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Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves
Computer Aided Geometric Design 28 (2011) 436–445
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Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves ✩ Rida T. Farouki a,∗ , Takis Sakkalis b a b
Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA Mathematics Laboratory, Agricultural University of Athens, 75 Iera Odos, Athens 11855, Greece
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 November 2010 Received in revised form 27 July 2011 Accepted 28 July 2011 Available online 3 August 2011
A rotation-minimizing frame on a space curve r(t ) is an orthonormal basis (f1 , f2 , f3 ) for R3 , where f1 = r /|r | is the curve tangent, and the normal-plane vectors f2 , f3 exhibit no instantaneous rotation about f1 . Such frames are useful in spatial path planning, swept surface design, computer animation, robotics, and related applications. The simplest curves that have rational rotation-minimizing frames (RRMF curves) comprise a subset of the quintic Pythagorean-hodograph (PH) curves, and two quite different characterizations of them are currently known: (a) through constraints on the PH curve coefficients; and (b) through a certain polynomial divisibility condition. Although (a) is better suited to the formulation of constructive algorithms, (b) has the advantage of remaining valid for curves of any degree. A proof of the equivalence of these two different criteria is presented for PH quintics, together with comments on the generalization to higher-order curves. Although (a) and (b) are both sufficient and necessary criteria for a PH quintic to be an RRMF curve, the (non-obvious) proof presented here helps to clarify the subtle relationships between them. © 2011 Elsevier B.V. All rights reserved.
Keywords: Rotation-minimizing frames Pythagorean-hodograph curves Complex numbers Quaternions Hopf map Polynomial identities
1. Introduction An adapted orthonormal frame (f1 , f2 , f3 ) on a space curve r(t ) incorporates the curve tangent t = r /|r | as the vector f1 , while the vectors f2 , f3 span the curve normal plane. The variation of such a frame is described by its angular velocity ω through the differential equations
f 1 = ω × f1 ,
f 2 = ω × f2 ,
f 3 = ω × f3 ,
and the frame is said to be rotation-minimizing if the angular velocity has no component along f1 = t — i.e., ω · t ≡ 0, so f2 , f3 exhibit no instantaneous rotation about f1 = t. Such frames are of great importance in applications such as motion planning, computer animation, geometric design, and robotics — see Farouki (2002); Farouki et al. (2010a, 2010b); Farouki and Han (2003); Jüttler (1998); Jüttler and Mäurer (1999a, 1999b); Klok (1986); Sír and Jüttler (2005); Wang and Joe (1997); Wang et al. (2008) for further details. Considerable interest has recently emerged in identifying curves that have rational rotation-minimizing frames (or RRMF curves). Such curves must be Pythagorean-hodograph (PH) curves (Farouki, 2008), since only PH curves admit rational unit tangents. The construction of RRMF curves is thus essentially a matter of identifying constraints on PH curves, that are sufficient and necessary for rational RMFs. Progress in this endeavor may be summarized as follows. ✩
*
This paper has been recommended for acceptance by H. Prautzsch Corresponding author. E-mail addresses: [email protected] (R.T. Farouki), [email protected] (T. Sakkalis).
0167-8396/$ – see front matter doi:10.1016/j.cagd.2011.07.004
© 2011
Elsevier B.V. All rights reserved.
R.T. Farouki, T. Sakkalis / Computer Aided Geometric Design 28 (2011) 436–445
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Choi and Han (2002) introduced the Euler–Rodrigues frame (ERF), a rational adapted frame on spatial PH curves, that is a useful reference for identifying rational RMFs. Using the ERF, Han (2008) subsequently formulated a rational function identity as a criterion characterizing RRMF curves of any degree, and showed that RRMF cubics are necessarily degenerate (i.e., planar curves, or with non-primitive hodographs). The existence of non-degenerate RRMF quintics, characterized by certain constraints on the coefficients of the Hopf map representation of spatial PH curves, was first established in Farouki et al. (2009). These conditions were soon superseded (Farouki, 2010) by much simpler, symmetric coefficient constraints for both the quaternion and Hopf map PH curve representations. Finally, Han’s criterion (Han, 2008) was shown in Farouki and Sakkalis (2010) to be equivalent to a certain polynomial divisibility condition. The methods employed to derive the RRMF quintic coefficient constraints (Farouki, 2010) and the polynomial divisibility condition (Farouki and Sakkalis, 2010) are very different in nature. Although both are sufficient and necessary conditions for a spatial PH quintic to be an RRMF curve, there is no obvious connection between them. Hence, the goal of this paper is to give a formal proof of their equivalence, that helps in understanding the non-trivial relationships between these two conditions. This task amounts to showing that the satisfaction of certain constraints on the Bernstein coefficients of two complex quadratic polynomials α (t ), β(t ) is equivalent to proportionality of |α (t )β (t ) − α (t )β(t )|2 and |α (t )|2 + |β(t )|2 . The transformation of α (t ), β(t ) to “canonical form” is a key step in making the problem tractable. The coefficient constraints offer a constructive basis for designing RRMF curves, but are currently known only for quintics. Conversely, the divisibility condition is existential in nature but applies to RRMF curves of any degree. Although the equivalence of these distinct characterizations follows implicitly from their independent derivations as sufficient-and-necessary conditions for RRMF quintics, it is non-obvious and its formal demonstration is non-trivial. The proof presented herein offers new insight into the complicated non-linear structure of RRMF curves, that may prove useful in extending the coefficient constraints to higher-order RRMF curves, so as to obtain additional degrees of freedom for the geometric design of spatial motions. The organization of this paper is as follows. Following a brief review of the quaternion and Hopf map models for spatial PH curves in Section 2, the Euler–Rodrigues frame is introduced in Section 3, and its angular velocity properties are analyzed. Section 4 then describes conditions that characterize rational rotation-minimizing frames, in terms of both coefficient constraints and the divisibility criterion. By invoking a reduction of these conditions to a simplified canonical form, they are shown in Section 5 to be equivalent for the case of quintics. Finally, Section 6 summarizes the results of this paper, and identifies some topics that deserve further investigation. 2. Quaternion and Hopf map forms A Pythagorean-hodograph (PH) curve r(t ) = (x(t ), y (t ), z(t )) is a polynomial curve whose derivative r (t ) = (x (t ), y (t ),
z (t )) satisfies
x 2 (t ) + y 2 (t ) + z 2 (t ) = σ 2 (t )
(1)
for some polynomial σ (t ). The quaternion and Hopf map forms (Choi et al., 2002) are two alternative (equivalent) models for the construction of PH curves. The former generates a Pythagorean hodograph r (t ) from a quaternion polynomial1
A(t ) = u (t ) + v (t )i + p (t )j + q(t )k,
(2)
and its conjugate A∗ (t ) = u (t ) − v (t )i − p (t )j − q(t )k through the product
r (t ) = A(t ) i A∗ (t ) = u 2 (t ) + v 2 (t ) − p 2 (t ) − q2 (t ) i
+ 2 u (t )q(t ) + v (t ) p (t ) j + 2 v (t )q(t ) − u (t ) p (t ) k.
(3)
The latter generates a Pythagorean hodograph from complex polynomials
α (t ) = u (t ) + iv (t ),
β(t ) = q(t ) + ip (t )
(4)
through the expression
2 2 r (t ) = α (t ) − β(t ) , 2 Re
α (t )β(t ) , 2 Im α (t )β(t ) .
(5)
The equivalence of (3) and (5) can be verified by setting A(t ) = α (t ) + kβ(t ), the imaginary unit i being identified with the quaternion element i. We shall find it convenient to transform back and forth between the representations (3) and (5) — see Farouki (2008) for a more thorough treatment of them. The parametric speed σ (t ) = |r (t )| of the curve r(t ) defined by integrating r (t ), i.e., the derivative of its arc length s with respect to the parameter t, is the polynomial
1 Calligraphic characters such as A denote quaternions, their scalar and vector parts being indicated by scal(A) and vect(A). Bold symbols denote complex numbers or vectors in R3 — the meaning should be clear from the context.
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2
2
2
σ (t ) = A(t ) = α (t ) + β(t ) = u 2 (t ) + v 2 (t ) + p 2 (t ) + q2 (t ).
(6)
The derivative of the parametric speed is
σ (t ) =
r (t ) · r (t )
σ (t )
= 2 u (t )u (t ) + v (t ) v (t ) + p (t ) p (t ) + q(t )q (t ) .
(7)
We consider here only regular curves, satisfying σ (t ) = 0 for all real t. Since all planar PH curves are trivially RRMF curves, it is useful to have a means to identify when (3) and (5) define a planar PH curve, rather than a true space curve. This occurs when x (t ), y (t ), z (t ) are linearly dependent. We now give a simplified version of the criteria stated in Farouki and Sakkalis (2010), that identify degeneration to a plane curve. This is based on the following result, which makes use of the easily-verified fact that two real polynomials a(t ), b(t ) are linearly dependent if and only if they satisfy ab = a b. Proposition 1. Three real polynomials a(t ), b(t ), c (t ) are linearly dependent if and only if they satisfy
ac − a c bc − b c = ac − a c bc − b c .
