A Note on Implicitization and Normal Parametrization of Rational Curves Rosario Rubio
∗
Departamento de Ingenier´ıa Industrial Universidad Antonio de Nebrija 28040 Madrid
[email protected]
J. Miguel Serradilla∗
M. Pilar Velez ´ ∗
Departamento de Ingenier´ıa Industrial Universidad Antonio de Nebrija 28040 Madrid
Departamento de Ingenier´ıa Informatica ´ Universidad Antonio de Nebrija 28040 Madrid
[email protected]
ABSTRACT
all points of the curve are in the image of the parametrization, which means finding a normal parametrization. Normality is an interesting problem in CAGD. For instance, to plot a curve or surface in a scientific computer system or in geometric modelling. Any parametrization is dominant, but it is not necessarily surjective, i.e. normal, and hence some points of the variety are missing. Normality was first tackled by [6] for any algebraic variety of arbitrary dimension over an algebraically closed field of characteristic zero. Their test of normality is based on Ritt-Wu’s decomposition algorithm and they provide normal parametrizations for conics and some quadrics. The remaining quadrics are presented in [4]. A much simpler test for rational planar curves was presented by [9, 16] using a computation of one greatest common divisor. A test for planar curves over any field of characteristic zero is introduced in [16]. And when possible, the author provides a reparametrization based on the ideas in [1, 2]. Another central problem in CAGD is finding algebraic equations of a given parametrized curve or surface, the so called implicitization problem. For instance, to check if a set of points lies in a specific algebraic variety, to draw a curve or surface nearby a singularity, to compute autointersection of offsets and drafts, or to compute the intersection of varieties. The importance of having efficient implicitization algorithms in Solid Modelling has given rise to a recent extensive study of this topic. This is basically an elimination problem that has been approached, mainly, from three different points of view, by means of resultants, Gr¨ obner basis and moving curves and surfaces. Sederberg and Arnon were the first ones to discuss the implicitization problem for planar curves using various resultant theories [12, 3]. Methods of implicitization for rational parametric equations were introduced in [5, 17], using the Gr¨ obner basis method. A geometrical method in implicitating a rational planar curve is the moving curves method [15, 13, 14]. Due to the intensive research activity in this topic, we refer to the surveys [8, 10] for more details. This work pretends to contribute slightly to the implicitization problem. We present a method which computes implicit equations of rational parametric curves in affine spaces over algebraically closed fields. This method requires normal parametrizations with a concrete property and it is based on the computation of generalized resultants. Thus we give a complete characterization of normal parametrizations and a reparametrization over algebraically closed fields. In Section 2 we find an algebraic variety, which contains the curve, associated to a rational parametrization by com-
In this paper we present a method to compute an implicitization of a rational parametrized curve in an affine space over an algebraically closed field. This method is the natural generalization of the resultant method for planar curves. For this purpose we need some normality assumptions on the parametrization of the curve. Furthermore, we provide a test to decide whether a parametrization is normal and if not, we compute a normal parametrization.
Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms—Algebraic algorithms
General Terms Algorithms
Keywords Rational curves, implicitization, normal parametrizations, generalized resultants
1.
[email protected]
INTRODUCTION
The interest in the study of algebraic curves has increased in the last decades, due mainly to the creation of Computer Aided Geometric Design tools (CAGD), which are used almost in every branch of Engineering and Industrial Design. CAGD programs use, generally, parametric representation of curves, and this is why the study and manipulation of curves from a parametrization is so important. A classic problem, from a mathematical point of view, is finding optimal parametrization for algebraic curves. A possible optimization is determining a parametrization where ∗Supported by MTM 2005-02568
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t0 ∈ A such that Fi (t0 , ai ) = 0 for all i. Note that fiD (t0 ) = 0 for all i, since the numerators and denominators of the fi ’s are coprime. Therefore, a ∈ Imψ. In the case lct (G)(a2 , . . . , an , u) = 0, the proof is similar. For the last inclusion, R(x, u) ∈< F1 , . . . , Fn > ∩k[x, u]. Then, for all α, mα ∈< F1 , . . . , Fn > ∩k[x] and consequently < mα >⊂ I(C).
