Geometric meanings of the parameters on rational conic segments

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1 Science in China Ser. A Mathematics 005 Vol.48 No Geometric meanings of the parameters on rational conic segments HU Qianqian & WANG Guojin Department of Mathematics, Zhejiang University, Hangzhou 3007, China; State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 3007, China Correspondence should be addressed to Wang Guojin wgj@math.zju.edu.cn) Received June 4, 004; revised December 3, 004 Abstract Using algebraic and geometric methods, functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve. That is, the inverse mappings of the mappings represented by the expressions of rational conic segments are given. These formulae relate some triangular areas or some angles, determined by the selected point on the curve and the control points of the curve, as well as by the weights of the rational Bézier curve. Also, the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment, as well as by the weights of the rational Bézier curve. These results are greatly useful for optimal parametrization, reparametrization, etc., of rational Bézier curves and surfaces. Keywords: rational Bézier curve, conic segment, ellipse, hyperbola, parabola, parameterization. DOI: 0.360/04ys068 Introduction In geometric shape design and machine building, we always need to construct and express the shape of the combinatorial curves consisting of conic segments, line segments and free-form curves. Therefore, circular arcs and conic segments, two simple elements commonly used in shape expression and mechanical accessory cartography, play an important role in modeling systems. In early CAGD systems in the world, the mathematical representations of circular arcs and conic segments were implicit functions, geometric splines, Ball curves, etc. After introducing NURBS into modeling systems, rational Bézier curves became the core models in curve design systems, because they can express not only polynomial curves e.g. lengthways curves on airframe) but also conic segments e.g. curves of cross section on airframe). Since then, a great number of treatises on low degree rational Bézier representation of conics have come forth [ 0]. As we know, the rational Bézier curve which represents the same conic segment and adopts the same control points is not unique, because there are different parametrization Copyright by Science in China Press 005

2 0 Science in China Ser. A Mathematics 005 Vol. 48 No methods. In order to choose the optimal parameter of the conic segment in CAD, it is important for us to thoroughly understand the geometric relationship between the point on the curve and the parameter. Besides, this relationship must be used in the algorithms of subdivision of rational Bézier curves, geometric continuity of rational Bézier surfaces, etc. And also sometimes people need to seek correct approaches and ways to the change of the parametrization, because conic segments are usually regarded as skeleton curves to construct blending surfaces. However, up to now, only Shi Fazhong [0,] has derived one of the geometric meanings of the parameter corresponding to the point on the rational quadratic conic segment, and obtained a functional relationship between the point on the circular arc and the corresponding parameter expressed by its central angle, when the circular arc is represented by a standard rational quadratic Bézier curve. As for elliptic, hyperbolic, or parabolic segments, represented by rational quadratic or cubic Bézier curves, the functional relationship between the point on the curve and the corresponding parameter is not well studied yet. Based on deriving these other geometric meanings of the parameter, corresponding to the point on the rational quadratic conic segments, this paper gives all the function relationships which are mentioned above, by using trigonometric, geometric principles and the necessary and sufficient conditions for the rational cubic Bézier representation of conics [8]. In other words, we formulate the inverse mappings of the mappings, represented by the expressions of rational conic segments. These formulae are expressed by some triangular areas or some angles, determined by the selected point on the curve and the control points of the curve, as well as by the weights of the rational Bézier curve. Also, the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment, as well as by the weights of the rational Bézier curve. These results are greatly useful for optimal parametrization, reparametrization, etc., of rational Bézier curves and surfaces. Relation between the point on rational quadratic conic segment and the corresponding point in its parametric field. Rational quadratic conic segments The rational quadratic Bézier representation of conic segment is represented by see Fig. ) Rt)= Bi t)ω ir i i=0, 0 t. ) Bi t)ω i i=0 Where Rt) is the point on the conic segment, corresponding to parameter t; R i =x i, y i ), i = 0, ), are the three control points; ω i > 0, i = 0,, are the three weights; Bi t) = t) i t i are the bases of quadratic Bernstein polynomial. i Copyright by Science in China Press 005

