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A Characterization of Helical Polynomial Curves of Any Degree

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A Characterization of Helical Polynomial Curves of Any Degree A CHARACTERIZATION OF HELICAL POLYNOMIAL CURVES OF ANY DEGREE J. MONTERDE Abstract. We give a full characterization of helical polynomial curves of any degree and a simple way to construct them. Existing results about Hermite interpolation are revisited. A simple method to select the best quintic interpolant among all possible solutions is suggested. 1. Introduction The notion of helical polynomial curves, i.e, polynomial curves which made a constant angle with a ¯xed line in space, have been studied by di®erent authors. Let us cite the papers [5, 6, 7] where the main results about the cubical and quintic cases are stablished. In [5] the authors gives a necessary condition a polynomial curve must satisfy in order to be a helix. The condition is expressed in terms of its hodograph, i.e., its derivative. If a polynomial curve, ®, is a helix then its hodograph, ®0, must be Pythagorean, i.e., jj®0jj2 is a perfect square of a polynomial. Moreover, this condition is su±cient in the cubical case: all PH cubical curves are helices. In the same paper it is also stated that not only ®0 must be Pythagorean, but also ®0^®00. Following the same ideas, in [1] it is proved that both conditions are su±cient in the quintic case. Unfortunately, this characterization is no longer true for higher degrees. It is possible to construct examples of polynomial curves of degree 7 verifying both conditions but being not a helix. The aim of this paper is just to show how a simple geometric trick can clarify the proofs of some previous results and to simplify the needed compu- tations to solve some related problems as for instance, the Hermite problem using helical polynomial curves. The Dietz-Hoschek-JÄuttlertheorem (see [2] or [3]) relates the Pythagorean condition to the Hopf map H : C2 ! R3. A polynomial curve ® is Pythagorean 0 if and only if there exist z1; z2 such that ® = H(z1; z2) where z1; z2 are two polynomial complex functions. The geometric trick is based on the following facts: ¯rst, the normalized Hopf map is related to the stereographic projection (the usual stereographic projection from the two dimensional unit sphere, S2, into the complex plane, Date: October 6, 2006. 2000 Mathematics Subject Classi¯cation. Primary 53A04; Secondary 53C40. Key words and phrases. generalized polynomial helices, Pythagorean hodograph curves. 1 2 J. MONTERDE not the generalized stereographic projection from the three-dimensional pro- jective space, using homogeneous coordinates, to S2, as it appears in Remark 1 of [5], or in Remark 2 of [6]); second, the tangent indicatrix of a helical curve describes a circle on the unit sphere and, ¯nally, the image by the stereographic projection of a circle on the unit sphere is a line-or-circle in the complex plane. Joining up these arguments, we obtain a full characterization of polyno- mial helices of any degree in terms of the quotient z1 . The condition is of z2 geometric nature: ® is an helix if and only if z1 is a rational parametriza- z2 tion of a piece of a line or a circle in C depending whether or not the vector (0; 0; 1) belongs to the hodograph, respectively. According to this approach, the already known results ([6, 7]) about Her- mite interpolation using helical curves are revisited and their proofs sim- pli¯ed. Moreover, the classi¯cation into two classes of helical PH quintics introduced in [5] is also clari¯ed, and a result concerning a su±cient condi- tion on the quaternions generating a PH helical curve stated and proved for quintics in the cited reference is here generalized to any dimension (see Cor. 2). Finally, a simple method to choose the best among all possible general PH quintic helices solving the ¯rst order Hermite problem is suggested. In the cited references, the choice among all the possible solutions involves the computation of some integral, the energy associated with the rotation- minimizing frame. Here we simply control the values of two parameters involved in our method (see Prop. 1). 