puting generalized resultants. We also characterize the rational parametrizations which associated algebraic varieties are equal to the curve. In Section 3 we note that the previous characterization implies the normality of the parametrization. Consequently we reparametrize the curve in order to get a new normal parametrization and an implicitization of the curve. Finally we find a complete characterization of normality. Section 4 is devoted to describe the algorithms based on these results. From now on, we use the following notations. Let k be an algebraically closed field and An the n-dimensional affine space over k. A rational parametrization is a map ψ such that ψ : A −→ t −→
Next result gives a necessary and sufficient condition for W = ∅. Lemma 1. Let di = degt (fi ), pi = degt ( fiN ) and qi = degt ( fiD ) for all i. Let I = {i ∈ {2, . . . , n} / di = d} where d = max{degt (fi ) / i = 2, . . . , n}. Then, pi > qi for some i ∈ {1} ∪ I if and only if W = ∅.
An (f1 (t), . . . , fn (t))
Proof. Assume pi > qi for some i ∈ {1} ∪ I. If i = 1, then lct (F1 (t, x1 )) is a nonzero constant. If i ∈ I, then the coefficient of lct (G(t, x2 , . . . , xn , u)) with respect to ui is a nonzero constant. Hence, W is the empty set. Conversely, suppose that pi ≤ qi for all i ∈ {1} ∪ I. Then, W = {(x1 , . . . , xn ) / lct (Fi (t, xi )) = 0 for i ∈ {1} ∪ I}. Notice that for i ∈ {1} ∪ I, lct (Fi (t, xi )) is a non constant linear polynomial depending only on xi . Therefore, we have W = ∅.
where fi ∈ k(t), ∀i = 1, . . . , n. Let C = Imψ (Zariski closure of the image of ψ) the rational curve parametrized by ψ. The degree of a rational function f is max{deg(fN ), deg(fD )}, fN , fN , fD ∈ k[t] and gcd(fN , fD ) = 1. where f = fD
2.
IMPLICITIZATION
Theorem 1. If either pi > qi for some i ∈ {1} ∪ I or I = {2, . . . , n} and pi ≤ qi for all i then C = V (mα ).
In this section we identify an algebraic variety which contains the rational curve and after an appropiate reparametrization it will give us some implicit equations. The computation of the new parametrization is analysed in Section 3. We introduce the generalized resultants of the implicitization polynomials associated to ψ.
Proof. The first case is straightforward. For the second one, note that W is a point, then by Proposition 1 we are done.
Definition 1. Let f ∈ k(t). We define the implicitization polynomial associated to f as:
3. NORMALIZATION In this section we give a complete test of normality that allows us to reparametrize to obtain the condition of Lemma 1. The results related to normality in this section are a generalization to n variables of [16] and complete some results of [6, 1]. The following proposition is proved in [6], but we include here a new proof based on the previous results.
F (t, x) = fN (t) − xfD (t) where f =
fN , fN , fD ∈ k[t] and gcd(fN , fD ) = 1. fD
Definition 2. Let ψ = (f1 , . . . , fn ) be a parametrization of C. We consider Fi (t, xi ) = fiN (t) − xi fiD (t). We define their generalized resultants as mα (x1 , . . . , xn ) where R(x, u)
Proposition 2. Let p1 = degt ( f1N ) and q1 = degt ( f1D ). If p1 > q1 , then the parametrization ψ is normal.
= R(x1 , . . . , xn , u2 , . . . , un ) = Res t (F1 , u2 F2 + · · · + un Fn ) . = α mα uα
Proof. It is obvious by Theorem 1.
For more details see [7]. Proposition 1. Let W = {(a1 , . . . , an ) ∈ An |lct (F1 )(a1 ) = lct (u2 F2 (t, x2 ) + · · · + un Fn (t, xn ))(a2 , . . . , an , u) = 0}. Then V (mα ) \ W ⊂ Imψ ⊂ C ⊂ V (mα ). Proof. Let G(t, x2 , . . . , xn , u) = u2 F2 + · · · + un Fn . For the first inclusion, let a = (a1 , . . . , an ) ∈ V (mα ) \ W . If lct (F1 )(a1 ) = 0, due to the behaviour of the resultant under the evaluation homomorphism (see [11, 18]),
The rational function f1 in the above proposition, can be replaced by any other rational function of the parametrization since we are working with the resultant Rest (F1 , u2 F2 + · · · + un Fn ) and it is not relevant the order of the associated implicitization polynomials. It is important to remark that if we reorder the rational functions we are working with different generalized resultants mα .
