Let 𝒜 be the algebra of quaternions ℍ or octonions 𝕆. In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial
) ∈ 𝒜
has a ...root in 𝒜. As a consequence, the Jacobian determinant |
)| is always nonnegative in 𝒜. Moreover, using the idea of the topological degree we show that a regular polynomial
) over 𝒜 has also a root in 𝒜. Finally, utilizing multiplication (*) in 𝒜, we prove various results on the topological degree of products of maps. In particular, if
is the unit sphere in 𝒜 and
,
:
are smooth maps, it is shown that deg(
*
) = deg(
) + deg(
).
•Methods to construct rational curves with rational arc lengths by direct integration are developed.•A comprehensive analysis of the case of rational curves with simple points at infinity is ...presented.•The method directly generates low-degree curves in cases where alternative methods require symbolic factorizations.•The method is illustrated with a selection of representative computed examples.
A methodology for the construction of rational curves with rational arc length functions, by direct integration of hodographs, is developed. For a hodograph of the form r′(ξ)=(u2(ξ)−v2(ξ),2u(ξ)v(ξ))/w2(ξ), where w(ξ) is a monic polynomial defined by prescribed simple roots, we identify conditions on the polynomials u(ξ) and v(ξ) which ensure that integration of r′(ξ) produces a rational curve with a rational arc length function s(ξ). The method is illustrated by computed examples, and a generalization to spatial rational curves is also briefly discussed. The results are also compared to existing theory, based upon the dual form of rational Pythagorean-hodograph curves, and it is shown that direct integration produces simple low-degree curves which otherwise require a symbolic factorization to identify and cancel common factors among the curve homogeneous coordinates.
A quaternion polynomial f(t) in the single variable t, is one whose coefficients are in the skew field H of quaternions. In this manuscript an elementary proof is given of the fact that such an f has ...a root in H. Moreover, an algorithm is proposed for finding all roots ζ of f(t), along with their multiplicities. The algorithm is based on computing the real part of ζ first, and then using the multiplication rule in H, the imaginary part of ζ is computed via a linear quaternion equation. Several numerical examples are also presented to illustrate the performance of the method.
This paper aims to review methods for computing orthogonal projection of points onto curves and surfaces, which are given in implicit or parametric form or as point clouds. Special emphasis is place ...on orthogonal projection onto conics along with reviews on orthogonal projection of points onto curves and surfaces in implicit and parametric form. Except for conics, computation methods are classified into two groups based on the core approaches: iterative and subdivision based. An extension of orthogonal projection of points to orthogonal projection of curves onto surfaces is briefly explored. Next, the discussion continues toward orthogonal projection of points onto point clouds, which spawns a different branch of algorithms in the context of orthogonal projection. The paper concludes with comments on guidance for an appropriate choice of methods for various applications.
An adapted orthonormal frame
(
f
1
,
f
2
,
f
3
)
on a space curve
r
(
t
)
, where
f
1
=
r
′
/
|
r
′
|
is the curve tangent, is
rotation-minimizing if its angular velocity satisfies
ω
⋅
f
1
≡
0
, ...i.e., the normal-plane vectors
f
2
,
f
3
exhibit no instantaneous rotation about
f
1
. The simplest space curves with
rational rotation-minimizing frames (RRMF curves) form a subset of the quintic spatial
Pythagorean-hodograph (PH)
curves, identified by certain non-linear constraints on the curve coefficients. Such curves are useful in motion planning, swept surface constructions, computer animation, robotics, and related fields. The condition that identifies the RRMF quintics as a subset of the spatial PH quintics requires a rational expression in four quadratic polynomials
u
(
t
)
,
v
(
t
)
,
p
(
t
)
,
q
(
t
)
and their derivatives to be reducible to an analogous expression in just two polynomials
a
(
t
)
,
b
(
t
)
. This condition has been analyzed, thus far, in the case where
a
(
t
)
,
b
(
t
)
are also quadratic, the corresponding solutions being called
Class I RRMF quintics. The present study extends these prior results to provide a complete categorization of all possible PH quintic solutions to the RRMF condition. A family of
Class II RRMF quintics is thereby newly identified, that correspond to the case where
a
(
t
)
,
b
(
t
)
are linear. Modulo scaling/rotation transformations, Class II curves have five degrees of freedom, as with the Class I curves. Although Class II curves have rational RMFs that are only of degree 6–as compared to degree 8 for Class I curves–their algebraic characterization is more involved than for the latter. Computed examples are used to illustrate the construction and properties of this new class of RRMF quintics. A novel approach for generating RRMF quintics, based on the sum-of-four-squares decomposition of positive real polynomials, is also introduced and briefly discussed.
