HIGHER DIMENSIONAL MINKOWSKI PYTHAGOREAN HODOGRAPH CURVES

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1 J. Appl. Math. & Computing Vol. 14(2004), No. 1-2, pp HIGHER DIMENSIONAL MINKOWSKI PYTHAGOREAN HODOGRAPH CURVES GWANG-IL KIM AND SUNHONG LEE Abstract. Rational parameterization of curves and surfaces is one of the main topics in computer-aided geometric design because of their computational advantages. Pythagorean hodograph (PH) curves and Minkowski Pythagorean hodograph (MPH) curves have attracted many researcher s interest because they provide for rational representation of their offset curves in Euclidean space and Minkowski space, respectively. In [10], Kim presented the characterization of the PH curves in the Euclidean space and analyzed the relation between the class of PH curves and the class of rational curves. In this paper, we extend the characterization of PH curves in [10] into that of MPH curves in the general Minkowski space and consider some generalized MPH curves satisfying this characterization. AMS Mathematics Subject Classification : 65D17, 68U05. Key words and phrases : Minkowski Pythagorean hodograph curves, extended normalizer, extended generalized stereographic projection 1. Introduction In the field of computer aided design, the rational parameterization of curves and surface is strongly preferred for a reason of computation. But many important geometric objects are not expressed in the rational parameterization: for example, the offsets to a given plane curve, i.e., the loci of points at a given fixed distance from a given curve. In their seminal paper [8], Farouki and Sakkalis introduced the Pythagorean hodograph (PH) curves in the Euclidean space R 2, which have polynomial functions as their speed or hodograph. The PH curves have advantageous computational properties. For example, the arc length of a PH curve can be represented in a closed form and the offset curves of the PH curve are rational. Therefore the Received April 11, Revised September 7, c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 405

2 406 Gwang-Il Kim and Sunhong Lee class of PH curves are very useful in computer-aided design and manufacturing applications ([1], [4], [5], [6], [7], [8] and [9]). When we deal with a domain in the plane, the medial axis transform of the domain is a powerful tool [3]. The medial axis transform of a planar domain is the set of pairs consisting of centers and radii of the circles maximally inscribed in the domain. The envelop of the medial axis circles gives the boundary or the offset curves of the original domain. If γ(t) =(x(t),y(t),r(t)) is a segment of the medial axis transform, the envelop curve ( x(t), ỹ(t)) may be written x(t) =x(t)+r(t) r (t)x (t) x (t) 2 + y (t) 2 r (t) 2 y (t) x (t) 2 + y (t) 2, ỹ(t) =y(t)+r(t) r (t)y (t) ± x (t) 2 + y (t) 2 r (t) 2 x (t) x (t) 2 + y (t) 2. Motivated by this property, Moon introduced the Minkowski Pythagorean hodograph (MPH) curves in the Minkowski space R 2,1 [11]. Let γ(t) be a one parameter family of spheres in R 3, whose envelop is a canal surface. In [12], Peternell and Pottmann showed that any canal surface with polynomial spine curve and polynomial radius function has a rational parameterization if there are two polynomial functions f 1 (t) and f 2 (t) such that x (t) 2 + y (t) 2 + z (t) 2 r (t) 2 = f 1 (t) 2 + f 2 (t) 2. Choi et al. define such polynomial curve γ(t) =(x(t),y(t),z(t),r(t)) a MPH curve in the Minkowski space R 3,1 [2]. The PH curves and the MPH curves have been studied in two or tree dimensional Euclidean and Minkowski spaces. Recently, Choi et al. presented a novel approach to the PH curves, based on Clifford algebra method, that unifies all known incarnations of PH curves into a single coherent framework [2]. On the other hand, the first author presented the characterization of the PH curves in the Euclidean space R n+1 for every positive integer n [10]. To obtain the results, he manages two maps: the extended normalizer and the extended generalized stereogrphic projection, and analyzed the relation between the class of Pythagorean curves in the Euclidean space R n+1 and the class of rational curves in R n. In this paper we extend the characterization of the PH curves in the Euclidean space R n+1, given by the first author [10], into that of the class of MPH curves in the general Minkowski space R m+1,n for non-negative integers m and n with m + n 1. We also consider some generalization of the MPH curves and obtain their characterization. 2. The Extended Normalizer and the Extended Generalized Stereographic Projection