2. Spatial Pythagorean hodograph curves Initially, we will use the quaternion representation of spatial PH curves. Given a quaternion polynomial A(t) = u(t) + iv(t) + jp(t) + kq(t); the product ®0(t) = A(t)iA¤(t) de¯nes (the asterisk means quaternionic conjugate) a spatial Pythagorean hodograph, ®0, whose components are x0 = u2 + v2 ¡ p2 ¡ q2; (2.1) y0 = 2(up + vq); z0 = 2(vp ¡ uq); and such that jj®0jj2 = (x0)2 + (y0)2 + (z0)2 = (u2 + v2 + p2 + q2)2. In terms of the Hopf map (see [3], theorem 4.2) H : C2 ! R3 2 2 de¯ned by H(z1; z2) = (2z1z¹2; jz1j ¡ jz2j ), and taking (2.2) z1(t) = u(t) + iv(t); z2(t) = p(t) + iq(t); CHARACTERIZATION OF HELICAL CURVES OF ANY DEGREE 3 the derivative of the curve can be written as 0 ® (t) = H(z1(t); z2(t)): 0 Theorem 1. A Pythagorean polynomial curve ® with ® (t) = H(z1(t); z2(t)) is a helix if and only the curve in the complex plane t ! z1(t) traces a line- z2(t) or-circle. Proof. A curve ® is a general helix if and only if its tangent indicatrix is a subset of a circle on the unit sphere S2. Let us de¯ne the normalized Hopf map Hn : C2 ! S2 by n 1 2 2 H (z1; z2) = 2 2 (2z1z¹2; jz1j ¡ jz2j ): jz1j + jz2j 0 If ® (t) = H(z1(t); z2(t)), then its tangent indicatrix is ®0(t) = Hn(z (t); z (t)): jj®0(t)jj 1 2 Notice that Hn(z ; z ) = Hn( z1 ; 1), and that for any z 2 C, Hn(z; 1) = 1 2 z2 ¡1 2 pst (z) where pst : S ! C[f1g denotes the extended stereographic projec- tion from the north pole of the whole unit sphere to the extended complex plane with the assumption pst(0; 0; 1) = 1. Therefore, ® is a general helix if and only if p¡1( z1(t) ) 2 S2 belongs to a st z2(t) circle in the unit sphere. But the image by the stereographic projection of a circle in the unit sphere is a line-or-circle in the extended complex plane (see Proposition 7.23 in [8]). ¤ 0 Corollary 1. A Pythagorean polynomial curve ® with ® (t) = H(z1(t); z2(t)) with (0; 0; 1) in the trace of tangent indicatrix is a helix if and only the curve in the complex plane t ! z1(t) traces a piece of a straight line. z2(t) Proof. Recall that a circle on the unit sphere passes trough the north pole if and only if its image by the extended stereographic projection is a straight line of the complex plane. ¤ Figure I. Representation of the tangent indicatrix of an helical curve and of its stereographic projection. 4 J. MONTERDE The next result was proved for the quadratic case in [5]. We shall prove it in its full generality. n Corollary 2. Let us suppose that fAigi=0 are a set of n + 1 quaternionic control points of a B¶eziercurve such that the interior control points are linear combinations of the exterior ones, then the associated quaternionic B¶eziercurve is the hodograph of a helical curve. Proof. If all the interior quaternionic control points are linear combinations Pn n of fA0; Ang, then A(t) = i=0 Bi (t)Ai is a parametrization of a piece of the quaternionic straight line B(t) = (1 ¡ t)A0 + tAn. What we need to prove is that B(t)iB(t)¤ is the hodograph of an helical curve, or equivalently that if z1(t) = a + tb; z2(t) = c + td; a; b; c; d 2 C then z1(t) belongs to a line-or-circle. z2(t) An straightforward computation shows that if Im(c ¢ d) 6= 0 then z1(t) z2(t) ¹ belongs to a circle with center p = i b¢c¹¡a¢d and radius r = j a ¡ pj. 2Im(c¢d) c If Im(c ¢ d) = 0, then we can write z2(t), after an a±ne change of the parameter t, as z2(t) = t¹ where ¹ 2 C ¡ f0g. Again a simple computa- ¹ tion shows that z1(t) belongs to a line passing through the point a¹+b with z2(t) ¹ a direction ¡ 2¹ . ¤ 2.1. Examples. It is just a matter of computation to check that the two examples of quintic helices in [5] verify the condition written in the statement of Th. 1. As it is said in the cited reference, they are, respectively, examples of the two classes of helical PH quintics. 2.1.1. Monotone-helical PH quintic example. The ¯rst example is de¯ned by the four quadratic polynomials u(t) = t2¡3t; v(t) = t2¡5t+10; p(t) = t2¡9t+10; q(t) = ¡2t2+3t+5: Let us recall that for this case the polynomials z1 and z2 share a common factor corresponding to the root 1 + 2i. So, we can simplify the quotient z1(t) and to check that it belongs to the circumference with center ( 3 ; 1 ) z2(t) p 10 10 2 and radius r = 2 . Moreover, let us see for this example which kind of rational parametriza- tion of a straight line is z1(t) when the hodograph is rotated in the space in z2(t) such a way that the tangent vector at t = 0 is (0; 0; 1). After a rotation with y-axis and angle arccos( p1 ) and a rotation with p 17 17 x-axis and angle arccos( 9 ) the hodograph curve is transformed into µ ¶ 2t(7t ¡ 15) 2t(71t ¡ 225) 1 ®0(t) = (5 ¡ 2t + t2) p ; p ; (405 ¡ 288t + 43t2) : 17 9 17 9 CHARACTERIZATION OF HELICAL CURVES OF ANY DEGREE 5 ¡! ®0(0) Now, it veri¯es t (0) = jj®0(0)jj = (0; 0; 1).
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