3 Geometric meanings of parameters on rational conic segments Fig.. The conic segment represented by a rational quadratic Bézier curve and the point P on the curve. We first prove the necessary and sufficient condition for setting a point on the rational quadratic Bézier curve in the plane, and then present some geometric meanings of the parameter, corresponding to an arbitrary point on the curve. Now we assume that R 0, R, R are not collinear, otherwise the curve Rt) is a degenerative conic section, that is, a linear segment. Theorem. Suppose P is an arbitrary point in the triangle R 0 R R, and 0,, are the directed areas of PR R, R 0 PR, R 0 R P respectively. The necessary and sufficient condition for the point P being on the rational quadratic Bézier curve ) is as follows: 4 0 = ω ω. ) Proof. Let be the directed area of R 0 R R. If the point P is on curve ), then there exists a real number t [0, ], such that P= t) t)tω R 0 + R + Bi t)ω i Bi t)ω i i=0 i=0 t ω R. Bi t)ω i By the definition of barycentric coordinates, the above formula implies that t) i=0 B i t)ωi, t)tω i=0 B i t)ωi, t ω i=0 0 = i=0 B i t)ωi ) T. Then relation ) holds. On the contrary, if ) holds, then there is ) ω = 4 ω 0. Find the ratio of 0 to in the ω above formula, and let the parameter t= 0 +ω 0 [0, ], then we have ) t), t)tω, t T 4 ω ω = 0 + ω 0 ) 0.

4 Science in China Ser. A Mathematics 005 Vol. 48 No So we have Rt)= 0 R 0 + R + R )=P, namely the point P is on the curve ). The proof of Theorem is completed. Theorem. Suppose the point P is on the rational quadratic Bézier curve ). Then its corresponding parameter must be represented by ω ω0 t= = = +ω 0 ω +ω ω0 +, 3) ω 0 where the meanings of 0,, are the same as those in Theorem. ) ω Proof. By Theorem, we know R 0 +ω 0 = P. And also because there is no double point on a rational quadratic Bézier curve, there must exist a one-to-one ω correspondence between the point P and the parameter t = 0 +ω 0. Next we can rewrite 3) as t= + ω = 0 + =. 4) ω ω 0 + ω And by ), it is easy to know that ω 0 = 0 =. ω Then 4) is proved, thereby 3) holds and the proof of Theorem is completed. ω ω It must be pointed out that the third formula in 3) was first put forward by Shi Fazhong [0]. However, his proof had to apply the results from Faux et al. [] Our proof can not only be directly acquired from Theorem, but also incidentally obtain the other two formulae in 3) at the same time. Theorem 3. Suppose the point P is on the rational quadratic Bézier curve ). Then its corresponding parameter is represented by t= ), 5) ω cot β cot C + cot α cot A where α= PR 0 R, β = PR R 0, A= R R 0 R, C = R R R 0 see Fig. ). Fig.. The triangle consisting of the linked lines between the point on the rational conic segment and the control points. Copyright by Science in China Press 005