3 Higher dimensional Minkowski Pythagorean Hodograph curves 407 In this section, we will introduce two maps which explain the relation between the set P(R m+1,n ) of Minkowski Pythagorean curves in the Minkowski space R m+1,n and the set R(R m,n ) of rational curves in the Minkowski space R m,n. Let m and n be non-negative integers with m + n 1. Let R m+1,n be the space of (m + n + 1)-tuples of real numbers. On the space R m+1,n we consider the Minkowski inner product Q as follows: for x =(x 0,...,x m+n ) and y =(y 0,...,y m+n )inr m+1,n, Q(x, y) =(x 0 y x m y m ) (x m+1 y m x m+n y m+n ). For x in R m+1,n, we use the notation x 2 m+1,n = Q(x, x). The space Rm+1,n with the Minkowski inner product is called a Minkowski space. A polynomial curve γ(t) =(γ 0 (t),...,γ m+n (t)) in R m+1,n is called a Minkowski Pythagorean (MP) curve if the curve γ(t) satisfies the condition γ(t) 2 m,n = γ 0(t) γ m (t) 2 γ m+1 (t) 2 γ m+n (t) 2 = σ(t) 2 for some real polynomial function σ(t). A polynomial curve γ(t) =(γ 0 (t),...,γ m+n (t)) in R m+1,n is called a Minkowski Pythagorean hodograph (MPH) curve if its hodograph γ (t) satisfies the condition γ (t) 2 m,n = τ(t) 2 for some real polynomial τ(t). Remark 1. In [2], Choi et al. define the MPH curves in R 3,1 as follows: A polynomial curve γ(t) =(x(t),y(t),z(t),r(t)) in R 3,1 is a MPH curve if there exist polynomials f 1 (t) and f 2 (t) such that γ (t) 2 3,1 = f 1(t) 2 + f 2 (t) 2. We consider this definition in some general setting in Section 4. Let H m+1,n be the subset of the Minkowski space R m+1,n, defined by H m+1,n = (x 0,...,x m+n ) R m+1,n : x 2 i x 2 m+j =1. For every MP curve α(t) inp(r m+1,n ), we take a polynomial function σ(t) such that α i (t) 2 α m+j (t) 2 = σ(t) 2. We consider the map N : P(R m+1,n )\{(0,...,0)} R(H m+1,n ), which is defined by N (α(t)) = α(t) σ(t). We call this map N the extended normalizer.

4 408 Gwang-Il Kim and Sunhong Lee Let R(R m,n ) be the set of rational curves in the Minkowski space R m,n. The extended generalized stereographic projection S : R(H m+1,n )\{(1, 0,...,0)} R(R m,n ) is defined by ( ϕ1 (t) S(ϕ(t)) = 1 ϕ 0 (t),..., ϕ m (t) 1 ϕ 0 (t), ϕ m+1(t) 1 ϕ 0 (t),..., ϕ ) m+n(t), 1 ϕ 0 (t) where ϕ(t) =(ϕ 0 (t),ϕ 1 (t),...,ϕ m (t),ϕ m+1 (t),...,ϕ m+n (t)). We note that the inverse map S 1 of the extended generalized stereographic projection S is defined by S 1 (γ(t)) = ( γ(t) 2 m,n 1 γ(t) 2 m,n +1, 2γ 1 (t) γ(t) 2 m,n +1,..., ) 2γ m+n (t) γ(t) 2 m,n +1 for a rational curve γ(t) =(γ 1 (t),...,γ m+n (t)) in the Minkowski space R m,n. By Lemma 1 in Section 3, for a rational curve γ(t) inr(r m,n ), there is a polynomial curve β(t) =(β 0 (t),...,β m+n (t)) with gcd(β 0 (t),...,β m+n (t)) = 1 so that every member α(t) of the class [N 1 S 1 (γ(t))] [N 1 S 1 ( γ(t))] is α(t) = a(t)β(t) for some polynomial function a(t). Such curve β(t) is unique up to nonzero constants. We call this curve β(t) the representative of the class [N 1 S 1 (γ(t))] [N 1 S 1 ( γ(t))]. 3. The Characterization of the MPH curves In this section we classify the set of MPH curves in R m+1,n using the extended normalizer and the extended generalized stereographic projection. Let γ(t) =(γ 1 (t),...,γ m+n (t)) be an arbitrary rational curve in R(R m,n ). We may write γ k (t) = r(t) s(t) fk(t) g k (t), 1 k m + n where gcd(r(t)f k (t),s(t)g k (t)) = 1, 1 k m + n, and gcd(f 1 (t),...,f m+n (t)) = 1 = gcd(g 1 (t),...,g m+n (t)). We set Since λ(t) = m+n k=1 g k (t), δ k (t) = λ(t) g k (t), 1 k m + n. γ(t) 2 m,n = r(t) 2 ( m i=1 (δ i(t)f i (t)) 2 n (δ m+j(t)f m+j (t)) 2 ) (s(t) λ(t)) 2,

