Polynomial Space Curves Research Articles

On Polynomial Space Curves with Flc-frame

The first and second derivatives of a curve provide us fundamental
 information in the study of the behavior of curve near a point. However,
 if a curve is a polynomial space curve of degree n, we don’t know what
 is the geometric meaning of the n-th derivative of the curve? There is no
 doubt that the Frenet frame is not suitable for this purpose because it is
 constructed by using first and second derivatives of a curve. On the other
 hand, in this paper by using a new frame called as Flc-frame we are able
 to give the geometric meaning of the n-th derivative of a curve. Moreover,
 we explore some basic concepts regarding polynomial space curves from
 point of view of Flc-frame in three dimensional Euclidean space.

ON RULED SURFACES CONSTRUCTED BY THE EVOLUTION OF A POLYNOMIAL SPACE CURVE

In this paper, we present the evolutions of the ruled surfaces constructed by the tangent, normal-like, and binormal-like vector fields of a polynomial space curve. These evolutions of the ruled surfaces depend on the evolutions of their directrices using the Flc (Frenet like curve) frame along a polynomial space curve. Therefore, the evolutions of a polynomial curve are expressed in the first step of this study. Then, some geometric properties of the special ruled surfaces are investigated and examples of these surfaces are given and their graphics are drawn using the Mathematica 9 program.

Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

Given a polynomial space curve r(ξ) that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves r˜(ξ) with the same rotation–minimizing frame as r(ξ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u(ξ)=r′(ξ)×r″(ξ) and distances from the origin specified in terms of a rational function f(ξ) as f(ξ)/‖u(ξ)‖. An explicit characterization of the rational curves r˜(ξ) generated by a given RRMF curve r(ξ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f(ξ), obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(ξ). Criteria for identifying low–degree instances of the curves r˜(ξ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples.

Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues frames

A minimal twist frame ( f 1 ( ξ ) , f 2 ( ξ ) , f 3 ( ξ ) ) on a polynomial space curve r ( ξ ) , ξ ∈ [ 0 , 1 ] is an orthonormal frame, where f 1 ( ξ ) is the tangent and the normal-plane vectors f 2 ( ξ ) , f 3 ( ξ ) have the least variation between given initial and final instances f 2 ( 0 ) , f 3 ( 0 ) and f 2 ( 1 ) , f 3 ( 1 ) . Namely, if ω = ω 1 f 1 + ω 2 f 2 + ω 3 f 3 is the frame angular velocity, the component ω 1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist frames, based on the Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues frames ( e 1 ( ξ ) , e 2 ( ξ ) , e 3 ( ξ ) ) — i.e., the normal-plane vectors e 2 ( ξ ) , e 3 ( ξ ) have no rotation about the tangent e 1 ( ξ ) . A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f 2 ( ξ ) , f 3 ( ξ ) are then obtained from e 2 ( ξ ) , e 3 ( ξ ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω 1 = constant ) can be accurately approximated.

C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

The problem of constructing a rational adapted frame (f1(ξ),f2(ξ),f3(ξ)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r(ξ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler–Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C1 rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C2 rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.

On new types of rational rotation-minimizing frame space curves

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.

Geometry of the ringed surfaces in [formula omitted] that generate spatial Pythagorean hodographs

A Pythagorean-hodograph (PH) curver(t)=(x(t),y(t),z(t)) has the distinctive property that the components of its derivative r′(t) satisfy x′2(t)+y′2(t)+z′2(t)=σ2(t) for some polynomial σ(t). Consequently, the PH curves admit many exact computations that otherwise require approximations. The Pythagorean structure is achieved by specifying x′(t),y′(t),z′(t) in terms of polynomials u(t),v(t),p(t),q(t) through a construct that can be interpreted as a mapping from R4 to R3 defined by a quaternion product or the Hopf map. Under this map, r′(t) is the image of a ringed surface S(t,ϕ) in R4, whose geometrical properties are investigated herein. The generation of S(t,ϕ) through a family of four-dimensional rotations of a “base curve” is described, and the first fundamental form, Gaussian curvature, total area, and total curvature of S(t,ϕ) are derived. Furthermore, if r′(t) is non-degenerate, S(t,ϕ) is not developable (a non-trivial fact in R4). It is also shown that the pre-images of spatial PH curves equipped with a rotation-minimizing orthonormal frame (comprising the tangent and normal-plane vectors with no instantaneous rotation about the tangent) are geodesics on the surface S(t,ϕ). Finally, a geometrical interpretation of the algebraic condition characterizing the simplest non-trivial instances of rational rotation-minimizing frames on polynomial space curves is derived.