R(a, u) = Rest (F1 , G)(a, u) = lct (F1 )(a1 )l Rest (F1 (t, a1 ), G(t, a2 , . . . , an , u))
Corollary 1. Let pi = degt ( fiN ) and qi = degt ( fiD ) for all i. If there exists i = 1, . . . , n such that pi > qi , the parametrization is normal. The generalized resultants R(x, u) = Rest (Fi (t, xi ), G) where G = u1 F1 (t, x1 ) + · · · + ui−1 Fi−1 (t, xi−1 )+ui+1 Fi+1 (t, xi+1 )+· · ·+un Fn (t, xn ) give an implicit representation of C.
where l = degt (G) − degt (G(a, u)). Since a ∈ V (mα ), we have Rest (F1 (t, a1 ), G(t, a2 , . . . , an , u)) = 0. So there exists
The corollary gives a sufficient condition for normality. For a complete test we need the next lemma.
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Lemma 2. Let C be a rational curve parametrized by ψ. Then Imψ = {a ∈ An | gcd(F1 (t, a1 ), . . . , Fn (t, an )) = 1}.
Corollary 2. Let C be a rational curve parametrized σt + β such that by ψ. Then there exists a unit u(t) = t ψ = ψ ◦ u(t) is a normal parametrization of C and {mα } gives an implicit representation of C.
Proof. Let a ∈ Imψ ⊂ C, then there exists t0 ∈ A such that a = ψ(t0 ). This implies that F1 (t, a1 ), . . . , Fn (t, an ) have a nontrivial common divisor. Conversely, if gcd(F1 (t, a1 ), . . . , Fn (t, an ) = 1, there exists t0 ∈ A a common root of the Fi (t, ai )’s. We have that fiD (t0 ) = 0 for all i, since the numerators and denominators of the fi ’s are coprime. Therefore fi (t0 ) = ai for all i.
Proof. By Theorem 1, the mα generated by the implicitization polynomials associated to ψ describe an implicitization if W = ∅. By Proposition 3 we can test whether ψ is normal. If these conditions are true then u(t) = t. Otherwise, we can choose σ as in the proof of Proposition 3 and 1 + σt u(t) = . t
Proposition 3. Let di = degt (fi ), pi = degt ( fiN ), qi = degt ( fiD ), N (i) = coeff( fiN , tdi ) and D(i) = coeff( fiD , tdi ), for all i. Then the parametrization is normal if and only if either for some i = 1, . . . , n, or pi ≤ qi , ∀i and pi > qi (1) N N (n) gcd f1N − (1) f1D , . . . , fnN − (n) fnD = 1. D D If ψ is not normal the only point of C that cannot be gene (1) N (n) N rated by ψ is , . . . , (n) . D(1) D
4. ALGORITHMS In this section we collect all the results of previous sections in two different algorithms. The first algorithm computes some implicit equations from any parametrization. The second algorithm decides if a parametrization is normal; and if it is not, it reparametrizes. Theorem 1 and Corollary 2 are the tools for the implicitization of a rational curve with a parametrization.
Proof. By Proposition 2, if pi > qi for some i, the parametrization is normal. Therefore, we can suppose that (1) (1) Np1 tp1 + · · · + N0 and define di = qi for all i. Let f1 = (1) (1) Dq1 tq1 + · · · + D0 G = u2 F2 + · · · + un Fn . . Then ψ˜ = Let σ be a rootof the denominator of f1 1 + σt 1 + σt , . . . , fn is normal: (f˜1 , . . . , f˜n ) = f1 t t f˜1 = = (1)
Dq1
=
(1)
1 + σt t
q1
Algorithm 1 (implicitization). Input: ψ = (f1 , . . . , fn ) ∈ k(t)n a parametrization. Output: An implicitization of the curve defined by ψ. Step 1: Compute F1 , . . . , Fn the implicitization polynomials associated to (f1 , . . . , fn ). Step 2: If degt ( fiN ) > degt ( fiD ) for some i, take j an integer such that fj has the lowest degree and satisfies degt ( fjN ) > degt ( fjD ). Go to Step 7.