(8)
Proof. Suppose first that a(t ), b(t ), c (t ) are linearly dependent — i.e., there exist real numbers λ, μ, ν , not all zero, such that λa(t ) + μb(t ) + ν c (t ) ≡ 0. If λ = 0 we have a(t ) = mb(t ) + nc (t ) with m = −μ/λ and n = −ν /λ. Thus, a(t ) − mb(t ) = nc (t ), so (a − mb)c = (a − mb )c and ac − a c = m(bc − b c ). Since (ac − a c ) = ac − a c and (bc − b c ) = bc − b c, the latter implies that (ac − a c )(bc − b c ) = (ac − a c )(bc − b c ), which is condition (8). If λ = 0, on the other hand, b(t ) and c (t ) must be multiples of each other, so that bc − b c = bc − b c = 0 and (8) still holds. Conversely, suppose (8) holds. Noting that (ac − a c ) = ac − a c and (bc − b c ) = bc − b c again, we infer that ac − a c and bc − b c are linearly dependent, and this implies that a, b, c are linearly dependent as well. 2 3. Euler–Rodrigues frame Along the PH curve defined by (2)–(3), a rational adapted orthonormal frame — the Euler–Rodrigues frame (ERF) — may be defined (Choi and Han, 2002) by
e1 (t ), e2 (t ), e3 (t ) =
(A(t ) i A∗ (t ), A(t ) j A∗ (t ), A(t ) k A∗ (t )) . |A(t )|2
(9)
Here e1 (t ) is the curve tangent while e2 (t ), e3 (t ) span the normal plane. The vectors (9) can be expressed in terms of the components of (2) as
e1 = e2 = e3 =
(u 2 + v 2 − p 2 − q2 )i + 2(uq + vp )j + 2( vq − up )k , u 2 + v 2 + p 2 + q2 2( vp − uq)i + (u 2 − v 2 + p 2 − q2 )j + 2(uv + pq)k u 2 + v 2 + p 2 + q2 2(up + vq)i + 2( pq − uv )j + (u 2 − v 2 − p 2 + q2 )k u 2 + v 2 + p 2 + q2
The variation of the ERF is characterized by its angular velocity
e1 = ω × e1 , and if
e2 = ω × e2 ,
e3 = ω × e3 ,
, .
(10)
ω through (11)
ω is written in terms of the basis (e1 , e2 , e3 ) as
ω = ω1 e1 + ω2 e2 + ω3 e3 ,
(12)
its components are given by
ω1 = e3 · e2 = −e2 · e3 = ω2 = e1 · e3 = −e3 · e1 = ω3 = e2 · e1 = −e1 · e2 =
2(uv − u v − pq + p q) u 2 + v 2 + p 2 + q2
2(up − u p + vq − v q) u 2 + v 2 + p 2 + q2
2(uq − u q − vp + v p ) u 2 + v 2 + p 2 + q2
The relations (11) can thus be cast in matrix form as
, , .
(13)
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⎡
⎤
⎡ e1 0 ⎢ ⎥ ⎣ ⎣ e2 ⎦ = −ω3 e3
ω2
ω3
−ω2
0
ω1
−ω1
0
⎤⎡
439
⎤
e1 ⎦ ⎣ e2 ⎦ . e3
(14)
Now the combinations of u , v , p , q and u , v , p , q in (7) and (13) are generated by the product
A∗ A = uu + v v + pp + qq + uv − u v − pq + p q i + up − u p + vq − v q j + uq − u q − vp + v p k,
(15)
namely,
σ = 2 uu + v v + pp + qq ,
σ ω1 = 2 uv − u v − pq + p q ,
σ ω2 = 2 up − u p + vq − v q ,
σ ω3 = 2 uq − u q − vp + v p .
On the other hand, we also have
A A∗ = uu + v v + pp + qq + uv − u v + pq − p q i + up − u p − vq + v q j + uq − u q + vp − v p k,
(16)
and from (10) and (12)–(13), one can verify that
σ ω = 2 uv − u v + pq − p q i + 2 up − u p − vq + v q j + 2 uq − u q + vp − v p k. Hence, we observe that
σ = 2 scal A∗ A = 2 scal A A∗ ,
σ ω = 2 vect A A∗ ,
and consequently
2
σ 2 + σ 2 |ω|2 = 4|A|2 A .
(17)
4. Rational rotation-minimizing frames Fig. 1 compares the variation of the normal-plane vectors for the Frenet frame (Kreyszig, 1959) and the rotationminimizing frame, along a quintic RRMF curve. The RMF evidently offers a more “natural” means of orienting a rigid body along this path, when one principal axis of the body is constrained to remain aligned with the path tangent.
Fig. 1. The Frenet frame (Kreyszig, 1959) normal-plane vectors along a quintic RRMF curve (left), and the RMF normal-plane vectors along the same path (right). The latter frame vectors exhibit no instantaneous rotation about the curve tangent, and have a rational dependence on the curve parameter.
Since satisfaction of (1) is necessary for a rational unit tangent, all RRMF curves must be PH curves. Han (2008) determined a criterion for PH curves to possess rational RMFs using the quaternion representation (3) — namely, relatively prime polynomials a(t ), b(t ) must exist, such that the components u (t ), v (t ), p (t ), q(t ) of A(t ) satisfy
uv − u v − pq + p q u 2 + v 2 + p 2 + q2
=
ab − a b a2 + b 2
(18)
.
This is equivalent to the existence of a rational normal-plane rotation
1 f2 (t ) = 2 f3 (t ) a (t ) + b2 (t )
a2 (t ) − b2 (t ) −2a(t )b(t ) 2a(t )b(t ) a2 (t ) − b2 (t )
e2 (t ) e3 (t )
(19)
that maps the ERF vectors e2 (t ), e3 (t ) onto the RMF vectors f2 (t ), f3 (t ) and exactly cancels the ω1 component of the ERF angular velocity. In terms of the Hopf map representation, condition (18) can be interpreted as requiring the existence of a complex polynomial w(t ) = a(t ) + ib(t ), with gcd(a(t ), b(t )) = constant, such that
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¯ ) ¯ + ββ Im(αα |α
|2
+ |β|2
=
¯ ) Im(ww | w |2
(20)
.
The existence of rational RMFs on spatial PH quintics has been studied thus far (Farouki, 2010; Farouki et al., 2009) under the assumption that conditions (18)–(20) are satisfied in the case where u 2 + v 2 + p 2 + q2 = a2 + b2 or |α |2 + |β|2 = |w|2 . It was shown in Farouki and Sakkalis (2010) that a necessary condition for this is the requirement that the polynomials
2
2
ρ = up − u p + vq − v q + uq − u q − vp + v p
2
(21)
and
η = uu + v v + pp + qq
2 + uv − u v − pq + p q
(22)
should both be divisible by the polynomial σ defined by (6). This condition becomes sufficient in the case where gcd(uv − u v − pq + p q, u 2 + v 2 + p 2 + q2 ) = 1. We henceforth refer to this as the generic case. Note that (21) and (22) can be more compactly expressed in terms of (4) as
2
2
¯ t )β (t ) ρ (t ) = α (t )β (t ) − α (t )β(t ) , η(t ) = α¯ (t )α (t ) + β(
(23)
and they satisfy the relation
ρ (t ) + η(t ) = σ (t )ζ (t ), |α (t )|2
where we define ζ (t ) = the product (15) becomes
(24)
+ |β (t )| . In the quaternion model, setting A(t ) = α (t ) + kβ(t ) 2
and A∗ (t )
= α¯ (t ) − kβ(t ),
A∗ (t )A (t ) = α¯ (t ) − kβ(t ) α (t ) + kβ (t ) = α¯ (t )α (t ) + α¯ (t )kβ (t ) − kβ(t )α (t ) − kβ(t )kβ (t ), ¯ t ) we have ¯ (t )k = kα (t ), β(t )k = kβ( and by noting that α
¯ t )β (t ) + k α (t )β (t ) − α (t )β(t ) . A∗ (t )A (t ) = α¯ (t )α (t ) + β(
Hence, the relation (24) is seen to be equivalent to (17). Remark 1. According to item 2 of Remark 5.1 in Farouki and Sakkalis (2010), a necessary condition for the satisfaction of (18) is that deg(u 2 + v 2 + p 2 + q2 ) deg(a2 + b2 ). As already noted above, rational RMFs on spatial PH curves have thus far been investigated (Farouki, 2010; Farouki et al., 2009; Farouki and Sakkalis, 2010) in the case where deg(u 2 + v 2 + p 2 + q2 ) = deg(a2 + b2 ), which naturally includes the generic case. For quadratic polynomials u , v , p , q defining an RRMF quintic, deg(u 2 + v 2 + p 2 + q2 ) = 4, so (18) can be satisfied in the case where deg(a2 + b2 ) = 4, 2, or 0. In this paper we shall follow the practice of prior studies in focusing on the satisfaction of (18) in the generic case. Preliminary investigations suggest that a comprehensive analysis of the non-generic cases is a non-trivial task, and for this reason we defer it to a future study. 5. Conditions for RRMF quintics The Bernstein basis for polynomials of degree m on t ∈ [0, 1] is defined by
bm k (t ) =
m k
(1 − t )m−k t k ,
k = 0, . . . , m .
We focus here on PH quintics, specified in the quaternion form (3) by quadratic quaternion polynomials
A(t ) = A0 b20 (t ) + A1 b21 (t ) + A2 b22 (t ),
(25)
or, equivalently, in the Hopf map form (5) by quadratic complex polynomials
α (t ) = α 0 b20 (t ) + α 1 b21 (t ) + α 2 b22 (t ), β(t ) = β 0 b20 (t ) + β 1 b21 (t ) + β 2 b22 (t ).
(26)
It was shown in Farouki (2010) that a spatial PH quintic has a rational RMF if and only if the coefficients of (25) satisfy
vect A2 iA∗0 = A1 iA∗1 ,
(27)
or, equivalently, the coefficients of (26) satisfy
¯ 2 − β 0 β¯ 2 ) = |α 1 |2 − |β 1 |2 , Re(α 0 α
α 0 β¯ 2 + α 2 β¯ 0 = 2α 1 β¯ 1 .