5 Higher dimensional Minkowski Pythagorean Hodograph curves 409 the components of ϕ(t) =S 1 (γ(t)) are given as follows: ( m r(t) 2 i=1 (δ i(t)f i (t)) 2 ) n (δ m+j(t)f m+j (t)) 2 s(t) 2 λ(t) 2 ϕ 0 (t) = ( m r(t) 2 i=1 (δ i(t)f i (t)) 2 ) n (δ m+j(t)f m+j (t)) 2 + s(t) 2 λ(t) 2 and 2r(t)s(t)λ(t)δ k (t)f k (t) ϕ k (t) = ( m r(t) 2 i=1 (δ i(t)f i (t)) 2 ) n (δ m+j(t)f m+j (t)) 2 + s(t) 2 λ(t) 2 for 1 k m + n. Consider the polynomial curve α(t) =(α 0 (t),α 1 (t),...,α m+n (t)) in R m+1,n, given by α 0 (t) =r(t) 2 (δ i (t)f i (t)) 2 (δ m+j (t)f m+j (t)) 2 s(t) 2 λ(t) 2 i=1 and α k (t) =2r(t)s(t)λ(t)δ k (t)f k (t), 1 k m + n. We set ξ(t) = gcd(δ 1 (t)f 1 (t),...,δ m+n (t)f m+n (t)) and η(t) = gcd(ξ(t),s(t)λ(t)). Then we may write ξ(t) = η(t)a(t), s(t)λ(t) = η(t)b(t), and δ k (t)f k (t) = ξ(t)θ k (t), 1 k m + n for some polynomial functions a(t), b(t), and θ k (t), 1 k m + n with gcd(θ 1 (t),...,θ m+n (t)) = 1. We set Θ(t) = θ i (t) 2 θ m+j (t) 2, ρ(t) = gcd(θ(t),b(t)). i=1 so that Θ(t) = ρ(t)τ(t) and b(t) = ρ(t)v(t) for some polynomial functions τ(t) and v(t). Then we obtain α 0 (t) =ρ(t)η(t) 2 [r(t) 2 a(t) 2 τ(t) ρ(t)v(t) 2 ], α k (t) =ρ(t)η(t) 2 [2r(t)a(t)v(t)θ k (t)], 1 k m + n. Lemma 1. The polynomial curve β(t) =(β 0 (t),...,β m+n (t)), given by β 0 (t) =r(t) 2 a(t) 2 τ(t) v(t) 2 ρ(t), β k (t) =2r(t)a(t)v(t)θ k (t), 1 k m + n, is the representative of the class [N 1 S 1 (γ(t))] [N 1 S 1 ( γ(t))].

