Spatial Pythagorean-hodograph Curves Research Articles

Polynomial curves with projections to PH curves

Despite the fact that the orthogonal projection of a spatial Pythagorean hodograph (PH) curve into the plane is not a planar PH curve in general, we can find special cases such that the PH property is preserved when the curve is projected. In Farouki et al. (2021) the authors studied how to generate spatial PH curves with planar PH projections. Their approach and presented results motivated us to continue and extend this investigation. We study geometric conditions under which a spatial curve is projected to a PH curve. For this purpose, we introduced a suitable geometric characterization of the curves with PH property via intersection multiplicity of the associated curves described by the hodograph mapping with the absolute conic. As a consequence we will show that a generic polynomial curve of degree higher than five possesses no parallel projection to a PH curve. On contrary, for a spatial cubic there are finitely many ways how to orthogonally project it to a planar PH cubic. And the same holds for oblique parallel projections of spatial quintics. Hence these cases are examined in more detail.

On the construction of polynomial minimal surfaces with Pythagorean normals

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.

G1 Hermite interpolation method for spatial PH curves with PH planar projections

The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G1 Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G1 interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G1 data in general, however, we show the feasibility of the system for the G1 data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves.

Planar projections of spatial Pythagorean-hodograph curves

Although the orthogonal projection of a spatial Pythagorean–hodograph (PH) curve on to a plane is not (in general) a planar PH curve, it is possible to construct spatial PH curves so as to ensure that their orthogonal projections on to planes of a prescribed orientation are planar PH curves. The construction employs an analysis of the root structure of the components of the quaternion polynomials that generate spatial PH curves, and it encompasses both helical and non–helical spatial PH curves. An initial characterization for orthogonal projections of spatial PH curves on to the coordinate planes provides the basis for a generalization to projections of arbitrary direction, based on unit quaternion rotation transformations of R3.

МОДЕЛЮВАННЯ ПРОСТОРОВИХ ІЗОТРОПНИХ КРИВИХ БЕЗЬЄ НА ОСНОВІ КРИВИХ ЗА ГОДОГРАФОМ ПІФАГОРА

Minimal surfaces are essential for geometric modeling, computer science, and for various branches of engineering, and the determination of the conditions for their construction has been studied in many works and includes a large number of methods. However, this topic has not yet been exhausted when using isotropic curves to construct minimal surfaces. These studies focus on methods for determining zero-length curves. For interactive control of such curves, it is advisable to use curves constructed on the basis of characteristic polygons, therefore, studies are based on Bézier curves. The paper highlights the experience of previous studies in the field of constructing spatial isotropic curves and plane Pythagorean-Hodograph curves. The modeling of spatial Pythagorean-Hodograph curves is based on the theory of quaternions. The authors of the paper present a different approach, namely, the construction of an isotropic spatial curve based on a plain Bézier Pythagorean-Hodograph curve. The transition to the spatial curve is carried out on the basis of determining the third coordinate from the condition that the length is equal to zero. In this case, a plain curve is built on the basis of real values, and the third coordinate will be purely imaginary. The authors carry out studies based on fifth-order curves. In this case, it is advisable to choose quadratic functions as basic polynomials. Three options are proposed for setting the initial values ​​of the coordinates for determining the Pythagorean-Hodograph curve. The problem of the boundary points of the curve and the determination of the intermediate points of the Bezier curve based on the given dependences have the most practical importance. To prove the reliability of the proposed provisions, the calculation of the points of the curve is performed and examples of simulated curves are given. Further research involves the use of the constructed curves for modeling minimal surfaces and isotropic portions of Bézier. Key words: isotropic curve, Bézier curve, Pythagorean-Hodograph curve.

Controlling extremal Pythagorean hodograph curves by Gauss–Legendre polygons

The problem of constructing spatial Pythagorean hodograph (PH) curves with a given Gauss–Legendre polygon is addressed. For planar/spatial PH curves of degree 2n+1, the Gauss–Legendre polygon, which consists of n+1 edges, obtained by evaluating the hodograph at the nodes of the Gauss–Legendre quadrature, is the rectifying polygon, which has the same length as the PH curve. On the other hand, if a planar polygon with n+1 edges is given, there are 2n planar PH curves whose Gauss–Legendre polygon is the given polygon. We here generalize this result to the spatial PH curves. For a given spatial polygon with n+1 edges, we construct n parameter family of PH curves of degree 2n+1. Among those PH curves, we identify 2n extremal solutions by choosing the quaternion preimages of the hodograph to have the maximal or the minimal distances from the adjacent quaternion solutions. We show that the extremal PH curves are one possible generalization of 2n planar PH curves with the planar Gauss–Legendre polygon by proving the planarity condition: the extremal PH curves are planar if the provided polygon is planar.

Gauss–Lobatto polygon of Pythagorean hodograph curves

The Gauss–Legendre polygon of Pythagorean hodograph (PH) curves can be used as the rectifying control polygon, which has (i) the end point interpolation property, (ii) the rectifying property, and (iii) the same degree of freedom as the PH curve. These properties make the Gauss–Legendre polygon a nice tool to control the shape of the PH curve. A drawback of the Gauss–Legendre polygon is that it does not determine the end tangent vectors. In this paper, we introduce the Gauss–Lobatto polygon as an alternative to the Gauss–Legendre polygon. Since the Gauss–Lobatto quadrature has the end points as the predetermined nodes, the Gauss–Lobatto polygon naturally determines the end tangent vectors of the PH curve. We analyze the rectifying property of the Gauss–Lobatto polygon for both planar and spatial PH curves. Concerning the degree of freedom, we show that the Gauss–Lobatto polygons of planar PH curves are not the rectifying control polygons. For spatial PH curves, we identify the relation between the degree of a PH curve and the number of edges in the Gauss–Lobatto polygon that makes the Gauss–Lobatto polygon a rectifying control polygon. We also provide the method to compute the spatial septic PH curve with the given Gauss–Lobatto polygon, and the algorithm for the deformation of the spatial septic PH curves.