(28)
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Remark 2. We emphasize that the coefficient constraints (27)–(28) identify only quintic RRMF curves, but the characterization in terms of divisibility of the polynomials (21)–(22) by (6) holds for RRMF curves of any degree. We focus henceforth on the Hopf map form (5). Our aim is to show that, for RRMF quintics, satisfaction of the constraints (28) is equivalent to the divisibility of (23) by (6). Our main result (whose proof will be deferred until certain preliminary results have been established) may be stated as follows.
¯ ), |α |2 + |β|2 ) = 1. ¯ + ββ Proposition 2. Assume that the quadratic complex polynomials (26) satisfy the condition2 gcd(Im(αα Then |α β − α β|2 is divisible by |α |2 + |β|2 if and only if their coefficients satisfy (28). To facilitate the proof of Proposition 2, we introduce a “canonical form” to simplify the analysis. For the quadratic polynomials (26) used in the Hopf map form (5), we have
α (t )β (t ) − α (t )β(t ) = 2(α 0 β 1 − α 1 β 0 )b20 (t ) + (α 0 β 2 − α 2 β 0 )b21 (t ) + 2(α 1 β 2 − α 2 β 1 )b22 (t ). Then |α (t )β (t ) − α (t )β(t )|2 and |α (t )|2 + |β(t )|2 are both real quartics, and they must be proportional if the former is to be divisible by the latter, i.e.,
α (t )β (t ) − α (t )β(t )2 = k α (t )2 + β(t )2
(29)
for some positive real k. Now given a quadratic complex polynomial c(t ) = c0 b20 (t ) + c1 b21 (t ) + c2 b22 (t ), the real quartic |c(t )|2 has Bernstein coefficients
|c0 |2 ,
2|c1 |2 + Re(c2 c0 )
Re(c0 c1 ),
3
,
Re(c1 c2 ),
|c2 |2 .
Hence, the condition (29) is equivalent to the coefficient constraints
4|α 0 β 1 − α 1 β 0 |2 = k |α 0 |2 + |β 0 |2 ,
¯ 0 β¯ 2 − α¯ 2 β¯ 0 ) = k Re(α 0 α¯ 1 + β 0 β¯ 1 ), 2 Re (α 0 β 1 − α 1 β 0 )(α 2 ¯ 0 β¯ 1 − α¯ 1 β¯ 0 ) = k 2 |α 1 |2 + |β 1 |2 + Re(α 2 α¯ 0 + β 2 β¯ 0 ) , 2α 0 β 2 − α 2 β 0 + 4 Re (α 1 β 2 − α 2 β 1 )(α ¯ 1 β¯ 2 − α¯ 2 β¯ 1 ) = k Re(α 1 α¯ 2 + β 1 β¯ 2 ), 2 Re (α 0 β 2 − α 2 β 0 )(α 4|α 1 β 2 − α 2 β 1 |2 = k |α 2 |2 + |β 2 |2 ,
(30) (31) (32) (33) (34)
for some positive real k. Now the identification of quintic RRMF curves as a subset of all spatial PH quintics is simplified (Farouki, 2010) by reducing the polynomials (26) to canonical form through a scaling/rotation transformation that does not influence the RRMF property. The canonical values (α 0 , β 0 ) = (1, 0) were used in Farouki (2010), but these values can be assigned to any chosen coefficients (αk , β k ). For complex numbers (μ, ν ) = (0, 0) consider the transformation
μ −ν¯ → β(t ) ν μ¯
α (t )
α (t ) β(t )
,
α (t ), β(t ) with inverse defined by
1 α (t ) μ¯ ν¯ α (t ) → . β(t ) β(t ) |μ|2 + |ν |2 −ν μ
(35)
of the polynomials
(36)
Under this transformation,
α (t )2 + β(t )2 → |μ|2 + |ν |2 α (t )2 + β(t )2 , α (t )β (t ) − α (t )β(t ) → |μ|2 + |ν |2 α (t )β (t ) − α (t )β(t ) , ¯ t )β (t ) → |μ|2 + |ν |2 α¯ (t )α (t ) + β( ¯ t )β (t ) , α¯ (t )α (t ) + β(
(37)
and the divisibility of (23) by (6) is unaltered. The map (35) is implemented by applying it to the Bernstein coefficients of α (t ) and β(t ),
αk βk
2
→
μ −ν¯ ν μ¯
αk βk
,
k = 0, 1, 2.
This assumption is equivalent to restricting our attention to the generic case of RRMF quintics (see Remark 1).
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Specifically, for any prescribed index k, the choices
α¯ k |αk |2 + |β k |2
μ=
−β k |αk |2 + |β k |2
ν=
and
(38)
in (35) yield (αk , β k ) → (1, 0). We choose the canonical case (α 1 , β 1 ) = (1, 0) here, since this gives more symmetrical arguments. For this canonical case, the conditions (30)–(34) that ensure the divisibility of |α (t )β (t )− α (t )β(t )|2 by |α (t )|2 + |β(t )|2 reduce to
4|β 0 |2 = k |α 0 |2 + |β 0 |2 ,
(39)
2 Re (α 2 β 0 − α 0 β 2 )β¯ 0 = k Re(α 0 ),
¯ 0 + β 2 β¯ 0 ) , 2|α 0 β 2 − α 2 β 0 | − 4 Re(β 2 β¯ 0 ) = k 2 + Re(α 2 α 2 Re (α 0 β 2 − α 2 β 0 )β¯ 2 = k Re(α 2 ), 4|β 2 |2 = k |α 2 |2 + |β 2 |2 , 2
(40) (41) (42) (43)
while the quintic RRMF coefficients constraints (28) become
¯ 2 − β 0 β¯ 2 ) = 1, Re(α 0 α
α 0 β¯ 2 + α 2 β¯ 0 = 0.
(44)
Lemma 1. The canonical-form RRMF quintic conditions (44) are satisfied if and only if α 2 , β 2 can be written in terms of α 0 , β 0 as
α 2 = γ α 0 and β 2 = −γ¯ β 0 ,
(45)
where γ is a complex value with real part
Re(γ ) =
1
|2
|α 0 + |β 0 |2
(46)
.
Proof. By substituting into conditions (44), one can easily verify that they are satisfied when α 2 , β 2 are specified by (45) and (46). Conversely, suppose that conditions (44) are satisfied. Then for some complex number γ we have
α 2 = γ α 0 and β¯ 2 = −γ β¯ 0 . Writing
γ = ζ + iη, substituting for α 2 , β 2 into the first of Eqs. (44), and equating real and imaginary parts, then yields
ζ= Thus
1
|α 0 |2 + |β 0 |2
.
α 2 , β 2 must be of the form (45)–(46) if conditions (44) hold. 2
Having established these preliminary results, we are now ready to present the proof of Proposition 2. Proof of Proposition 2. We need consider only the canonical case (α 1 , β 1 ) = (1, 0) for which conditions (28) reduce to (44), and the divisibility condition amounts to satisfaction of (39)–(43) for some positive real k. Suppose conditions (44) are satisfied. Then Lemma 1 indicates that α 2 , β 2 must be of the form (45)–(46). Consequently, conditions (39) and (43) are satisfied with
k=
4|β 0 |2
|α 0 |2 + |β 0 |2
=
4|β 2 |2
|α 2 |2 + |β 2 |2
.
Using this k, and noting from (45)–(46) that we have
α2β 0 − α0β 2 =
2α 0 β 0
|α 0 |2 + |β 0 |2
,
the left-hand sides of (40) and (42) reduce to k Re(α 0 ) and k Re(α 2 ) — in agreement with the right-hand sides. Finally, the left- and right-hand sides of (41) both yield the value
k 1+
2 | α 0 |2
|α 0 |2 + |β 0 |2
,
and thus agree. Hence, satisfaction of (44) implies satisfaction of (39)–(43). Conversely, suppose conditions (39)–(43) are satisfied. We assume β 0 , β 2 = 0 since β 0 = 0 ⇒ α 0 = 0 and β 2 = 0 ⇒ α 2 = 0 from (39) and (43), and it is then impossible to satisfy (41) with k > 0. We consider the following cases:
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Case 1: α 0 = 0 or α 2 = 0. Suppose that α 0 = 0. Then we must have β 0 = 0, since otherwise (41) yields 0 = 2k, contradicting the assumption that k > 0. Thus, since (39) and (43) imply that
|α 0 |2 ||β 2 |2 = |α 2 |2 |β 0 |2 ,
(47)
we obtain α 2 = 0. By similar arguments, α 2 = 0 ⇒ α 0 = 0. Hence, under conditions (39)–(43), (39)–(43) with α 0 = α 2 = 0 yield k = 4 and Re(β 2 β¯ 0 ) = −1, and (44) is then obviously satisfied. Case 2:
α 0 = 0 ⇔ α 2 = 0. Now
α 0 , α 2 = 0. In this case (39) and (43) again imply (47), and hence α 2 , β 2 must be of the form
α 2 = δ eiφ α 0 and β 2 = δ eiθ β 0
(48)
α 2 , β 2 into (40) and (42) and simplifying, these equations imply that Re(α 0 ) = Re eiφ α 0 − Re eiθ α 0 Re eiφ α 0 .
for real values δ , φ , θ with δ > 0. Substituting for
Re(α 0 ) − Re ei(φ−θ ) α 0
α 0 = |α 0 |eiψ then yields cos ψ − cos(φ − θ + ψ) cos ψ = cos(φ + ψ) − cos(θ + ψ) cos(φ + ψ).
Setting
Expanding the terms involving θ and using standard trigonometric relations, we finally have
sin θ =
cos2 ψ − cos2 (φ + ψ) sin(φ + 2ψ)
= sin φ.