6 410 Gwang-Il Kim and Sunhong Lee Proof. Since β 2 m+1,n =[r(t)2 a(t) 2 τ(t)+v(t) 2 ρ(t)] 2, the polynomial curve β(t) is a MP curve in R m+1,n. Now it suffices to show that gcd(β 0 (t),...,β m+n (t)) = 1. Suppose that gcd(β 0 (t),...,β m+n (t)) 1. Then there exist a non-constant polynomial function d(t) which is a irreducible divisor of the greatest common divisor. Since gcd(θ 1 (t),...,θ m+n (t)) = 1, d(t) divides r(t)a(t) orv(t). Suppose that d(t) divides r(t)a(t). Then since d(t) divides β 0 (t), d(t) divides v(t) 2 ρ(t). It implies that d(t) divides b(t) =v(t)ρ(t). This contradicts to gcd(r(t)a(t),b(t)) = 1. Suppose that d(t) does not divides r(t)a(t). Then d(t) divide v(t). It implies that d(t) divides τ(t). This contradicts to gcd(τ(t),v(t)) = 1. The proof is done. Every constant function A(t) =(c 0,...,c m+n ) in the Minkowski space R m+1,n is clearly MPH curve. We present the main result: Theorem 1. Let A(t) be a non-constant polynomial curves in the Minkowski space R m+1,n. The polynomial curve A(t) is a MPH curve if and only if there exist polynomial functions s(t),u(t),v(t),τ(t),ρ(t), and θ k (t) for 1 k m + n with m i=1 θ i(t) 2 n θ m+j(t) 2 = ρ(t)τ(t) and gcd(τ(t),v(t)) = gcd(θ 1 (t),...,θ m+n (t)) = gcd(u(t),v(t)ρ(t)) = 1, so that the hodograph α(t) :=A (t) =(α 0 (t),...,α m+n (t)) of A(t) is expressed as follows: α 0 (t) =s(t)[u(t) 2 τ(t) v(t) 2 ρ(t)], α k (t) =s(t)[2u(t)v(t)θ k (t)], 1 k m + n. Proof. Suppose that α(t) is given as above. Then we obtain α(t) 2 m+1,n = s(t) 2 [u(t) 2 τ(t)+v(t) 2 ρ(t)] 2, which implies that α(t) is a MP curve in the Minkowski space R m+1,n. Therefore A(t) is a MPH curve in the space R m+1,n. Suppose that the non-constant polynomial curve A(t) is a MPH curve in the space R m+1,n.ifa(t) =(t, 0,...,0), we may choose v(t) 0 and s(t) u(t) τ(t) 1. Let one of θ k,1 k m + n be the constant function with value 1 and the others the constant function of value 0. We may choose ρ(t) 1orρ(t) 1 so that m i=1 θ i(t) 2 n θ m+j(t) 2 = ρ(t)τ(t). Now assume that A(t) (t, 0,...,0). Then its hodograph α(t) :=A (t) is a MP curve in R m+1,n. For the rational curve γ(t) =S N(α(t)) in R m+n, the hodograph α(t) is a member of the class N 1 S 1 (γ(t)) N 1 S 1 ( γ(t)). By Lemma 1, there exists

7 Higher dimensional Minkowski Pythagorean Hodograph curves 411 a polynomial function s(t) such that α(t) = s(t)β(t) where the representative β(t) =(β 0 (t),...,β m+n (t)) is given as follows: β 0 (t) =u(t) 2 τ(t) v(t) 2 ρ(t), β k (t) =2u(t)v(t)θ k (t), 1 k m + n, where the polynomial functions u(t), v(t), τ(t), ρ(t), and θ k (t), 1 k m + n, satisfy the conditions m i=1 θ i(t) 2 n θ m+j(t) 2 = ρ(t)τ(t) and gcd(τ(t),v(t)) = gcd(θ 1 (t),...,θ m+n (t)) = gcd(u(t),v(t)ρ(t)) = 1. Therefore, α(t) = s(t)β(t). The proof is done. 4. The Generalization of the MPH curves In this section we define some generalization of MPH curves and find out that Theorem 1 implies the characterization of these MPH curves. First, we find a one-to-one correspondence between the set of MPH curves in R m+1,n and the set of MPH curves in R n+1,m. Let A(t) be a MPH curve in R m+1,n. We may choose a polynomial curve σ(t) 2 such that α i (t) 2 α m+j (t) 2 = σ(t) 2, where α(t) is the hodograph of A(t). Since σ(t) 2 + α m+j (t) 2 α i (t) 2 = α 0 (t) 2, the polynomial curve (σ(t),α m+1 (t),...,α m+n (t),α 1 (t),...,α m (t)) becomes a hodograph of a MPH curve in R n+1,m. This argument have the result: Theorem 2. There is a one-to-one correspondence between the set of MPH curves in R m+1,n and the set of MPH curves in R n+1,m. We consider some general Minkowski Pythagorean hodograph curves. i=1 Definition 1. A polynomial curve A(t)inR m+1,n is called a Minkowski Pythagorean hodograph (MPH) curve of type k if there exist polynomial functions σ 1 (t),...,σ k (t) such that k (1) α i (t) 2 α m+j (t) 2 = σ l (t) 2, where α(t) is the hodograph of A(t). l=1