5 Geometric meanings of parameters on rational conic segments 3 Proof. Using the sine law, we obtain PR = sin α PR 0 sin β, R R = sin A R 0 R sin C. Thus 0 = area PR R ) area R 0 R P) = R P sin R PR ) PR = R R sin R R P) R P sin R PR 0 ) PR 0 R 0 R sin R R 0 P) sin α sin β = sin A sinc β) sin C sina α) sin α sin β = cot β cot C cot α cot A. 6) Then 5) is obtained from the third formula in 4). That is the end of the proof. Theorem and Theorem 3 essentially formulate the inverse mapping R R t [0, ], of the mapping t [0, ] R R, represented by ) for the rational quadratic conic segment. That is, using the expression ) we can determine the position of the point on the curve, for every given parameter t; conversely, using the formula 3) or 5) we can evaluate the parameter t for every given point on the curve. The quantities affecting the value of the parameter t are the weights of the conic segment, as well as some geometric quantities such as the triangular areas, the angles, etc, determined by the control points and the selected point on the conic segment. Next we discuss the geometric meanings of the parameter, when the curve is an elliptic segment, hyperbolic segment, or parabolic segment respectively. The analysis has shown that the parameter value can be uniquely determined by the weights of the conic segment and the parametric angles, of the selected point and two endpoints on the curve. Now we define the location angle as follows: Definition. Let the point P be on the conic segment, the point O be the origin in coordinate system. The directed angle from the positive axis of abscissas x to the ray OP is defined as the location angle of the point P on the conic segment see Fig. ).. Rational quadratic elliptic, hyperbolic and parabolic segments First we consider rational quadratic elliptic segments. By the geometric invariability of rational Bézier curves, we might as well establish the coordinate system whose origin is the elliptic center, and the coordinate axes x, y are the major axis and minor axis of the ellipse respectively. Denote the elliptic equation by x /a + y /b = a b > 0), and let θ = ϕ, ξ, ηsee Fig. 3) be the location angles of the selected point and two endpoints on the curve segment respectively, whose corresponding parametric angles are θ = ϕ, ξ, η respectively. And they satisfy the relationship of tan θ = b/a tan θ. Then the parametric equation of the elliptic segment is represented by ) a cos ϕ f ϕ)=, 0 ξ ϕ η π. 7) b sin ϕ ) Here, at two endpoints the elliptic segment has R 0)=f ϕ=ξ t=0 = a sin ξ b cos ξ ) dϕ t=0, R ) = f ϕ=η t= = a sin η, respectively. According to t= b cos η

6 4 Science in China Ser. A Mathematics 005 Vol. 48 No Fig. 3. The elliptic segment represented by a rational quadratic Bézier curve. the formula of the derived vector on the endpoint of rational quadratic Bézier curve [9], we have a cos ξ aω0 ω sin ξ a cos η + aω ω t=0 sin η t= R = =. b sin ξ + bω0 ω ξ b sin η t=0 cos bω ω η t= cos Now let kmn) be the slope of the straight line MN, it is easy to compute kr 0 P)= b a cot ξ + ϕ, kr 0 R )= b a cot ξ + η, kpr )= b a cot ϕ + η, Then kr 0 R )= b a cot ξ, kr R )= b cot η. a tan α= kr 0R ) kr 0 P) + kr 0 R ) kr 0 P) = ab sin η ϕ a sin ξ+η sin ξ+ϕ + b cos ξ+ϕ and tan β = the same way. cos ξ+η tan A= kr 0R ) kr 0 R ) + kr 0 R ) kr 0 R ) = ab sin η ξ, a sin ξ sin ξ+η + b cos ξ cos ξ+η kpr) kr0r) +kpr ) kr 0R ), tan C = krr) kr0r) +kr R ) kr 0R ) By the four aforementioned formulae, and also noticing that sin η ξ sin ϕ+η = sin ϕ ξ sin η [ cos ϕ ξ + η ) cos ξ+ϕ [ = cos ξ+ϕ cos ϕ ξ and similar formulae, we have η )] =sin η ϕ can also be computed in )] [ + cos ϕ ξ + η ) cos ϕ ξ η )] sin ξ+η, cot β cot C cot α cot A = sin η ϕ. sin ϕ ξ This shows, by applying Theorem 3, that the parameter corresponding to the point P = f ϕ)ξ ϕ η) on the curve is represented by t=tϕ)= Copyright by Science in China Press 005 sin ϕ ξ sin ϕ ξ + sin η ϕ ω. 8),