This has solutions θ = (2m + 1)π − φ and θ = 2mπ + φ for integer m — i.e., eiθ = −e−iφ and eiθ = eiφ . We consider each case in turn. Case 2a: eiθ = −e−iφ . Setting
δ cos φ =
1
|α 0 |2 + |β 0 |2
Then, substituting for
α 2 = δ eiφ α 0 , β 2 = −δ e−iφ β 0 in (39)–(40), we obtain .
α 2 and β 2 into conditions (44), one can easily verify that this relation implies their satisfaction.
Case 2b: eiθ = eiφ . Taking α 2 = δ eiφ α 0 , β 2 = δ eiφ β 0 gives α 0 β 2 − α 2 β 2 = 0. By (40) and (42), this implies that Re(α 0 ) = Re(α 2 ) = 0. Writing α 0 = iα (where α = 0) we have α 2 = δ eiφ iα = δ α (− sin φ + i cos φ). Consequently, Re(α 2 ) = −δ α sin φ = 0 implies that sin φ = 0, since δ > 0 and α = 0. We thus have eiφ = ±1. Setting (α 2 , β 2 ) = ±(δ α 0 , δβ 0 ) in (39) and (41) gives
δ=
∓1 . |α 0 |2 + |β 0 |2
The negative value, corresponding to eiφ = +1, contradicts δ > 0, so this choice is invalid. For the positive value, corresponding to eiφ = −1, we have α 2 = −δ α 0 and β 2 = −δβ 0 and (recalling that α 1 = 1, β 1 = 0) hence
α (t ) = α 0 b20 (t ) + 1 · b21 (t ) −
α0
b2 (t ), |α 0 |2 + |β 0 |2 2 β0 β(t ) = β 0 b20 (t ) + 0 · b21 (t ) − b2 (t ). 2 |α 0 | + |β 0 |2 2
Using Re(α 0 ) = 0, one can then verify that
¯ = −α ¯ + ββ Im αα |α |2 + |β|2 =
(|α 0 |2 + |β 0 |2 )b20 (t ) + 0 · b21 (t ) + 1 · b22 (t ) , |α 0 |2 + |β 0 |2
[(|α 0 |2 + |β 0 |2 )b20 (t ) + 0 · b21 (t ) + 1 · b22 (t )]2 , |α 0 |2 + |β 0 |2
¯ ), |α |2 + |β|2 ) = 1, so this choice is also invalid. ¯ + ββ which contradicts the assumption that gcd(Im(αα
2
We conclude with an example to illustrate the equivalence of the RRMF conditions based on coefficient constraints and the divisibility criterion.
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Example 1. The curve defined through (5) by the coefficients
√
α 0 = −1 − 2i,
α 2 = −2 + i,
2
1
β 0 = 2i,
1
α 1 = 2 − √ i,
β1 = √ ,
β 2 = 2 − i,
2
for the polynomials (26) satisfies the coefficient constraints (28), since
¯ 2 − β 0 β¯ 2 ) = |α 1 |2 − |β 1 |2 = 2, Re(α 0 α
α 0 β¯ 2 + α 2 β¯ 0 = 2α 1 β¯ 1 = 2 − i.
In this case, we have
u (t ) = Re
v (t ) = Im
√
√
α (t ) = −(3 + 2 2 )t 2 + (2 + 2 2 )t − 1,
√
√
α (t ) = ( 2 − 1)t 2 + (4 − 2 )t − 2,
p (t ) = Im β(t ) = t 2 − 4t + 2, q(t ) = Re β(t ) = (2 −
√
2 )t 2 +
√
2t ,
and hence
2
√
2
√
σ (t ) = α (t ) + β(t ) = u 2 (t ) + v 2 (t ) + p 2 (t ) + q2 (t ) = (27 + 6 2 )t 4 − (52 + 6 2 )t 3 + 62t 2 − 36t + 9, u (t ) v (t ) − u (t ) v (t ) − p (t )q (t ) + p (t )q(t ) = 2(9 +
√
2 )t 2 − 6(3 +
√
√
2 )t + 3 2.
In particular, since gcd(uv − u v − pq + p q, u 2 + v 2 + p 2 + q2 ) = 1, there is no cancellation in (18), and we have a generic RRMF quintic. Satisfaction of the divisibility condition in this case is verified by noting that
2
ρ (t ) = α (t )β (t ) − α (t )β(t ) = 10σ (t ),
√ √ √ 2 18 + 4 2 ¯ t )β (t ) = η(t ) = α¯ (t )α (t ) + β( 438t 2 − (444 − 50 2 )t + 171 − 38 2 σ (t ). 73
One can then verify that the RRMF condition (18) is satisfied with
a(t ) = 4t 2 − 6t + 3,
b(t ) = −(3 +
√
2 )t 2 +
√
2t .
These two polynomials define the rational rotation (19) that maps the ERF onto the RMF. Fig. 2 illustrates this curve with its rational RMF.
Fig. 2. The curve of Example 1 shown with its rational RMF. For clarity, only the normal-plane RMF vectors are shown: the tangent vector is omitted.
6. Closure The equivalence of two distinct characterizations for a subset of spatial PH quintic curves with rational rotationminimizing frames (i.e., RRMF curves) has been demonstrated. The first characterization is based on the coefficient constraints (27)–(28) in the quaternion and Hopf map representations. The second is phrased in terms of divisibility of the polynomials (21) and (22) by the parametric speed polynomial (6). A key step in the demonstration of this equivalence
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445
is the canonical-form reduction of the quaternion and Hopf map representations, a transformation that eliminates nonessential degrees of freedom without compromising the RRMF property. The characterization and construction of RRMF curves is an inherently difficult task, due to the non-linear nature of the RRMF condition that must be superposed on the quaternion or Hopf map form of spatial PH curves. In the simplest non-trivial instance — namely, the generic RRMF quintics — the equivalence proof presented herein corroborates the validity of two very different approaches to their characterization, and offers possible clues to the development of a more universal theory of RRMF curves. Although considerable progress has recently been made in elucidating the basic theory of RRMF curves, and in developing algorithms for their practical use in various applications, much remains to be done. On the theoretical side, satisfaction of the RRMF condition (18) or (20) in the non-generic cases, i.e., with non-trivial factors common to the numerators and denominators, has not yet been categorized. Moreover, the available characterizations of RRMF quintics are purely algebraic, and a geometrically more-intuitive insight into them — which would facilitate higher-order generalizations — is desirable. Preliminary studies on the design of rational rotation-minimizing rigid-body motions by interpolating position/orientation data using RRMF curves (Farouki et al., 2010a, 2010b) suggest that the quintics, which have barely sufficient degrees of freedom, do not yield interpolants of optimal shape, thus necessitating consideration of higher-order curves. References Choi, H.I., Han, C.Y., 2002. Euler–Rodrigues frames on spatial Pythagorean-hodograph curves. Comput. Aided Geom. Design 19, 603–620. Choi, H.I., Lee, D.S., Moon, H.P., 2002. Clifford algebra, spin representation, and rational parameterization of curves and surfaces. Adv. Comput. Math. 17, 5–48. Farouki, R.T., 2002. Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves. Graph. Models 64, 382–395. Farouki, R.T., 2008. Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin. Farouki, R.T., 2010. Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves. Adv. Comput. Math. 33, 331–348. Farouki, R.T., Han, C.Y., 2003. Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves. Comput. Aided Geom. Design 20, 435–454. Farouki, R.T., Sakkalis, T., 2010. Rational rotation-minimizing frames on polynomial space curves of arbitrary degree. J. Symbolic Comput. 45, 844–856. Farouki, R.T., Giannelli, C., Manni, C., Sestini, A., 2009. Quintic space curves with rational rotation-minimizing frames. Comput. Aided Geom. Design 26, 580–592. Farouki, R.T., Giannelli, C., Manni, C., Sestini, A., 2010a. Design of rational rotation-minimizing rigid body motions by Hermite interpolation, Math. Comp., doi:10.1090/S0025-5718-2011-02519-6, in press. Farouki, R.T., Giannelli, C., Sestini, A., 2010b. Geometric design using space curves with rational rotation-minimizing frames. In: Daehlen, M., et al. (Eds.), Lecture Notes in Computer Science, vol. 5862. Springer, pp. 194–208. Han, C.Y., 2008. Nonexistence of rational rotation-minimizing frames on cubic curves. Comput. Aided Geom. Design 25, 298–304. Jüttler, B., 1998. Generating rational frames of space curves via Hermite interpolation with Pythagorean hodograph cubic splines. In: Geometric Modelling and Processing ’98. Bookplus Press, pp. 83–106. Jüttler, B., Mäurer, C., 1999a. Cubic Pythagorean hodograph spline curves and applications to sweep surface modelling. Computer-Aided Design 31, 73–83. Jüttler, B., Mäurer, C., 1999b. Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics. J. Geom. Graph. 3, 141–159. Klok, F., 1986. Two moving coordinate frames for sweeping along a 3D trajectory. Comput. Aided Geom. Design 3, 217–229. Kreyszig, E., 1959. Differential Geometry. University of Toronto Press. Sír, Z., Jüttler, B., 2005. Spatial Pythagorean hodograph quintics and the approximation of pipe surfaces. In: Martin, R., Bez, H., Sabin, M. (Eds.), Mathematics of Surfaces XI. Springer, Berlin, pp. 364–380. Wang, W., Joe, B., 1997. Robust computation of the rotation minimizing frame for sweep surface modelling. Computer-Aided Design 29, 379–391. Wang, W., Jüttler, B., Zheng, D., Liu, Y., 2008. Computation of rotation minimizing frames. ACM Trans. Graphics 27 (1), 1–18. Article 2.
