The problem of designing smoothly rounded right-angle corners with Pythagorean-hodograph (PH) curves is addressed. A G1 corner can be uniquely specified as a single PH cubic segment, closely approximating a circular arc. Similarly, a G2 corner can be uniquely constructed with a single PH quintic segment having a unimodal curvature distribution. To obtain G2 corners incorporating shape freedoms that permit a fine tuning of the curvature profile, PH curves of degree 7 are required. It is shown that degree 7 PH curves define a one-parameter family of G2 corners, facilitating precise control over the extremum of the unimodal curvature distribution, within a certain range of the parameter. As an alternative, a G2 corner construction based upon splicing together two PH quintic segments is proposed, that provides two free parameters for shape adjustment. The smooth corner shapes constructed through these schemes can exploit the computational advantages of PH curves, including exact computation of arc length, rational offset curves, and real-time interpolator algorithms for motion control in manufacturing, robotics, inspection, and similar applications. © 2014 Elsevier B.V.

A unique feature of polynomial Pythagorean–hodograph (PH) curves is the ability to interpolate G 1 Hermite data (end points and tangents) with a specified total arc length. Since their construction involves the solution of a set of non–linear equations with coefficients dependent on the specified data, the existence of such interpolants in all instances is non–obvious. A comprehensive analysis of the existence of solutions in the case of spatial PH quintics with end derivatives of equal magnitude is presented, establishing that a two–parameter family of interpolants exists for any prescribed end points, end tangents, and total arc length. The two free parameters may be exploited to optimize a suitable shape measure of the interpolants, such as the elastic bending energy.

A reduced difference polynomial f(u,v)=(p(u)−p(v))/(u−v) may be associated with any given univariate polynomial p(t), t ∈ [0, 1] such that the locus f(u,v)=0 identifies the pairs of distinct values u and v satisfying p(u)=p(v). The Bernstein coefficients of f(u, v) on [0, 1]2 can be determined from those of p(t) through a simple algorithm, and can be restricted to any subdomain of [0, 1]2 by standard subdivision methods. By constructing the reduced difference polynomials f(u, v) and g(u, v) associated with the coordinate components of a polynomial curve r(t)=(x(t),y(t)), a quadtree decomposition of [0, 1]2 guided by the variation-diminishing property yields a numerically stable scheme for isolating real solutions of the system f(u,v)=g(u,v)=0, which identify self-intersections of the curve r(t). Through the Kantorovich theorem for guaranteed convergence of Newton–Raphson iterations to a unique solution, the self-intersections can be efficiently computed to machine precision. By generalizing the reduced difference polynomial to encompass products of univariate polynomials, the method can be readily extended to compute the self-intersections of rational curves, and of the rational offsets to Pythagorean–hodograph curves.

In a recent paper (Lee et al., 2014) a family of rational Pythagorean-hodograph (PH) curves is introduced, characterized by constraints on the coefficients of a truncated Laurent series, and used to solve the first-order Hermite interpolation problem. Contrary to a claim made in this paper, it is shown that these rational PH curves have rational arc length functions only in degenerate cases, where the center of the Laurent series is a real value.

Real-time CNC interpolators achieving a constant or variable feedrate V along a parametric curve r(ξ) are usually based on truncated Taylor series expansions defining the time-dependence of the curve parameter ξ. Since the feedrate should be specified as a function of a physically meaningful variable (such as time t, path arc length s, or curvature κ) rather than ξ, successive applications of the differentiation chain rule are necessary to determine Taylor series coefficients beyond the linear term. The closed-form expressions for the higher-order coefficients are increasingly cumbersome to derive and implement, and consequently error-prone. To address this issue, the use of Richardson extrapolation as a simple means to compute rapidly convergent approximations to the higher-order coefficients is investigated herein. The methodology is demonstrated in the context of (1) an arc-length-dependent feedrate for cornering motions; (2) direct real-time offset curve interpolation; and (3) a curvature-dependent feedrate. All of these examples admit simple implementations that circumvent the need for tedious symbolic calculations of higher-order coefficients, and are compatible with real-time controllers with millisecond sampling intervals.

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.

The problem of constructing a plane polynomial curve with given end points and end tangents, and a specified arc length, is addressed. The solution employs planar quintic Pythagorean–hodograph (PH) curves with equal-magnitude end derivatives. By reduction to canonical form it is shown that, in this context, the problem can be expressed in terms of finding the real solutions to a system of three quadratic equations in three variables. This system admits further reduction to just a single univariate biquadratic equation, which always has positive roots. It is found that this construction of G1 Hermite interpolants of specified arc length admits two formal solutions — of which one has attractive shape properties, and the other must be discarded due to undesired looping behavior. The algorithm developed herein offers a simple and efficient closed-form solution to a fundamental constructive geometry problem that avoids the need for iterative numerical methods.

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.

Recent developments in the basic theory, algorithms, and applications for curves with rational rotation-minimizing frames (RRMF curves) are reviewed, and placed in the context of the current state-of-the-art by highlighting the many significant open problems that remain. The simplest non-trivial RRMF curves are the quintics, characterized by a scalar condition on the angular velocity of the Euler-Rodrigues frame (ERF). Two different classes of RRMF quintics can be identified. The first class of curves may be characterized by quadratic constraints on the quaternion coefficients of the generating polynomials; by the root structure of those polynomials; or by a certain polynomial divisibility condition. The second class has a strictly algebraic characterization, less well-suited to geometrical construction algorithms. The degree 7 RRMF curves offer more shape freedoms than the quintics, but only one of the four possible classes of these curves has been satisfactorily described. Generalizations of the adapted rotation-minimizing frames, for which the angular velocity has no component along the tangent, to directed and osculating frames (with analogous properties relative to the polar and binormal vectors) are also discussed. Finally, a selection of applications for rotation-minimizing frames are briefly reviewed - including construction of swept surfaces, rigid-body motion planning, 5-axis CNC machining, and camera orientation control.