Rotation-minimizing osculating frames

An orthonormal frame (f1,f2,f3) is rotation-minimizing with respect to fi if its angular velocity ω satisfies ω⋅fi≡0 — or, equivalently, the derivatives of fj and fk are both parallel to fi. The Frenet frame (t,p,b) along a space curve is rotation-minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation-minimizing with respect to the tangent t have attracted much interest. This study is concerned with rotation-minimizing osculating frames (f,g,b) incorporating the binormal b, and osculating-plane vectors f, g that have no rotation about b. These frame vectors may be defined through a rotation of t, p by an angle equal to minus the integral of curvature with respect to arc length. In aeronautical terms, the rotation-minimizing osculating frame (RMOF) specifies yaw-free rigid-body motion along a curved path. For polynomial space curves possessing rational Frenet frames, the existence of rational RMOFs is investigated, and it is found that they must be of degree 7 at least. The RMOF is also employed to construct a novel type of ruled surface, with the property that its tangent planes coincide with the osculating planes of a given space curve, and its rulings exhibit the least possible rate of rotation consistent with this constraint.

Sparse Hessian factorization in curved trajectories for unconstrained minimization

Curved trajectories algorithm (CTA) is a package for the minimization of unconstrained functions of several variables with intervals on the variables. The core algorithm is novel in that steps may follow polynomial space curves instead of straight lines. The space curves result from truncations of a Taylor series expansion of the Gradient inverse function. A critical item in the efficiency of CTA is the factorization of the sparse Hessian matrix and handling a non-positive definite Hessian. This paper describes a new approach for non-positive definite Hessians that has given robust results especially for ill-conditioned problems and the CTA package compares very favourably with other minimization packages using a sample of large Cuter problems.

Curve-induced distortion of polynomial space curves, flat-mapped extension modeling, and their impact on ANCF thin-plate finite elements

The Absolute Nodal Coordinate Formulation (ANCF) is a relatively new nonlinear finite element type that uses Hermite splines for shape functions. In this investigation, the ANCF is examined as a possible tool for use in modeling the media in flexible media transport systems, such as printers, copy machines, and roll-to-roll systems. However, it is demonstrated using an example of a thin plate-type ANCF finite element that these elements can suffer from significant membrane locking, which can be problematic for paper or paper-like media. One source of this locking is identified to be a property of all parametric curves that are composed of polynomials. The property is that for parametric polynomial curves, changes in the state of curvature of the curve cause changes in the distribution of points along the curve. This property is labeled Curve-Induced Distortion (CID) by the authors of this paper. CID can cause axial and membrane strain distortion in elements, causing them to be overly stiff. A new solution method is proposed to directly counteract CID in finite elements that use cubic Hermite curves for shape functions, specifically for modeling problems in which bending occurs primarily around one axis, such as paper in printing and media transport machinery. This method is labeled Flat-Mapped Extension Modeling (FMEM). FMEM is a mixed field method that uses a 1D Hermite polynomial kinematically linked to the 3D Hermite curve to represent the axial displacement field. FMEM significantly reduces the effect of CID in the ANCF element tested here. This investigation demonstrates using a single ANCF plate element type that the ANCF’s accuracy can be significantly improved by FMEM with only a small increase in computational cost. It is shown with this plate-element example that without correcting CID, the ANCF element tested is computationally much slower than contemporary methods like the co-rotational formulation for similar accuracy. But with FMEM, the ANCF is significantly faster than the co-rotational formulation for similar accuracy.

Rational rotation-minimizing frames on polynomial space curves of arbitrary degree

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.

Quintic space curves with rational rotation-minimizing frames

The existence of polynomial space curves with rational rotation-minimizing frames (RRMF curves) is investigated, using the Hopf map representation for PH space curves in terms of complex polynomials α ( t ) , β ( t ) . The known result that all RRMF cubics are degenerate (linear or planar) curves is easily deduced in this representation. The existence of non-degenerate RRMF quintics is newly demonstrated through a constructive process, involving simple algebraic constraints on the coefficients of two quadratic complex polynomials α ( t ) , β ( t ) that are sufficient and necessary for any PH quintic to admit a rational rotation-minimizing frame. Based on these constraints, an algorithm to construct RRMF quintics is formulated, and illustrative computed examples are presented. For RRMF quintics, the Bernstein coefficients α 0 , β 0 and α 2 , β 2 of the quadratics α ( t ) , β ( t ) may be freely assigned, while α 1 , β 1 are fixed (modulo one scalar freedom) by the constraints. Thus, RRMF quintics have sufficient freedoms to permit design by the interpolation of G 1 Hermite data (end points and tangent directions). The methods can also be extended to higher-order RRMF curves.