Contents lists available at ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves ✩ Rida T. Farouki a,∗ , Takis Sakkalis b a b
Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA Mathematics Laboratory, Agricultural University of Athens, 75 Iera Odos, Athens 11855, Greece
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 November 2010 Received in revised form 27 July 2011 Accepted 28 July 2011 Available online 3 August 2011
A rotation-minimizing frame on a space curve r(t ) is an orthonormal basis (f1 , f2 , f3 ) for R3 , where f1 = r /|r | is the curve tangent, and the normal-plane vectors f2 , f3 exhibit no instantaneous rotation about f1 . Such frames are useful in spatial path planning, swept surface design, computer animation, robotics, and related applications. The simplest curves that have rational rotation-minimizing frames (RRMF curves) comprise a subset of the quintic Pythagorean-hodograph (PH) curves, and two quite different characterizations of them are currently known: (a) through constraints on the PH curve coefficients; and (b) through a certain polynomial divisibility condition. Although (a) is better suited to the formulation of constructive algorithms, (b) has the advantage of remaining valid for curves of any degree. A proof of the equivalence of these two different criteria is presented for PH quintics, together with comments on the generalization to higher-order curves. Although (a) and (b) are both sufficient and necessary criteria for a PH quintic to be an RRMF curve, the (non-obvious) proof presented here helps to clarify the subtle relationships between them. © 2011 Elsevier B.V. All rights reserved.
Keywords: Rotation-minimizing frames Pythagorean-hodograph curves Complex numbers Quaternions Hopf map Polynomial identities
1. Introduction An adapted orthonormal frame (f1 , f2 , f3 ) on a space curve r(t ) incorporates the curve tangent t = r /|r | as the vector f1 , while the vectors f2 , f3 span the curve normal plane. The variation of such a frame is described by its angular velocity ω through the differential equations
f 1 = ω × f1 ,
f 2 = ω × f2 ,
f 3 = ω × f3 ,
and the frame is said to be rotation-minimizing if the angular velocity has no component along f1 = t — i.e., ω · t ≡ 0, so f2 , f3 exhibit no instantaneous rotation about f1 = t. Such frames are of great importance in applications such as motion planning, computer animation, geometric design, and robotics — see Farouki (2002); Farouki et al. (2010a, 2010b); Farouki and Han (2003); Jüttler (1998); Jüttler and Mäurer (1999a, 1999b); Klok (1986); Sír and Jüttler (2005); Wang and Joe (1997); Wang et al. (2008) for further details. Considerable interest has recently emerged in identifying curves that have rational rotation-minimizing frames (or RRMF curves). Such curves must be Pythagorean-hodograph (PH) curves (Farouki, 2008), since only PH curves admit rational unit tangents. The construction of RRMF curves is thus essentially a matter of identifying constraints on PH curves, that are sufficient and necessary for rational RMFs. Progress in this endeavor may be summarized as follows. ✩
*
This paper has been recommended for acceptance by H. Prautzsch Corresponding author. E-mail addresses: [email protected] (R.T. Farouki), [email protected] (T. Sakkalis).
0167-8396/$ – see front matter doi:10.1016/j.cagd.2011.07.004
© 2011
Elsevier B.V. All rights reserved.
R.T. Farouki, T. Sakkalis / Computer Aided Geometric Design 28 (2011) 436–445
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Choi and Han (2002) introduced the Euler–Rodrigues frame (ERF), a rational adapted frame on spatial PH curves, that is a useful reference for identifying rational RMFs. Using the ERF, Han (2008) subsequently formulated a rational function identity as a criterion characterizing RRMF curves of any degree, and showed that RRMF cubics are necessarily degenerate (i.e., planar curves, or with non-primitive hodographs). The existence of non-degenerate RRMF quintics, characterized by certain constraints on the coefficients of the Hopf map representation of spatial PH curves, was first established in Farouki et al. (2009). These conditions were soon superseded (Farouki, 2010) by much simpler, symmetric coefficient constraints for both the quaternion and Hopf map PH curve representations. Finally, Han’s criterion (Han, 2008) was shown in Farouki and Sakkalis (2010) to be equivalent to a certain polynomial divisibility condition. The methods employed to derive the RRMF quintic coefficient constraints (Farouki, 2010) and the polynomial divisibility condition (Farouki and Sakkalis, 2010) are very different in nature. Although both are sufficient and necessary conditions for a spatial PH quintic to be an RRMF curve, there is no obvious connection between them. Hence, the goal of this paper is to give a formal proof of their equivalence, that helps in understanding the non-trivial relationships between these two conditions. This task amounts to showing that the satisfaction of certain constraints on the Bernstein coefficients of two complex quadratic polynomials α (t ), β(t ) is equivalent to proportionality of |α (t )β (t ) − α (t )β(t )|2 and |α (t )|2 + |β(t )|2 . The transformation of α (t ), β(t ) to “canonical form” is a key step in making the problem tractable. The coefficient constraints offer a constructive basis for designing RRMF curves, but are currently known only for quintics. Conversely, the divisibility condition is existential in nature but applies to RRMF curves of any degree. Although the equivalence of these distinct characterizations follows implicitly from their independent derivations as sufficient-and-necessary conditions for RRMF quintics, it is non-obvious and its formal demonstration is non-trivial. The proof presented herein offers new insight into the complicated non-linear structure of RRMF curves, that may prove useful in extending the coefficient constraints to higher-order RRMF curves, so as to obtain additional degrees of freedom for the geometric design of spatial motions. The organization of this paper is as follows. Following a brief review of the quaternion and Hopf map models for spatial PH curves in Section 2, the Euler–Rodrigues frame is introduced in Section 3, and its angular velocity properties are analyzed. Section 4 then describes conditions that characterize rational rotation-minimizing frames, in terms of both coefficient constraints and the divisibility criterion. By invoking a reduction of these conditions to a simplified canonical form, they are shown in Section 5 to be equivalent for the case of quintics. Finally, Section 6 summarizes the results of this paper, and identifies some topics that deserve further investigation. 2. Quaternion and Hopf map forms A Pythagorean-hodograph (PH) curve r(t ) = (x(t ), y (t ), z(t )) is a polynomial curve whose derivative r (t ) = (x (t ), y (t ),
z (t )) satisfies
x 2 (t ) + y 2 (t ) + z 2 (t ) = σ 2 (t )
(1)
for some polynomial σ (t ). The quaternion and Hopf map forms (Choi et al., 2002) are two alternative (equivalent) models for the construction of PH curves. The former generates a Pythagorean hodograph r (t ) from a quaternion polynomial1
A(t ) = u (t ) + v (t )i + p (t )j + q(t )k,
(2)
and its conjugate A∗ (t ) = u (t ) − v (t )i − p (t )j − q(t )k through the product
r (t ) = A(t ) i A∗ (t ) = u 2 (t ) + v 2 (t ) − p 2 (t ) − q2 (t ) i
+ 2 u (t )q(t ) + v (t ) p (t ) j + 2 v (t )q(t ) − u (t ) p (t ) k.
(3)
The latter generates a Pythagorean hodograph from complex polynomials
α (t ) = u (t ) + iv (t ),
β(t ) = q(t ) + ip (t )
(4)
through the expression
2 2 r (t ) = α (t ) − β(t ) , 2 Re
α (t )β(t ) , 2 Im α (t )β(t ) .
(5)
The equivalence of (3) and (5) can be verified by setting A(t ) = α (t ) + kβ(t ), the imaginary unit i being identified with the quaternion element i. We shall find it convenient to transform back and forth between the representations (3) and (5) — see Farouki (2008) for a more thorough treatment of them. The parametric speed σ (t ) = |r (t )| of the curve r(t ) defined by integrating r (t ), i.e., the derivative of its arc length s with respect to the parameter t, is the polynomial
1 Calligraphic characters such as A denote quaternions, their scalar and vector parts being indicated by scal(A) and vect(A). Bold symbols denote complex numbers or vectors in R3 — the meaning should be clear from the context.
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2
2
2
σ (t ) = A(t ) = α (t ) + β(t ) = u 2 (t ) + v 2 (t ) + p 2 (t ) + q2 (t ).
(6)
The derivative of the parametric speed is
σ (t ) =
r (t ) · r (t )
σ (t )
= 2 u (t )u (t ) + v (t ) v (t ) + p (t ) p (t ) + q(t )q (t ) .
(7)
We consider here only regular curves, satisfying σ (t ) = 0 for all real t. Since all planar PH curves are trivially RRMF curves, it is useful to have a means to identify when (3) and (5) define a planar PH curve, rather than a true space curve. This occurs when x (t ), y (t ), z (t ) are linearly dependent. We now give a simplified version of the criteria stated in Farouki and Sakkalis (2010), that identify degeneration to a plane curve. This is based on the following result, which makes use of the easily-verified fact that two real polynomials a(t ), b(t ) are linearly dependent if and only if they satisfy ab = a b. Proposition 1. Three real polynomials a(t ), b(t ), c (t ) are linearly dependent if and only if they satisfy
ac − a c bc − b c = ac − a c bc − b c .