8 412 Gwang-Il Kim and Sunhong Lee Let A(t) be a MPH curve of type k with polynomial functions σ 1 (t),...,σ k (t) satisfying (1). Then since k 1 α i (t) 2 α m+j (t) 2 σ l (t) 2 = σ k (t) 2, the polynomial curve (α 0 (t),...,α m+n,σ 1 (t),...,σ k 1 (t)) becomes a hodograph of a MPH curve in R m+1,n+k 1. Therefore the characterization of the set of MPH curves in R m+1,n+k 1 implies that of the set of MPH curves of type k in R m+1,n : Theorem 3. Let A(t) be a polynomial curves in the Minkowski space R m+1,n. The polynomial curve A(t) is a MPH curve of type k if and only if there exist polynomial functions s(t),u(t),v(t),τ(t),ρ(t), and θ l (t) for 1 l m+n+k 1 with m i=1 θ i(t) 2 n+k 1 θ m+j (t) 2 = ρ(t)τ(t) and gcd(τ(t),v(t)) = gcd(θ 1 (t),...,θ m+n+k 1 (t)) = gcd(u(t),v(t)ρ(t)) = 1, so that the hodograph α(t) :=A (t) =(α 0 (t),...,α m+n (t)) of A(t) is expressed as follows: α 0 (t) =s(t)[u(t) 2 τ(t) v(t) 2 ρ(t)], α k (t) =s(t)[2u(t)v(t)θ k (t)], 1 k m + n. Here we consider some canal surfaces as an application of Theorem 3. Consider a canal surface Φ, which is the envelope of one parameter family of spheres Σ(t) : (x, y, z) (x(t),y(t),z(t)) R 3 = r(t), where γ(t) =(x(t),y(t),z(t)) be a polynomial spine curve and r(t) is a nonnegative polynomial. Peternell and Pottmann [12] proved that such canal surface has a rational parameterization by showing that there exist polynomials f 1 (t) and f 2 (t) such that x (t) 2 + y (t) 2 + z (t) 2 r (t) 2 = f 1 (t) 2 + f 2 (t) 2. Using Theorem 3, we give a simple example of a one parameter family of spheres: Set u(t) =1,v(t) =t, τ(t) =2t 2 +3,ρ(t) =1, θ 1 (t) =t 2 +2,θ 2 (t) =1,θ 3 (t) =t 2 +1,θ 4 (t) =1. Then we can choose a MPH curve A(t) =(x(t),y(t),z(t),r(t)) =(t 3 /3+3t, t 4 /2+2t 2,t 2,t 4 + t 2 ) l=1

9 Higher dimensional Minkowski Pythagorean Hodograph curves 413 in R 3,1, which represents a one parameter family of spheres. Acknowledgements The authors wish to express their sincere thanks to the referees for their invaluable comments and suggestions. References [1] G. Albrecht and R. T. Farouki Construction of C 2 Pythagorean hodograph interpolation splines by the homotopy method, Adv. Comput. Math. 5 (1996), [2] H. I. Choi, D. S. Lee, and H. P. Moon, Clifford algebra, spin representation, and rational parameterization of curves and surfaces, Adv. Comput. Math. 17 (2002), [3] H. I. Choi, S. W. Choi, and H. P. Moon, Mathematical theory of medial axis transform Pacific J. Math. 181(1) (1997), [4] R. Dietz, J. Hoschek and B. Jüttler, An algebraic approach to curves and surfaces on the sphere and on other quadrics, Comput. Aided Geom. Design 10 (1993), [5] R. T. Farouki, Pythagorean hodograph curves in practical use, Geometry Processing for Design and Manufacturing, SIAM, Philadelphia (1992), [6] R. T. Farouki, The conformal map z z 2 of the hodograph plane, Comput. Aided Geom. Design 11 (1994), [7] R. T. Farouki and C. A. Neff Hermite interpolating by Pythagorean hodograph quintics, Math. Comput. 64 (1995), [8] R. T. Farouki and T. Sakkalis, Pythagorean hodographs, IBM J. Res. Development 34 (1990), [9] R. T. Farouki and T. Sakkalis, Pythagorean-hodograph space curves, Adv. Comput. Math. 2 (1994), [10] G.-I. Kim, Higher dimensional PH curves, Proc. Japan Acad., 78 (2002), Ser. A. [11] H. P. Moon, Minkowski Pythagorean hodographs, Comput. Aided Geom. Design 16 (1999), [12] M. Peternell and H. Pottman, Computing rational parametrizations of canal surfaces, J. Symbolic Comput. 23 (1997), Gwang-Il Kim is a professor in Department of Mathematics, Gyeongsang National University, Jinju, Korea. He received his M. Sc from POSTECH in 1991 and Ph.D from the same University in His research interests focus on computer aided geometric design and dynamical systems. Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju, Korea gikim@gsnu.ac.kr Sunhong Lee is a lecturer in Department of Mathematics, Gyeongsang National University, Jinju, Korea. He received his Ph.D from POSTECH in His research interests focus on complex analysis and related topics. Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju, Korea sunhong@gsnu.ac.kr