7 Geometric meanings of parameters on rational conic segments 5 Then its inverse function on ϕ is represented by ϕ = ϕt) = arctan t) sin ξ + t sin η ω. 9) t) cos ξ + t cos η ω Next we consider rational quadratic hyperbolic segments. When the curve is one segment of a hyperbola x /a y /b =a, b>0, x>0), and let the meanings of θ = ϕ, ξ, η and θ = ϕ, ξ, ηsee Fig. 4) be similar to those for elliptic segments. Then θ and θ satisfy the relationship of tan θ = b/a sin θ, and the parametric equation of the hyperbolic segment is represented by ) a/ cos ϕ f ϕ)=, arctan b b tan ϕ a <ξ ϕ η <arctan b a. 0) Fig. 4. The hyperbolic segment represented by a rational quadratic Bézier curve. Using the same method as for elliptic segments, we know that the parameter corresponding to the point P=f ϕ)ξ ϕ η) on the curve is represented by t=tϕ)= sin ϕ ξ ω cos ξ cos η. ) sin ϕ ξ + sin η ϕ Then the inverse function of the parameter t on the parametric angle ϕ is represented by t) sin ξ + t sin η ϕ=ϕt)= arctan t) cos ξ + t cos η ω cos ξ cos η ω cos ξ cos η. ) Finally we look at rational quadratic parabolic segments. When the curve is one segment of a parabola x = y /4p p p > 0), and let the meanings of θ = ϕ, ξ, η, θ = ϕ, ξ, ηsee Fig. 5) be similar to those for elliptic segments. Then θ and θ satisfy the relationship of θ = θ, and the parametric equation of the parabolic segment is represented by p ) cot ϕ f 3 ϕ)= ), 0<ξ ϕ η <π. 3) p cot ϕ

8 6 Science in China Ser. A Mathematics 005 Vol. 48 No Fig. 5. The parabolic segment represented by a rational quadratic Bézier curve. Using the same method as for elliptic segments, we can see that the parameter corresponding to the point P=f 3 ϕ)ξ ϕ η) on the curve is represented by t=tϕ)= sin ϕ ξ sin η sin ϕ ξ sin η η ϕ + sin sin ξ ω. 4) Then the inverse function of the parameter t on the parametric angle ϕ is represented by ϕ=ϕt)= arctan Summarizing the aforementioned contents, we get t) + t ω t) cot ξ + t cot. 5) η ω Theorem 4. When the rational quadratic Bézier curve ) represents an elliptic, hyperbolic, and parabolic segment given by 7), 0), 3) respectively, the geometric meanings of the parameter, corresponding to an arbitrary point P on the curve, are represented by 8), ), 4) or 9), ), 5) respectively..3 Applications In curve modeling or curve conversion, sometimes the user of a designed system does not satisfy the corresponding parametric value t, which is calculated on a certain point P of the known rational quadratic Bézier curve ), according to one of the above formulae 3), 5), 8), ) and 4). So we hope to calculate another parametric value u 0, ). It is easy to show that if introducing the fractional linear parameter transformation as follows: t= u + u / ω0 ω ω u), 0 u, 6) for the curve ) under the condition ω = ω 0ω, 7) ω ω the parameter t can be changed to u by converting the curve ) into the rational quadratic Bézier curve R u)0 u ), whose control points and shape are all invariable, parameter is u, and the weights are ωi, i = 0,,. The process is as follows: first, rewrite t in formulae 3), 5), 8) or ), 4)) as u, ω i as ωi, i = 0,,, then we can solve the Copyright by Science in China Press 005