Helical polynomial curves and double Pythagorean hodographs I. Quaternion and Hopf map representations

For regular polynomial curves r ( t ) in R 3 , relations between the helicity condition, existence of rational Frenet frames, and a certain “double” Pythagorean-hodograph (PH) structure are elucidated in terms of the quaternion and Hopf map representations of spatial PH curves. After reviewing the definitions and properties of these representations, and conversions between them, linear and planar PH curves are identified as degenerate spatial PH curves by certain linear dependencies among the coefficients. Linear and planar curves are trivially helical, and all proper helical polynomial curves are PH curves. All spatial PH cubics are helical, but not all PH quintics. The two possible types of helical PH quintic (monotone and general) are identified as subsets of the PH quintics by constraints on their quaternion coefficients. The existence of a rational Frenet frame and curvature on polynomial space curves is equivalent to a certain “double” PH form, first identified by Beltran and Monterde, in which | r ′ ( t ) | and | r ′ ( t ) × r ′ ′ ( t ) | are both polynomials in t . All helical PH curves are double PH curves, which encompass all PH cubics and all helical PH quintics, although non-helical double PH curves of higher order exist. The “double” PH condition is thoroughly analyzed in terms of the quaternion and Hopf map forms, and their connections. A companion paper presents a complete characterization of all helical and non-helical double PH curves up to degree 7.

Characterization and construction of helical polynomial space curves

Helical space curves are characterized by the property that their unit tangents maintain a constant inclination with respect to a fixed line, the axis of the helix. Equivalently, a helix exhibits a circular tangent indicatrix, and constant curvature/torsion ratio. If a polynomial space curve is helical, it must be a Pythagorean-hodograph (PH) curve. The quaternion representation of spatial PH curves is used to characterize and construct helical curves. Whereas all spatial PH cubics are helical, the helical PH quintics form a proper subset of all PH quintics. Two types of PH quintic helix are identified: (i) the “monotone-helical” PH quintics, in which a scalar quadratic factors out of the hodograph, and the tangent exhibits a consistent sense of rotation about the axis; and (ii) general helical PH quintics, which possess irreducible hodographs, and may suffer reversals in the sense of tangent rotation. First-order Hermite interpolation is considered for both helical PH quintic types. The helicity property offers a means of fixing the residual degrees of freedom in the general PH quintic Hermite interpolation problem, and yields interpolants with desirable shape features.

Generation and Simulation of Robot Trajectories in a Virtual CAD-Based Off-Line Programming Environment

This paper presents a new approach to the generation and optimisation of the position and orientation trajectories in Cartesian task space, and simulation in a virtual CAD-based off-line programming environment. A novel concept of the robot pose ruled surface is proposed, and defined as a motion locus of the robot orientation vector, i.e. the vector of equivalent angular displacement. The planning for trajectories of robot end-effectors can be accomplished by generating the robot pose ruled surface and optimising its area under the constraints with good kinematics and dynamics performances. The established optimisation model is based on functional analysis and dynamics planning, and is simplified by using high-order polynomial space curves as the robotic position and orientation trajectories. The developed system, RoboSim, is a virtual integrated environment which can be used as a testbed for automatic motion planning and simulation for robots. Two examples are given to verify the feasibility of the proposed approach and models, and to demonstrate the capabilities of generation, optimisation, and simulation modules and libraries in the RoboSim package. The simulation results show that the approach and system are feasible and useful for motion planning, performance analysis and evaluation, and CAD-based prototyping and off-line programming in both the virtual and the real design and planning environment.

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bezier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.

Detecting cusps and inflection points in curves

In many applications it is desirable to analyze parametric curves for undesirable features like cusps and inflection points. Previously known algorithms to analyze such features are limited to cubics and in many cases are for planar curves only. We present a general purpose method to detect cusps in polynomial or rational space curves of arbitrary degree. If a curve has no cusp in its defining interval, it has a regular parametrization and our algorithm computes that. In particular, we show that if a curve has a proper parametrization then the necessary and sufficient condition for the existence of cusps is given by the vanishing of the first derivative vector. We present a simple algorithm to compute the proper parametrization of a polynomial curve and reduce the problem of detecting cusps in a rational curve to that of a polynomial curve. Finally, we use the regular parametrizations to analyze for inflection points.