p 1 + σt 1 (1) + · · · + N0 t p 1 + σt 1 (1) (1) + · · · + Dp1 + · · · + D0 t (1) Np1
(1)
(1)
(1)
(1)
Step 3: If deg fi for all i are equal, take j an integer such that fj has the lowest degree and satisfies degt ( fjN ) > degt ( fjD ). Go to Step 7.
Np1 (1 + σt)p1 te1 + · · · + N0 tq1
Step 4: If deg fi for all i ∈ {1, . . . , n} \ {j} are equal, go to Step 7.
Dq1 (1 + σt)q1 + · · · + Dp1 (1 + σt)p1 te1 + · · · + D0 tq1
where e1 = p1 − q1 > 0. Note that coeff(f˜1D , tdi ) = f1D (σ) is zero and coeff(f˜1N , tdi ) = f1N (σ) is non-zero. Then the degree of the numerator of f˜1 is greater than the degree of the denominator and the new parametrization is normal. If there exists t0 ∈ A \ {0} such that Fi (t0 , ai ) = 0 for all 1 + σt0 is a common root of the Fi (t, ai ). Hence the only i, t0 point that might not be generated by ψ is ψ(0). Analazing ψ(0) we have ⎧ (i) ⎪ N ⎨ if pi = qi (i) D f˜i (0) = 0 if pi < qi ⎪ ⎩ no defined if pi > qi By Lemma 2, ψ normal if and only if either ψ(0) is not defined or gcd(F1 (t, f˜1 (0)), . . . , Fn (t, f˜n (0))) = 1. Since pi > qi , gcd(F1 (t, f˜1 ), . . . , Fn (t, f˜n )) = N (1) N (n) gcd f1N − (1) f1D , . . . , fnN − (n) fnD = 1. D D If it is not normal, the only points that cannot be desN (1) N (n) cribed by ψ is ψ(0) = , . . . , (n) . D(1) D
Step 5: Let j be an integer such that fj has the lowest degree. . Let (f1 , . . . , fn ) = Step 6: Compute σ a rootof fjD 1 + σt 1 + σt , . . . , fn . f1 t t Compute F1 , . . . , Fn the implicitization polynomials associated to (f1 , . . . , fn ). Step 7: Let G = u1 F1 + · · · + uj−1 Fj−1 + uj+1 Fj+1 + · · · + un Fn . Step 8: Compute R = Rest (Fj , G) = α mα uα . Step 9: Return {mα = 0}. Remark 1. In Steps 2,3 and 5 of Algorithm 1 we choose fj with lowest degree in order to make a faster computation. Next algorithm decides whether a parametrization is normal with the computation of a greatest common divisor, see Proposition 3. If it is not, it returns a new normal parametrization.
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Algorithm 2 (Normalization). Input: (f1 , . . . , fn ) ∈ k(t)n .