A rotation-minimizing adapted frame on a space curve
r
(
t
)
is an orthonormal basis
(
f
1
,
f
2
,
f
3
)
for
R
3
such that
f
1
is coincident with the curve tangent
t
=
r
′
/
|
r
′
|
at each point and ...the normal-plane vectors
f
2
,
f
3
exhibit no instantaneous rotation about
f
1
. Such frames are of interest in applications such as spatial path planning, computer animation, robotics, and swept surface constructions. Polynomial curves with
rational rotation-minimizing frames (RRMF curves) are necessarily Pythagorean-hodograph (PH) curves–since only the PH curves possess rational unit tangents–and they may be characterized by the fact that a rational expression in the four polynomials
u
(
t
)
,
v
(
t
)
,
p
(
t
)
,
q
(
t
)
that define the quaternion or Hopf map form of spatial PH curves can be written in terms of just two polynomials
a
(
t
)
,
b
(
t
)
. As a generalization of prior characterizations for RRMF cubics and quintics, a sufficient and necessary condition for a spatial PH curve of arbitrary degree to be an RRMF curve is derived herein for the generic case satisfying
u
2
(
t
)
+
v
2
(
t
)
+
p
2
(
t
)
+
q
2
(
t
)
=
a
2
(
t
)
+
b
2
(
t
)
. This RRMF condition amounts to a divisibility property for certain polynomials defined in terms of
u
(
t
)
,
v
(
t
)
,
p
(
t
)
,
q
(
t
)
and their derivatives.
The existence of rational rotation-minimizing frames (RRMF) on polynomial space curves is characterized by the satisfaction of a certain identity among rational functions. In this note we prove that ...previously thought degree limitations on that condition are incorrect. In that regard, new types of RRMF curves are discovered.
A polynomial Pythagorean-hodograph (PH) curve r(t)=(x1(t),…,xn(t)) in Rn is characterized by the property that its derivative components satisfy the Pythagorean condition x1′2(t)+⋯+xn′2(t)=σ2(t) for ...some polynomial σ(t), ensuring that the arc length s(t)=∫σ(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R2 and R3, by means of the complex-number and the quaternion or Hopf map representations, and the basic theory and algorithms for their practical construction and analysis are currently well-developed. However, the case of PH curves in Rn for n>3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)-tuples when n>3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n=5 and n=9, and investigate some of their properties.
The existence of rational rotation-minimizing frames on polynomial space curves is characterized by the satisfaction of a certain identity among rational functions. Part 2 of Remark 5.1 in the ...original paper states an inequality among the degrees of the denominators of these rational functions, but the proof given therein was incomplete. A formal proof of this inequality, which is essential to the complete categorization of rational rotation-minimizing frames on polynomial space curves, appears to be a rather formidable task. Since all known examples and special cases suggest that the inequality is correct, it is restated here as a conjecture rather than a definitive result, and some preliminary steps towards the proof are presented.
The scalar–vector representation is used to derive a simple algorithm to obtain the roots of a quadratic quaternion polynomial. Apart from the familiar vector dot and cross products, this algorithm ...requires only the determination of the unique positive real root of a cubic equation, and special cases (e.g., double roots) are easily identified through the satisfaction of algebraic constraints on the scalar/vector parts of the coefficients. The algorithm is illustrated by computed examples, and used to analyze the root structure of quadratic quaternion polynomials that generate quintic curves with rational rotation-minimizing frames (RRMF curves). The degenerate (i.e., linear or planar) quintic RRMF curves correspond to the case of a double root. For polynomials with distinct roots, generating non-planar RRMF curves, the cubic always factors into linear and quadratic terms, and a closed-form expression for the quaternion roots in terms of a real variable, a unit vector, a uniform scale factor, and a real parameter τ∈−1,+1 is derived.
PDF