(8)
Proof. Suppose first that a(t ), b(t ), c (t ) are linearly dependent — i.e., there exist real numbers λ, μ, ν , not all zero, such that λa(t ) + μb(t ) + ν c (t ) ≡ 0. If λ = 0 we have a(t ) = mb(t ) + nc (t ) with m = −μ/λ and n = −ν /λ. Thus, a(t ) − mb(t ) = nc (t ), so (a − mb)c = (a − mb )c and ac − a c = m(bc − b c ). Since (ac − a c ) = ac − a c and (bc − b c ) = bc − b c, the latter implies that (ac − a c )(bc − b c ) = (ac − a c )(bc − b c ), which is condition (8). If λ = 0, on the other hand, b(t ) and c (t ) must be multiples of each other, so that bc − b c = bc − b c = 0 and (8) still holds. Conversely, suppose (8) holds. Noting that (ac − a c ) = ac − a c and (bc − b c ) = bc − b c again, we infer that ac − a c and bc − b c are linearly dependent, and this implies that a, b, c are linearly dependent as well. 2 3. Euler–Rodrigues frame Along the PH curve defined by (2)–(3), a rational adapted orthonormal frame — the Euler–Rodrigues frame (ERF) — may be defined (Choi and Han, 2002) by
e1 (t ), e2 (t ), e3 (t ) =
(A(t ) i A∗ (t ), A(t ) j A∗ (t ), A(t ) k A∗ (t )) . |A(t )|2
(9)
Here e1 (t ) is the curve tangent while e2 (t ), e3 (t ) span the normal plane. The vectors (9) can be expressed in terms of the components of (2) as
e1 = e2 = e3 =
(u 2 + v 2 − p 2 − q2 )i + 2(uq + vp )j + 2( vq − up )k , u 2 + v 2 + p 2 + q2 2( vp − uq)i + (u 2 − v 2 + p 2 − q2 )j + 2(uv + pq)k u 2 + v 2 + p 2 + q2 2(up + vq)i + 2( pq − uv )j + (u 2 − v 2 − p 2 + q2 )k u 2 + v 2 + p 2 + q2
The variation of the ERF is characterized by its angular velocity
e1 = ω × e1 , and if
e2 = ω × e2 ,
e3 = ω × e3 ,
, .
(10)
ω through (11)
ω is written in terms of the basis (e1 , e2 , e3 ) as
ω = ω1 e1 + ω2 e2 + ω3 e3 ,
(12)
its components are given by
ω1 = e3 · e2 = −e2 · e3 = ω2 = e1 · e3 = −e3 · e1 = ω3 = e2 · e1 = −e1 · e2 =
2(uv − u v − pq + p q) u 2 + v 2 + p 2 + q2
2(up − u p + vq − v q) u 2 + v 2 + p 2 + q2
2(uq − u q − vp + v p ) u 2 + v 2 + p 2 + q2
The relations (11) can thus be cast in matrix form as
, , .
(13)
R.T. Farouki, T. Sakkalis / Computer Aided Geometric Design 28 (2011) 436–445
⎡
⎤
⎡ e1 0 ⎢ ⎥ ⎣ ⎣ e2 ⎦ = −ω3 e3
ω2
ω3
−ω2
0
ω1
−ω1
0
⎤⎡
439
⎤
e1 ⎦ ⎣ e2 ⎦ . e3
(14)
Now the combinations of u , v , p , q and u , v , p , q in (7) and (13) are generated by the product
A∗ A = uu + v v + pp + qq + uv − u v − pq + p q i + up − u p + vq − v q j + uq − u q − vp + v p k,
(15)
namely,
σ = 2 uu + v v + pp + qq ,
σ ω1 = 2 uv − u v − pq + p q ,
σ ω2 = 2 up − u p + vq − v q ,
σ ω3 = 2 uq − u q − vp + v p .
On the other hand, we also have
A A∗ = uu + v v + pp + qq + uv − u v + pq − p q i + up − u p − vq + v q j + uq − u q + vp − v p k,
(16)
and from (10) and (12)–(13), one can verify that
σ ω = 2 uv − u v + pq − p q i + 2 up − u p − vq + v q j + 2 uq − u q + vp − v p k. Hence, we observe that
σ = 2 scal A∗ A = 2 scal A A∗ ,
σ ω = 2 vect A A∗ ,
and consequently
2
σ 2 + σ 2 |ω|2 = 4|A|2 A .
(17)
4. Rational rotation-minimizing frames Fig. 1 compares the variation of the normal-plane vectors for the Frenet frame (Kreyszig, 1959) and the rotationminimizing frame, along a quintic RRMF curve. The RMF evidently offers a more “natural” means of orienting a rigid body along this path, when one principal axis of the body is constrained to remain aligned with the path tangent.
Fig. 1. The Frenet frame (Kreyszig, 1959) normal-plane vectors along a quintic RRMF curve (left), and the RMF normal-plane vectors along the same path (right). The latter frame vectors exhibit no instantaneous rotation about the curve tangent, and have a rational dependence on the curve parameter.
Since satisfaction of (1) is necessary for a rational unit tangent, all RRMF curves must be PH curves. Han (2008) determined a criterion for PH curves to possess rational RMFs using the quaternion representation (3) — namely, relatively prime polynomials a(t ), b(t ) must exist, such that the components u (t ), v (t ), p (t ), q(t ) of A(t ) satisfy
uv − u v − pq + p q u 2 + v 2 + p 2 + q2
=
ab − a b a2 + b 2
(18)
.
This is equivalent to the existence of a rational normal-plane rotation
1 f2 (t ) = 2 f3 (t ) a (t ) + b2 (t )
a2 (t ) − b2 (t ) −2a(t )b(t ) 2a(t )b(t ) a2 (t ) − b2 (t )
e2 (t ) e3 (t )
(19)
that maps the ERF vectors e2 (t ), e3 (t ) onto the RMF vectors f2 (t ), f3 (t ) and exactly cancels the ω1 component of the ERF angular velocity. In terms of the Hopf map representation, condition (18) can be interpreted as requiring the existence of a complex polynomial w(t ) = a(t ) + ib(t ), with gcd(a(t ), b(t )) = constant, such that
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¯ ) ¯ + ββ Im(αα |α
|2
+ |β|2
=
¯ ) Im(ww | w |2
(20)
.
The existence of rational RMFs on spatial PH quintics has been studied thus far (Farouki, 2010; Farouki et al., 2009) under the assumption that conditions (18)–(20) are satisfied in the case where u 2 + v 2 + p 2 + q2 = a2 + b2 or |α |2 + |β|2 = |w|2 . It was shown in Farouki and Sakkalis (2010) that a necessary condition for this is the requirement that the polynomials
2
2
ρ = up − u p + vq − v q + uq − u q − vp + v p
2
(21)
and
η = uu + v v + pp + qq
2 + uv − u v − pq + p q
(22)
should both be divisible by the polynomial σ defined by (6). This condition becomes sufficient in the case where gcd(uv − u v − pq + p q, u 2 + v 2 + p 2 + q2 ) = 1. We henceforth refer to this as the generic case. Note that (21) and (22) can be more compactly expressed in terms of (4) as
2
2
¯ t )β (t ) ρ (t ) = α (t )β (t ) − α (t )β(t ) , η(t ) = α¯ (t )α (t ) + β(
(23)
and they satisfy the relation
ρ (t ) + η(t ) = σ (t )ζ (t ), |α (t )|2
where we define ζ (t ) = the product (15) becomes
(24)
+ |β (t )| . In the quaternion model, setting A(t ) = α (t ) + kβ(t ) 2
and A∗ (t )
= α¯ (t ) − kβ(t ),
A∗ (t )A (t ) = α¯ (t ) − kβ(t ) α (t ) + kβ (t ) = α¯ (t )α (t ) + α¯ (t )kβ (t ) − kβ(t )α (t ) − kβ(t )kβ (t ), ¯ t ) we have ¯ (t )k = kα (t ), β(t )k = kβ( and by noting that α
¯ t )β (t ) + k α (t )β (t ) − α (t )β(t ) . A∗ (t )A (t ) = α¯ (t )α (t ) + β(
Hence, the relation (24) is seen to be equivalent to (17). Remark 1. According to item 2 of Remark 5.1 in Farouki and Sakkalis (2010), a necessary condition for the satisfaction of (18) is that deg(u 2 + v 2 + p 2 + q2 ) deg(a2 + b2 ). As already noted above, rational RMFs on spatial PH curves have thus far been investigated (Farouki, 2010; Farouki et al., 2009; Farouki and Sakkalis, 2010) in the case where deg(u 2 + v 2 + p 2 + q2 ) = deg(a2 + b2 ), which naturally includes the generic case. For quadratic polynomials u , v , p , q defining an RRMF quintic, deg(u 2 + v 2 + p 2 + q2 ) = 4, so (18) can be satisfied in the case where deg(a2 + b2 ) = 4, 2, or 0. In this paper we shall follow the practice of prior studies in focusing on the satisfaction of (18) in the generic case. Preliminary investigations suggest that a comprehensive analysis of the non-generic cases is a non-trivial task, and for this reason we defer it to a future study. 5. Conditions for RRMF quintics The Bernstein basis for polynomials of degree m on t ∈ [0, 1] is defined by
bm k (t ) =
m k
(1 − t )m−k t k ,
k = 0, . . . , m .
We focus here on PH quintics, specified in the quaternion form (3) by quadratic quaternion polynomials
A(t ) = A0 b20 (t ) + A1 b21 (t ) + A2 b22 (t ),
(25)
or, equivalently, in the Hopf map form (5) by quadratic complex polynomials
α (t ) = α 0 b20 (t ) + α 1 b21 (t ) + α 2 b22 (t ), β(t ) = β 0 b20 (t ) + β 1 b21 (t ) + β 2 b22 (t ).
(26)
It was shown in Farouki (2010) that a spatial PH quintic has a rational RMF if and only if the coefficients of (25) satisfy
vect A2 iA∗0 = A1 iA∗1 ,
(27)
or, equivalently, the coefficients of (26) satisfy
¯ 2 − β 0 β¯ 2 ) = |α 1 |2 − |β 1 |2 , Re(α 0 α
α 0 β¯ 2 + α 2 β¯ 0 = 2α 1 β¯ 1 .