7. REFERENCES [1] C. Alonso, J. Gutierrez, and T. Recio. Real parametric curves: some symbolic algorithm issues. 14th IMACS World Symposium. Atlanta., 1994. [2] C. Andradas and T. Recio. Plotting missing points and branches of real parametric curves. To appear in AAECC, 2006. [3] D. S. Arnon and T. W. Sederberg. Implicit equation for a parametric surface by Gr¨ obner basis. In Proc. 1984 MACSYMA User’s Conf. (Golden, V. E. ed.) General Electric, Schenectady, New York, pages 431–436, 1984. [4] C. Bajaj and A. V. Royappa. Finite representation of real parametric curves and surfaces. Int. J. Comput. Geom. Appl., 5(3):313–326, 1995. [5] B. Buchberger. Application of Gr¨ obner bases in non-linear computational geometry. L.N.C.S., 296:52–80, Springer-Verlag 1987. [6] S. C. Chou and X. S. Gao. On the normal parametrization of curves and surfaces. Int. J. Comput. Geom. Appl., 1:125–136, 1991. [7] D. A. Cox, J. Little, and D. O’Shea. Ideal, varieties and algorithms. UTM, Springer-Verlag, New York, 1997. [8] X. S. Gao. Conversion between implicit and parametric representations of algebraic varieties. In Mathematics Mechanization and applications, Academic Press, pages 343–362, 2000. [9] J. Gutierrez, R. Rubio, and J.-T. Yu. D-resultant of rational functions. Proc. Amer. Math. Soc., 130(8):2237–2246, 2002. [10] Y. S. Kotsireas. Panorama of methods for exact implicitization of algebraic curves and surfaces. In Geometric Computation, Lecture Note Series on Computing, 11, pages 126–155, 2004. [11] B. Mishra. Algorithmic Algebra. Springer, 1993. [12] T. W. Sederberg, D. C. Anderson, and R. N. Goldman. Implicit representation of parametric curves and surfaces. Comput. Vision Graph. Image Process., 28:72–84, 1984. [13] T. W. Sederberg and F. Chen. Implicitization using moving curves and surfaces. In Proc. of SIGGRAPH’95, pages 301–308, 1995. [14] T. W. Sederberg, R. N. Goldman, and H. Du. Implicitizing rational curves by the methods of moving algebraic curves. J. Symbolic Comput., 23(2-3):153–175, 1997. [15] T. W. Sederberg, T. Saito, K. S. Qi, and D. Klimaszewski. Curve implicitization using moving lines. Comput. Aided Geom. Design, 11(6):687–706, 1994. [16] R. Sendra. Normal parametrizations of algebraic plane curves. J. Symbolic Comput., 33:863–885, 2002. [17] S. Shannon and M. Sweedler. Using Gr¨ obner bases to determine algebraic membership. J. Symbolic Comput., 6:267–273, 1988. [18] F. Winkler. Polynomial algorithms in computer algebra. Springer, 1996.
Output: True, if it is normal. On the other case, (f˜1 , . . . , f˜n ) a normal reparametrization of (f1 , . . . , fn ). Step 1: Let di = degt (fi ). If di > degt ( fiD ) for some i, return true. Step 2: Let pi = degt ( fiN ) and qi = degt ( fiD ), for all i. Step 3: Let N (i) = coeff( fiN , tdi ) and D(i) = coef f ( fiD , tdi ), for all i. N (1) N (n) Step 4: If gcd f1N − (1) f1D , . . . , fnN − (n) fnD is D D constant, return true. Step 5: Compute σ a root of f1D . 1 + σt 1 + σt , . . . , fn . Return f1 t t
5.
CONCLUSION
We have presented two methods, one to implicitizate and the other to normalize parametric rational curves and we have also shown a relationship between the two problems. This implicitization is based on the computation of some generalized resultants, which extends the planar curves resultant method to arbitrary affine spaces. There exists an intensive research on implicitization of planar curves (see [8, 10]), but as far as we know there are not implicitization methods for non planar curves except for Gr¨ obner bases based method and some other particular cases. The advantages of resultants over Gr¨ obner bases is the simplicity of the computation, whilst the output obtained from the resultant based method is rather larger than the one obtained from a Gr¨ obner basis. In many cases our implicitization method requires only the computation of a resultant, but sometimes we need to compute a root of a polynomial. Therefore it is necessary to work over an algebraically closed field. If the characteristic of the field is zero there exists a way to avoid the computation of this root and the algebraically closed property for the field. Remark 2. The set W can be empty, a point or a linear variety of dimension greater than 0. The last case is the only one that needs reparametrization (Theorem 1). In this last case pi ≤ qi for all i ∈ {1} ∪ I and I {2, . . . , n}. It is possible to make a linear change of coordinates in An to be in the second case. After a linear change xi → x i = ai2 x2 + . . . ain xn where aij = 0, ∀i, j ∈ {2, . . . , n}, the deg(fi ) for all i = 2, . . . , n are equal. This remark was pointed out by one of the anonymous referees.
6.
ACKNOWLEDGEMENTS
The authors would like to thank Rafael Sendra who suggested the reading of some very useful references. We are grateful to the referees for usefull comments and remarks that have improved this note.
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