed form for even degree..
In this paper, we propose another method for degree reduction with C 1 -constraints using properly modified Chebyshev polynomials of the second kind. It is not the best C 1 -constrained degree reduction and its uniform error bound is also larger than the Loo norm of the constrained Jacobi polynomial with C 1 -continuity numerically [10]. But our method presents the explicit form of a sharp error bound of the L^ norm for all degrees along with the coefficients. We also present a simple subdivision scheme using the uniform error bound of our method in closed form. We apply our method to the degree reduction of a plane Bezier curve and compare the numerical result to that of the best degree reduction method.
The outline of this paper is as follows. In Section 2, we introduce our method of C 1 -constrained degree reduction using constrained Chebyshev polynomials of the second kind with C 1 -continuity. We also present explicitly their uniform error bound and the control points in Bezier form, which are useful in actual CAD/CAM systems. Using the uniform error bound we give the subdivision scheme for the C 1 -constrained degree reduction within a specified tolerance. In Section 3, we give an example of the C 1 -constrained degree reduction of a plane Bezier curve of degree seven using our method, and compare its result to that of the best degree reduction by plotting the graphs of the degree reduced Bezier curves. In Section 4, we summarise our work.

Degree reduction with C 1 -constraints
In this section, we introduce our method for degree reduction of a Bezier curve with a C'-constraint at both endpoints. It is well-known [14] that the best C 1 -constrained degree reduction has the constrained Chebyshev polynomial with C 1 -continuity as an error function, but the polynomial can be obtained numerically using the Remes algorithm. A good C 1 -constrained degree reduction using the constrained Jacobi polynomial with C'-continuity [10] has the error bound explicitly for even degree. We use constrained Chebyshev polynomial of the second kind with C'-continuity as an error function for C 1 -constrained degree reduction, since the polynomial has explicit representation in terms of Bezier coefficients and uniform error bound for all degrees. The following are some well-known properties for Chebyshev polynomials of the second kind which we shall call on later in the paper. PROPERTY 2.1 (Refer to [6,20]). The Chebyshev polynomial of the second kind, U n {x) = sin((n + l)0)/sin(0), 6 = arccosx, of degree n, has the following properties: (a) U n (x) has leading coefficient 2"; (b) The zeros of £/"(*) are x = cos(kn/(n + 1)), it = 1 , . . . . n, and the largest zero of U n (x) is cos(n/(n + 1)) which is denoted by fi n in this paper; (c) 2~" U n (x) has the smallest L^ norm on [-1, 1] amongst all monic polynomials weighted by Vl - We define the constrained Chebyshev polynomial of the second kind with C l -continuity for n > 4 by t{tfor / 6 [0, 1], where mu n -i = cos(jr/(n -1)) is the largest zero of U n -zix). THEOREM 2.3. The constrained Chebyshev polynomial of the second kind with C 1continuity E n it) is a monic polynomial of degree n, has double zeros at t = 0, 1, and its uniform norm is bounded by In particular, equality holds for even n.
As an illustration, we plot the constrained Chebyshev polynomial of the second kind with C 1 -continuity, 4"" 1 cos"~2(n/(n -l))E n (t), uniformised by the error bound in Theorem 2.3, for 4 < n < 10, in Figure 1. As shown in Table 1, we compare the uniform error bound of £"(?) which is obtainable explicitly with the uniform

PROPOSITION 2.4. Let f (t) = £ " = 0 bjBfit) be the given Bezier curve of degree n having control points b t . Then the Bezier curve f (t) of degree (less than or equal to) n -1 given by f (t) := f (t) -A"boE n (t) is a degree reduction of f (t) with C 1 -constraint at both endpoints, and its error norm is bounded by
11/(0 -4"-' cos"-2 (n/(n -1)) ' where the nth forward difference A"b 0 = Yl"=o(~ l)'(")^n-/ ' •* equal to the leading coefficient of the polynomial f (t) of degree n.
PROOF. Since the n-th degree polynomials f (t) and A"b 0 E n (t) have the same leading coefficient, / (t) is a polynomial of degree less than or equal to n -1. The jfc-th order derivative £<*'(') = 0 for * = 0, 1, at r = 0, 1, yields that/ (i) (O = / w (t) and / (r) is a degree reduction of / (r) with a C 1 -constraint at both endpoints. The error bound is easily obtained from Theorem 2.3.
The following proposition gives the closed form of the control points of the constrained Chebyshev polynomials of the second kind with C 1 -continuity in Bezier form, which is needed to calculate / (t) as a Bezier curve or a segment of spline in CAD/CAM systems. Using the proposition above we represent the degree reduction / (f) of/ (r) in Bezier form: where fc,-, i = 0, . . . , n -1, are the Bezier coefficients of/ (t) of degree n -1.
Bezier coefficients hi of f (t) are given by
In the practical application, for a given tolerance it is necessary to subdivide the Bezier curve/ (t) of degree n into k pieces in order to approximate each piece by the lower degree Bezier curve within the tolerance. In the following theorem, we show how many subdivisions are required so that each piecewise degree reduction using our method has error less than the given tolerance. THEOREM 2.7. For a given tolerance s, the Bezier curve f (t) = ^2" i=o biB"(t) must be subdivided into k segments so that the degree reduction for each segment has uniform error less than s, where k is given by "-2 (7T/(n-l))j l/n" (2.4) and [x~\ denotes the smallest integer larger than x.
In order to satisfy that the uniform error bound be less than s, (2.4) holds by Theorem 2.3.

Example
In this section, we apply our method to reduce the degree of a plane Bezier curve of degree seven. Let the Bezier curve be given by [7]   given f(t) best f best (t) _ --*• -our method f(t) FIGURE 2. The best degree reduction and our method: the given Bezier curve / ( / ) , the best degree reduction /" be "(») and our method/(/) are plotted by solid lines, dashed lines, and dashed lines with crosses, respectively. The boxes, triangles and circles are the control points of each Bdzier curve, in order.
On the other hand, the best degree reduction by the constrained Chebyshev polynomial yields / best (r) = / By subdividing / (t) at t = 1/2 into two Bezier segments as shown in Figure 3, the uniform error bound is given by x 2 7 0.000263 < 0 001. and the degree reduction with C 1 -constraint for each Bezier segment is achieved within the specified tolerance.
given f(t) first segment •x--second segment FIGURE 3. Degree reduction using the subdivision scheme: the degree reductions using our method £"(/) for the first and second segments are plotted by dashed lines and dashed lines with crosses, respectively. The circles and boxes are the control points of the degree reduction for each subdivision Bezier segment, in order.

Comments
In this paper we presented a method for degree reduction of Bezier curves with the C 1 -constraint at both endpoints having the explicit form of the uniform error bound. The constrained Chebyshev polynomial of the second kind with C 1 -continuity E n (t) proposed in this paper has the minimum L^ error bound among the monic polynomials which have zeros of multiplicity two at both endpoints / = 0, 1, and for which the uniform error bounds are known explicitly. Even if E n (t) is not the best degree reduction with a C 1 -constraint and its error bound is larger than that of the constrained Jacobi polynomial with C 1 -continuity proposed by Kim and Ahn [10], it is more useful than those polynomials since no numerical calculations are needed. That is to say, the Bezier coefficients and the uniform error bound of E n (t) in explicit form can be obtained for any degree. We also gave a simple subdivision scheme and the numerical results for an example using our degree reduction method.