9 Geometric meanings of parameters on rational conic segments 7 ratio of ω /ω 0, ω /ω, or ω /ω 0 ; Further choose all the weights ω i, i = 0,, by applying the formula 7), and finally get a fractional linear parameter transformation 6). Therefore it is evident that Theorem, Theorem 3 and Theorem 4 provide a basis for reparametrization at a point which the user selects on a rational quadratic Bézier curve. For reparametrization of the curve ), in general, we have Corollary. For a conic segment represented by the rational quadratic Bézier curve ), let the value of ω/ω 0 ω ) be invariable, however, adjust the value of the ratio /ω, ω /ω, or /ω ; then the value of the parameter t corresponding to the point P on the curve will be increased reduced), when any value of the ratio above is increased reduced). 3 Relation between the point on rational cubic conic segment and the corresponding point in parametric field The rational cubic Bézier representation of a conic segment is represented by 3 Bi 3t)ω ir i i=0 Rt)=, 0 t, 8) 3 Bi 3 t)ω i i=0 where Rt) is the point on the conic segment corresponding to parameter t; R i = x i, y i ), i = 0,,, 3 are the four control points without any ) three points collinear; ω i > 0, i = 0,,, 3 are the four weights; and Bi 3t) = 3 t) 3 i t i are the bases of cubic i Bernstein polynomial. In order to analyze the geometric meaning of the parameter corresponding to each point on conic segments represented by rational cubic Bézier curves, first we need to introduce two lemmas as follows. The proofs of the lemmas can be found in ref [8]. Lemma. The rational cubic Bézier curve 8) is a conic segment, if and only if the system of equations { 3ω k + 3ω k ω 3 k 3 = 0 has a unique root k = ω0 ω R 0 3ω R k + 3ω R k ω 3 R 3 k 3 = 0 ) ) S = 3S 0 ω ω ) S S ) = ω ω 3 ) 3S3 S ), 9) where S i i = 0,,, 3) are the directed areas of the triangles R R R 3, R 0 R R 3, R 0 R R 3 and R 0 R R respectively see Fig. 6). Lemma. Suppose the rational cubic Bézier curve 8) is a conic segment. Then using the real number k given by 9), we can make a fractional linear transformation t=ku/ku + u)), 0)

10 8 Science in China Ser. A Mathematics 005 Vol. 48 No Fig. 6. The directed areas S 0, S, S, S 3. and change the parameter t to the parameter u in the curve 8), so that a rational quadratic Bézier curve Ru)= B 0 u)λ 0Q 0 + B u)λ Q + B u)λ Q B 0u)λ 0 + B u)λ + B u)λ, 0 u, ) whose shape is identical to the curve 8) see Fig. 7), can be generated, where λ 0, λ, λ )=, + 3ω k)/ = ω 3 k 3 + 3ω k )/, ω 3 k 3 ), ) Q 0, Q, Q )= R 0, R 0 + 3ω kr = ω ) 3k 3 R 3 + 3ω k R, R + 3ω k ω 3 k 3 + 3ω k 3. 3) Fig. 7. Two rational conic sections Rt), Ru) with the identical shape and their control points. By Lemma and Lemma, we have Theorem 5. Let Rt), Ru) be the rational cubic and quadratic conic segments given by Lemma respectively and let S i i=0,,, 3) be the directed areas given by Lemma. Let P be a point located on the conic segment Rt) and let i i = 0,, ) be the directed areas of PQ Q, Q 0 PQ, Q 0 Q P, respectively see Fig. 7). Suppose k is the real number given by 9). Then the parameter corresponding to the point P is represented by t= ) =. 4) + 6 ω 3 S 3 S k ω3 0 Furthermore, suppose the directed areas of the four triangles PR R 3, R R P, R 0 PR 3 and R 0 R P), which are generated by dividing the quadrangle R 0 R R R 3 with the linked lines between each vertex of the quadrangle and the point P, are δ i i = 0,,, 3) Copyright by Science in China Press 005