(28)
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Remark 2. We emphasize that the coefficient constraints (27)–(28) identify only quintic RRMF curves, but the characterization in terms of divisibility of the polynomials (21)–(22) by (6) holds for RRMF curves of any degree. We focus henceforth on the Hopf map form (5). Our aim is to show that, for RRMF quintics, satisfaction of the constraints (28) is equivalent to the divisibility of (23) by (6). Our main result (whose proof will be deferred until certain preliminary results have been established) may be stated as follows.
¯ ), |α |2 + |β|2 ) = 1. ¯ + ββ Proposition 2. Assume that the quadratic complex polynomials (26) satisfy the condition2 gcd(Im(αα Then |α β − α β|2 is divisible by |α |2 + |β|2 if and only if their coefficients satisfy (28). To facilitate the proof of Proposition 2, we introduce a “canonical form” to simplify the analysis. For the quadratic polynomials (26) used in the Hopf map form (5), we have
α (t )β (t ) − α (t )β(t ) = 2(α 0 β 1 − α 1 β 0 )b20 (t ) + (α 0 β 2 − α 2 β 0 )b21 (t ) + 2(α 1 β 2 − α 2 β 1 )b22 (t ). Then |α (t )β (t ) − α (t )β(t )|2 and |α (t )|2 + |β(t )|2 are both real quartics, and they must be proportional if the former is to be divisible by the latter, i.e.,
α (t )β (t ) − α (t )β(t )2 = k α (t )2 + β(t )2
(29)
for some positive real k. Now given a quadratic complex polynomial c(t ) = c0 b20 (t ) + c1 b21 (t ) + c2 b22 (t ), the real quartic |c(t )|2 has Bernstein coefficients
|c0 |2 ,
2|c1 |2 + Re(c2 c0 )
Re(c0 c1 ),
3
,
Re(c1 c2 ),
|c2 |2 .
Hence, the condition (29) is equivalent to the coefficient constraints
4|α 0 β 1 − α 1 β 0 |2 = k |α 0 |2 + |β 0 |2 ,
¯ 0 β¯ 2 − α¯ 2 β¯ 0 ) = k Re(α 0 α¯ 1 + β 0 β¯ 1 ), 2 Re (α 0 β 1 − α 1 β 0 )(α 2 ¯ 0 β¯ 1 − α¯ 1 β¯ 0 ) = k 2 |α 1 |2 + |β 1 |2 + Re(α 2 α¯ 0 + β 2 β¯ 0 ) , 2α 0 β 2 − α 2 β 0 + 4 Re (α 1 β 2 − α 2 β 1 )(α ¯ 1 β¯ 2 − α¯ 2 β¯ 1 ) = k Re(α 1 α¯ 2 + β 1 β¯ 2 ), 2 Re (α 0 β 2 − α 2 β 0 )(α 4|α 1 β 2 − α 2 β 1 |2 = k |α 2 |2 + |β 2 |2 ,
(30) (31) (32) (33) (34)
for some positive real k. Now the identification of quintic RRMF curves as a subset of all spatial PH quintics is simplified (Farouki, 2010) by reducing the polynomials (26) to canonical form through a scaling/rotation transformation that does not influence the RRMF property. The canonical values (α 0 , β 0 ) = (1, 0) were used in Farouki (2010), but these values can be assigned to any chosen coefficients (αk , β k ). For complex numbers (μ, ν ) = (0, 0) consider the transformation
μ −ν¯ → β(t ) ν μ¯
α (t )
α (t ) β(t )
,
α (t ), β(t ) with inverse defined by
1 α (t ) μ¯ ν¯ α (t ) → . β(t ) β(t ) |μ|2 + |ν |2 −ν μ
(35)
of the polynomials
(36)
Under this transformation,
α (t )2 + β(t )2 → |μ|2 + |ν |2 α (t )2 + β(t )2 , α (t )β (t ) − α (t )β(t ) → |μ|2 + |ν |2 α (t )β (t ) − α (t )β(t ) , ¯ t )β (t ) → |μ|2 + |ν |2 α¯ (t )α (t ) + β( ¯ t )β (t ) , α¯ (t )α (t ) + β(
(37)
and the divisibility of (23) by (6) is unaltered. The map (35) is implemented by applying it to the Bernstein coefficients of α (t ) and β(t ),
αk βk
2
→
μ −ν¯ ν μ¯
αk βk
,
k = 0, 1, 2.
This assumption is equivalent to restricting our attention to the generic case of RRMF quintics (see Remark 1).
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Specifically, for any prescribed index k, the choices
α¯ k |αk |2 + |β k |2
μ=
−β k |αk |2 + |β k |2
ν=
and
(38)
in (35) yield (αk , β k ) → (1, 0). We choose the canonical case (α 1 , β 1 ) = (1, 0) here, since this gives more symmetrical arguments. For this canonical case, the conditions (30)–(34) that ensure the divisibility of |α (t )β (t )− α (t )β(t )|2 by |α (t )|2 + |β(t )|2 reduce to
4|β 0 |2 = k |α 0 |2 + |β 0 |2 ,
(39)
2 Re (α 2 β 0 − α 0 β 2 )β¯ 0 = k Re(α 0 ),
¯ 0 + β 2 β¯ 0 ) , 2|α 0 β 2 − α 2 β 0 | − 4 Re(β 2 β¯ 0 ) = k 2 + Re(α 2 α 2 Re (α 0 β 2 − α 2 β 0 )β¯ 2 = k Re(α 2 ), 4|β 2 |2 = k |α 2 |2 + |β 2 |2 , 2
(40) (41) (42) (43)
while the quintic RRMF coefficients constraints (28) become
¯ 2 − β 0 β¯ 2 ) = 1, Re(α 0 α
α 0 β¯ 2 + α 2 β¯ 0 = 0.
(44)
Lemma 1. The canonical-form RRMF quintic conditions (44) are satisfied if and only if α 2 , β 2 can be written in terms of α 0 , β 0 as
α 2 = γ α 0 and β 2 = −γ¯ β 0 ,
(45)
where γ is a complex value with real part
Re(γ ) =
1
|2
|α 0 + |β 0 |2
(46)
.
Proof. By substituting into conditions (44), one can easily verify that they are satisfied when α 2 , β 2 are specified by (45) and (46). Conversely, suppose that conditions (44) are satisfied. Then for some complex number γ we have
α 2 = γ α 0 and β¯ 2 = −γ β¯ 0 . Writing
γ = ζ + iη, substituting for α 2 , β 2 into the first of Eqs. (44), and equating real and imaginary parts, then yields
ζ= Thus
1
|α 0 |2 + |β 0 |2
.
α 2 , β 2 must be of the form (45)–(46) if conditions (44) hold. 2
Having established these preliminary results, we are now ready to present the proof of Proposition 2. Proof of Proposition 2. We need consider only the canonical case (α 1 , β 1 ) = (1, 0) for which conditions (28) reduce to (44), and the divisibility condition amounts to satisfaction of (39)–(43) for some positive real k. Suppose conditions (44) are satisfied. Then Lemma 1 indicates that α 2 , β 2 must be of the form (45)–(46). Consequently, conditions (39) and (43) are satisfied with
k=
4|β 0 |2
|α 0 |2 + |β 0 |2
=
4|β 2 |2
|α 2 |2 + |β 2 |2
.
Using this k, and noting from (45)–(46) that we have
α2β 0 − α0β 2 =
2α 0 β 0
|α 0 |2 + |β 0 |2
,
the left-hand sides of (40) and (42) reduce to k Re(α 0 ) and k Re(α 2 ) — in agreement with the right-hand sides. Finally, the left- and right-hand sides of (41) both yield the value
k 1+
2 | α 0 |2
|α 0 |2 + |β 0 |2
,
and thus agree. Hence, satisfaction of (44) implies satisfaction of (39)–(43). Conversely, suppose conditions (39)–(43) are satisfied. We assume β 0 , β 2 = 0 since β 0 = 0 ⇒ α 0 = 0 and β 2 = 0 ⇒ α 2 = 0 from (39) and (43), and it is then impossible to satisfy (41) with k > 0. We consider the following cases:
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Case 1: α 0 = 0 or α 2 = 0. Suppose that α 0 = 0. Then we must have β 0 = 0, since otherwise (41) yields 0 = 2k, contradicting the assumption that k > 0. Thus, since (39) and (43) imply that
|α 0 |2 ||β 2 |2 = |α 2 |2 |β 0 |2 ,
(47)
we obtain α 2 = 0. By similar arguments, α 2 = 0 ⇒ α 0 = 0. Hence, under conditions (39)–(43), (39)–(43) with α 0 = α 2 = 0 yield k = 4 and Re(β 2 β¯ 0 ) = −1, and (44) is then obviously satisfied. Case 2:
α 0 = 0 ⇔ α 2 = 0. Now
α 0 , α 2 = 0. In this case (39) and (43) again imply (47), and hence α 2 , β 2 must be of the form
α 2 = δ eiφ α 0 and β 2 = δ eiθ β 0
(48)
α 2 , β 2 into (40) and (42) and simplifying, these equations imply that Re(α 0 ) = Re eiφ α 0 − Re eiθ α 0 Re eiφ α 0 .
for real values δ , φ , θ with δ > 0. Substituting for
Re(α 0 ) − Re ei(φ−θ ) α 0
α 0 = |α 0 |eiψ then yields cos ψ − cos(φ − θ + ψ) cos ψ = cos(φ + ψ) − cos(θ + ψ) cos(φ + ψ).
Setting
Expanding the terms involving θ and using standard trigonometric relations, we finally have
sin θ =
cos2 ψ − cos2 (φ + ψ) sin(φ + 2ψ)
= sin φ.