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.

Rational frames of minimal twist along space curves under specified boundary conditions

An adapted orthonormal frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent $\mathbf {f}_{1}(\xi ) =\mathbf {r}^{\prime }(\xi )/|\mathbf {r}^{\prime }(\xi )|$ and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal–plane vectors. Employing the Euler–Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.

Helical polynomial curves interpolating G1 data with prescribed axes and pitch angles

A helical curve, or curve of constant slope, offers a natural flight path for an aerial vehicle with a limited climb rate to achieve an increase in altitude between prescribed initial and final states. Every polynomial helical curve is a spatial Pythagorean-hodograph (PH) curve, and the distinctive features of the PH curves have attracted growing interest in their use for Unmanned Aerial Vehicle (UAV) path planning. This study describes an exact algorithm for constructing helical PH paths, corresponding to a constant climb rate at a given speed, between initial and final positions and motion directions. The algorithm bypasses the more sophisticated algebraic representations of spatial PH curves, and instead employs a simple “lifting” scheme to generate helical PH paths from planar PH curves constructed using the complex representation. In this context, a novel scheme to construct planar quintic PH curves that interpolate given end points and tangents, with exactly prescribed arc lengths, plays a key role. It is also shown that these helical paths admit simple closed-form rotation-minimizing adapted frames. The algorithm is simple, efficient, and robust, and can accommodate helical axes of arbitrary orientation through simple rotation transformations. Its implementation is illustrated by several computed examples.

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.

Identification and “reverse engineering” of Pythagorean-hodograph curves

Methods are developed to identify whether or not a given polynomial curve, specified by Bézier control points, is a Pythagorean-hodograph (PH) curve — and, if so, to reconstruct the internal algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean hodographs, respectively, from prescribed Bézier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved.

Geometric Hermite interpolation by monotone helical quintics

Monotone helical curves are polynomial helices whose unit tangent maintains a fixed sense of rotation about an axis. We investigate the interpolation of geometric Hermite data consisting of end points, tangents, and curvatures by monotone helical quintics. Based on the Hopf map model for spatial Pythagorean-hodograph curves, which subsume polynomial helices, we show that the geometric Hermite interpolation can be determined by solving a certain univariate polynomial equation of degree twelve.

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.

Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves

Simple criteria for the existence of rational rotation-minimizing frames (RRMFs) on quintic space curves are determined, in terms of both the quaternion and Hopf map representations for Pythagorean-hodograph (PH) curves in ℝ3. In both cases, these criteria amount to satisfaction of three scalar constraints that are quadratic in the curve coefficients, and are thus much simpler than previous criteria. In quaternion form, RRMF quintics can be characterized by just a single quadratic (vector) constraint on the three quaternion coefficients. In the Hopf map form, the characterization is in terms of one real and one complex quadratic constraint on the six complex coefficients. The identification of these constraints is based on introducing a “canonical form” for spatial PH curves and judicious transformations between the quaternion and Hopf map descriptions. The simplicity of these new characterizations for the RRMF quintics should help facilitate the development of algorithms for their construction, analysis, and practical use in applications such as animation, spatial motion planning, and swept surface constructions.

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.

A geometric product formulation for spatial Pythagorean hodograph curves with applications to Hermite interpolation

A novel formulation for spatial Pythagorean hodograph (PH) curves, based on the geometric product of vectors from Clifford algebra, is proposed. Compared to the established quaternion representation, in which a hodograph is generated by a continuous sequence of scalings/rotations of a fixed unit vector n ˆ , the new representation corresponds to a sequence of scalings/reflections of n ˆ . The two representations are shown to be equivalent for cubic and quintic PH curves, when freedom in choosing n ˆ is retained for the vector formulation. The latter also subsumes the original (sufficient) characterization of spatial Pythagorean hodographs, proposed by Farouki and Sakkalis, as a particular choice for n ˆ . In the context of the spatial PH quintic Hermite interpolation problem, variation of the unit vector n ˆ offers a geometrically more-intuitive means to explore the two-parameter space of solutions than the two free angular variables that arise in the quaternion formulation. This space is seen to have a decomposition into a product of two one-parameter spaces, in which one parameter determines the arc length and the other can be used to vary the curve shape at fixed arc length.

First order hermite interpolation with spherical Pythagorean-hodograph curves

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the Pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatialC1 Hermite data, we construct a spatial PH curve on a sphere that is aC1 Hermite interpolant of the given data as follows: First, we solveC1 Hermite interpolation problem for the stereographically projected planar data of the given data in ℝ3 with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in ℝ3 using the inverse general stereographic projection.

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.

Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves

An exact specification of the rotation-minimizing frame on a spatial Pythagorean-hodograph (PH) curve can be derived by integration of a rational function. The result is an angular function θ( t) of the curve parameter, comprising in general both rational and logarithmic terms, that specifies the orientation of the rotation-minimizing frame relative to the Frenet frame. For PH cubics and quintics, the solution employs only arithmetic operations on the curve coefficients and some complex square and cube root extractions. Moreover, the generalization to PH curves of arbitrary order entails only standard polynomial algorithms (i.e., arithmetic, greatest common divisors, and resultants), solution of a linear system, and a minimal element of polynomial root-solving. Rotation-minimizing frames are employed in computer animation, the construction of swept surfaces, and in robotics applications where the axis of a tool or probe should remain tangential to a given spatial path while minimizing changes of orientation about this axis.