11 Geometric meanings of parameters on rational conic segments 9 respectivelysee Fig. 7). Then the parameter corresponding to the point P is represented by t=. 5) ωω3 + k ω δ0 δ 3 Proof. Let be the directed area of Q 0 Q Q. By the definition of the barycentric coordinates, we have P= 0 Q 0 + Q + Q ). 6) Applying the third formula of eq. 3) and noticing ), we immediately know that the parameter corresponding to the point P can be represented by u= =. 7) λ + 0 λ 0 + k 3 ω 3 0 On the other hand, substituting 3) into 6), we have P= 0 + ω ) 0 R 0 + 3ω ) k +3ω k + 3ω k R + R 3 = 0 R 0 + 3ω ) ) k ω 3 k 3 +3ω k R ω3 k 3 + ω 3 k 3 +3ω k + R 3. Then by the uniqueness of the barycentric coordinates, we get And by 9), it is easy to show that 0 / =δ 0 /S, / =δ 3 /S. S ω0 =k ω S, k = 3 3. S ω ω 3 S 0 Substituting the four formulae mentioned above into 7) and applying ), we obtain u= =. S ωω3 S 0 + k4 ω δ0 δ 3 Finally, by 0) we compute the values of t corresponding to u, which are given in 4) and 5). The proof of Theorem 5 is completed. Theorem 5 essentially formulates the inverse mapping R R t [0, ], of the mapping t [0, ] R R, represented by 8) for the rational cubic conic segment. The quantities that influence the value of the parameter t are the weights of the conic segment, as well as the geometric quantities such as the triangular areas, etc, determined by the control points and the selected point on the conic segment. Comparing the equation 4) with the third formula in 4), we see that the parametric expression for the rational cubic conic segment is quite similar to that for the rational quadratic conic segment. The difference between them is only in the factor 0 / which is k ω 3 / or ω / in the second term of their denominators. Therefore by computing the value of the quantity k corresponding to the elliptic segment, hyperbolic segment, or parabolic segment, represented by rational cubic Bézier curves respectively, we can obtain the expressions of the parameters on the different kinds of rational cubic conic segments.

12 0 Science in China Ser. A Mathematics 005 Vol. 48 No In the following, we discuss the geometric meaning of the parameter on the curve that is an elliptic segment, hyperbolic segment, or parabolic segment, respectively. First we examine rational cubic elliptic segments. Establishing the coordinate system as in Subsection., then if the edge vectors R 0 R and R R 3 are non-parallel, there must be a cos ξ aλ0 Q = b sin ξ+ bλ0 λ du λ du sin ξ a cos η+ aλ u=0 = cos ξ b sin η bλ u=0 λ du λ du sin η u= cos η u=. 8) Regarding the above relation as simultaneous linear equations with two unknowns λ0 λ λ and du u=0 λ du, and solving it, we obtain u= λ 0 λ du = λ u=0 λ du =tan η ξ. 9) u= By 9) in which the parameter t is regarded as u, we know dϕ λ du = sin η ξ u=0 λ 0, dϕ λ0 du = sin η ξ u= λ. Substituting the above formulae into 9), it follows that λ =cos η ξ λ 0 λ. 30) Thus applying ), we have + 3ω k) ω 3 k 3 + 3ω k ) =4 cos η ξ ω 3 k 3. Cleaning up the above expression, we obtain an equation of second degree with one unknown k as follows: [ ω ω 3 k 3ω ω + 4 cos η ξ ) ] ω 3 k+ ω =0. 3 Then we extract and get where δ = k, = 3ω ω 3 + 6ω 4 cos η ξ ) ± δ, 3) 6 [ 9ω ω 0 cos η ξ ) ] [ 9ω ω 0 + cos η ξ ) ]. 3) ω 3 ω ω 3 ω Thus we know that, when the edge vectors R 0 R and R R 3 are non-parallel, the parameter of the point P on the rational cubic elliptic segment can be represented by t=tϕ)= sin ϕ ξ sin ϕ ξ + sin η ϕ in which the real numbers k are shown in 3). k ω3. 33) When the edge vectors R 0 R and R R 3 are parallel to each other, we have S 0 =S, S = S 3, and substituting these formulae into 9), we can obtain k = /3ω = 3ω /ω 3. The Copyright by Science in China Press 005