This has solutions θ = (2m + 1)π − φ and θ = 2mπ + φ for integer m — i.e., eiθ = −e−iφ and eiθ = eiφ . We consider each case in turn. Case 2a: eiθ = −e−iφ . Setting
δ cos φ =
1
|α 0 |2 + |β 0 |2
Then, substituting for
α 2 = δ eiφ α 0 , β 2 = −δ e−iφ β 0 in (39)–(40), we obtain .
α 2 and β 2 into conditions (44), one can easily verify that this relation implies their satisfaction.
Case 2b: eiθ = eiφ . Taking α 2 = δ eiφ α 0 , β 2 = δ eiφ β 0 gives α 0 β 2 − α 2 β 2 = 0. By (40) and (42), this implies that Re(α 0 ) = Re(α 2 ) = 0. Writing α 0 = iα (where α = 0) we have α 2 = δ eiφ iα = δ α (− sin φ + i cos φ). Consequently, Re(α 2 ) = −δ α sin φ = 0 implies that sin φ = 0, since δ > 0 and α = 0. We thus have eiφ = ±1. Setting (α 2 , β 2 ) = ±(δ α 0 , δβ 0 ) in (39) and (41) gives
δ=
∓1 . |α 0 |2 + |β 0 |2
The negative value, corresponding to eiφ = +1, contradicts δ > 0, so this choice is invalid. For the positive value, corresponding to eiφ = −1, we have α 2 = −δ α 0 and β 2 = −δβ 0 and (recalling that α 1 = 1, β 1 = 0) hence
α (t ) = α 0 b20 (t ) + 1 · b21 (t ) −
α0
b2 (t ), |α 0 |2 + |β 0 |2 2 β0 β(t ) = β 0 b20 (t ) + 0 · b21 (t ) − b2 (t ). 2 |α 0 | + |β 0 |2 2
Using Re(α 0 ) = 0, one can then verify that
¯ = −α ¯ + ββ Im αα |α |2 + |β|2 =
(|α 0 |2 + |β 0 |2 )b20 (t ) + 0 · b21 (t ) + 1 · b22 (t ) , |α 0 |2 + |β 0 |2
[(|α 0 |2 + |β 0 |2 )b20 (t ) + 0 · b21 (t ) + 1 · b22 (t )]2 , |α 0 |2 + |β 0 |2
¯ ), |α |2 + |β|2 ) = 1, so this choice is also invalid. ¯ + ββ which contradicts the assumption that gcd(Im(αα
2
We conclude with an example to illustrate the equivalence of the RRMF conditions based on coefficient constraints and the divisibility criterion.
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Example 1. The curve defined through (5) by the coefficients
√
α 0 = −1 − 2i,
α 2 = −2 + i,
2
1
β 0 = 2i,
1
α 1 = 2 − √ i,
β1 = √ ,
β 2 = 2 − i,
2
for the polynomials (26) satisfies the coefficient constraints (28), since
¯ 2 − β 0 β¯ 2 ) = |α 1 |2 − |β 1 |2 = 2, Re(α 0 α
α 0 β¯ 2 + α 2 β¯ 0 = 2α 1 β¯ 1 = 2 − i.
In this case, we have
u (t ) = Re
v (t ) = Im
√
√
α (t ) = −(3 + 2 2 )t 2 + (2 + 2 2 )t − 1,
√
√
α (t ) = ( 2 − 1)t 2 + (4 − 2 )t − 2,
p (t ) = Im β(t ) = t 2 − 4t + 2, q(t ) = Re β(t ) = (2 −
√
2 )t 2 +
√
2t ,
and hence
2
√
2
√
σ (t ) = α (t ) + β(t ) = u 2 (t ) + v 2 (t ) + p 2 (t ) + q2 (t ) = (27 + 6 2 )t 4 − (52 + 6 2 )t 3 + 62t 2 − 36t + 9, u (t ) v (t ) − u (t ) v (t ) − p (t )q (t ) + p (t )q(t ) = 2(9 +
√
2 )t 2 − 6(3 +
√
√
2 )t + 3 2.
In particular, since gcd(uv − u v − pq + p q, u 2 + v 2 + p 2 + q2 ) = 1, there is no cancellation in (18), and we have a generic RRMF quintic. Satisfaction of the divisibility condition in this case is verified by noting that
2
ρ (t ) = α (t )β (t ) − α (t )β(t ) = 10σ (t ),
√ √ √ 2 18 + 4 2 ¯ t )β (t ) = η(t ) = α¯ (t )α (t ) + β( 438t 2 − (444 − 50 2 )t + 171 − 38 2 σ (t ). 73
One can then verify that the RRMF condition (18) is satisfied with
a(t ) = 4t 2 − 6t + 3,
b(t ) = −(3 +
√
2 )t 2 +
√
2t .
These two polynomials define the rational rotation (19) that maps the ERF onto the RMF. Fig. 2 illustrates this curve with its rational RMF.
Fig. 2. The curve of Example 1 shown with its rational RMF. For clarity, only the normal-plane RMF vectors are shown: the tangent vector is omitted.
6. Closure The equivalence of two distinct characterizations for a subset of spatial PH quintic curves with rational rotationminimizing frames (i.e., RRMF curves) has been demonstrated. The first characterization is based on the coefficient constraints (27)–(28) in the quaternion and Hopf map representations. The second is phrased in terms of divisibility of the polynomials (21) and (22) by the parametric speed polynomial (6). A key step in the demonstration of this equivalence
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is the canonical-form reduction of the quaternion and Hopf map representations, a transformation that eliminates nonessential degrees of freedom without compromising the RRMF property. The characterization and construction of RRMF curves is an inherently difficult task, due to the non-linear nature of the RRMF condition that must be superposed on the quaternion or Hopf map form of spatial PH curves. In the simplest non-trivial instance — namely, the generic RRMF quintics — the equivalence proof presented herein corroborates the validity of two very different approaches to their characterization, and offers possible clues to the development of a more universal theory of RRMF curves. Although considerable progress has recently been made in elucidating the basic theory of RRMF curves, and in developing algorithms for their practical use in various applications, much remains to be done. On the theoretical side, satisfaction of the RRMF condition (18) or (20) in the non-generic cases, i.e., with non-trivial factors common to the numerators and denominators, has not yet been categorized. Moreover, the available characterizations of RRMF quintics are purely algebraic, and a geometrically more-intuitive insight into them — which would facilitate higher-order generalizations — is desirable. Preliminary studies on the design of rational rotation-minimizing rigid-body motions by interpolating position/orientation data using RRMF curves (Farouki et al., 2010a, 2010b) suggest that the quintics, which have barely sufficient degrees of freedom, do not yield interpolants of optimal shape, thus necessitating consideration of higher-order curves. References Choi, H.I., Han, C.Y., 2002. Euler–Rodrigues frames on spatial Pythagorean-hodograph curves. Comput. Aided Geom. Design 19, 603–620. Choi, H.I., Lee, D.S., Moon, H.P., 2002. Clifford algebra, spin representation, and rational parameterization of curves and surfaces. Adv. Comput. Math. 17, 5–48. Farouki, R.T., 2002. Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves. Graph. Models 64, 382–395. Farouki, R.T., 2008. Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin. Farouki, R.T., 2010. Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves. Adv. Comput. Math. 33, 331–348. Farouki, R.T., Han, C.Y., 2003. Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves. Comput. Aided Geom. Design 20, 435–454. Farouki, R.T., Sakkalis, T., 2010. Rational rotation-minimizing frames on polynomial space curves of arbitrary degree. J. Symbolic Comput. 45, 844–856. Farouki, R.T., Giannelli, C., Manni, C., Sestini, A., 2009. Quintic space curves with rational rotation-minimizing frames. Comput. Aided Geom. Design 26, 580–592. Farouki, R.T., Giannelli, C., Manni, C., Sestini, A., 2010a. Design of rational rotation-minimizing rigid body motions by Hermite interpolation, Math. Comp., doi:10.1090/S0025-5718-2011-02519-6, in press. Farouki, R.T., Giannelli, C., Sestini, A., 2010b. Geometric design using space curves with rational rotation-minimizing frames. In: Daehlen, M., et al. (Eds.), Lecture Notes in Computer Science, vol. 5862. Springer, pp. 194–208. Han, C.Y., 2008. Nonexistence of rational rotation-minimizing frames on cubic curves. Comput. Aided Geom. Design 25, 298–304. Jüttler, B., 1998. Generating rational frames of space curves via Hermite interpolation with Pythagorean hodograph cubic splines. In: Geometric Modelling and Processing ’98. Bookplus Press, pp. 83–106. Jüttler, B., Mäurer, C., 1999a. Cubic Pythagorean hodograph spline curves and applications to sweep surface modelling. Computer-Aided Design 31, 73–83. Jüttler, B., Mäurer, C., 1999b. Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics. J. Geom. Graph. 3, 141–159. Klok, F., 1986. Two moving coordinate frames for sweeping along a 3D trajectory. Comput. Aided Geom. Design 3, 217–229. Kreyszig, E., 1959. Differential Geometry. University of Toronto Press. Sír, Z., Jüttler, B., 2005. Spatial Pythagorean hodograph quintics and the approximation of pipe surfaces. In: Martin, R., Bez, H., Sabin, M. (Eds.), Mathematics of Surfaces XI. Springer, Berlin, pp. 364–380. Wang, W., Joe, B., 1997. Robust computation of the rotation minimizing frame for sweep surface modelling. Computer-Aided Design 29, 379–391. Wang, W., Jüttler, B., Zheng, D., Liu, Y., 2008. Computation of rotation minimizing frames. ACM Trans. Graphics 27 (1), 1–18. Article 2.