13 Geometric meanings of parameters on rational conic segments above formulae are just the special conditions where we take η ξ as π in 3), so the parameter expression 33) of the point P is also fit for the case in which the edge vectors R 0 R and R R 3 are parallel. Analogously, for rational cubic hyperbolic segments, establishing the coordinate system as in Subsection., then it is easy to show that the parameter of the point P on this curve can be represented by in which where δ = 9ω ω 3 ω t=tϕ)= sin ϕ ξ sin ϕ ξ + sin η ϕ k ω3 cos ξ cos η k, = 3ω + ω 0 4 cos ξ+η ) / cos ξ cos η) ω 3 6ω cos ) ξ+η cos ξ cos η 9ω ω 3 ω. 34) ± 6 δ, 35) ) ξ+η cos +. 36) cos ξ cos η Finally, for rational cubic parabolic segments, establishing the coordinate system as in Subsection., thus we can see that the parameter of the point P on this curve can be represented by in which t=tϕ)= k, = 3ω ω 3 ω ± sin ϕ ξ sin η sin ϕ ξ sin η η ϕ + sin sin ξ 9ω k ω3, 37) ω ) 0 ω ω ) 0. 38) ω 3 ω ω 3 ω Theorem 6. When the rational cubic Bézier curve 8) represents an elliptic, hyperbolic, and parabolic segment given by 7), 0), 3) respectively, the geometric meaning of the parameter corresponding to an arbitrary point P on the curve is represented by 33), 34), 37) respectively. From Theorem 6 we know that eqs. 33), 34), 37) give convenient and succinct methods to calculate the parametric value, for the rational cubic elliptic segment, hyperbolic segment, and parabolic segment, respectively. Comparing these formulae with the formula in Theorem 4, it is easy to find the striking similarity between them. The difference between them is only whether the coefficient k in the second term of their denominators is equal to ; Furthermore, the above formulae also indicate that, if we want to change the status of the parametrization of the curve segment, we just change the coefficient K = kω 3 / by changing the weights, while keeping the values ω / ω ) and ω /ω ω 3 ) constant. When the value of K is increased reduced), the value of the parameter t of the point P will be reduced increased). From eq. 9), we know that the value of the ratio ω i /ω i+ i=0,, ) is in direct proportion to k. Then we have

14 Science in China Ser. A Mathematics 005 Vol. 48 No Corollary. For a conic segment represented by the rational cubic Bézier curve 8), let the value of ω /ω ) and ω /ω ω 3 ) be invariable, however, adjust ω i+ /ω i mod4)), the values of the ratio of the weights; then the value of the original parameter t corresponding to the point P on the curve will be increased reduced), when the value of the ratio above is increased reduced). Corollary provides a basis for reparametrization and optimal parametrization on rational cubic conic segments. Acknowledgements This work was supported by the Foundation of State Key Basic Research 973 Item Grant No. 004CB79400), the National Natural Science Foundation of China Grant Nos & ), the National Natural Science Foundation for Innovative Research Groups Grant No. 6000). References. Faux, I. D., Pratt, M. J., Computational Geometry for Design and Manufacture, Chichester, UK: Ellis Horwood Limited, Forrest, A. R., The twisted cubic curves: A computer aided geometric design approach, Computer-Aided Design, 980, 4): Farin, G., Algorithms for rational Bézier curves, Computer-Aided Design, 983, 5): Pigel, L., Representation of quadric primitives by rational polynomials, Computer Aided Geometric Design, 985, 3): Lee, E. T. Y., The rational Bézier representation for conic, in Geometric Modeling: Algorithms and New Trends ed. Farin, G. E.), SIAM, 987, Pigel, L., Tiller, W., A menagerie of rational B-spline circles, IEEE Computer Graphics and Application, 989, 9): Wang, G. J., Rational cubic circular arcs and their application in CAD, Computers in Industry, 99, 63): Wang, G. J., Wang, G. Z., The rational cubic Bézier representation of conics, Computer Aided Geometric Design, 99, 96): Farin, G., Curves and Surfaces for Computer Aided Geometric Design, A Practical Guide, 3rd ed., Boston: Academic Press, Shi, F. Z., Computer aided geometric design and non-uniform rational B-spline CAGD&NURBS) in Chinese), Beijing: Beijing Aeronautics and Astronautics University Press, Shi, F. Z., On parameterization of circular arc represented by standard rational Bézier curves, in Proceedings of the First National Academic Conference for Geometric Design and ComputingCSIAM) eds. Zhang, C. M., Fang, Y.) in Chinese), Dongying Shandong): Petroleum University Press, 00. Copyright by Science in China Press 005