Pythagorean-hodograph Cubics Research Articles

GEOMETRIC HERMITE INTERPOLATION FOR PLANAR PYTHAGOREAN-HODOGRAPH CUBICS

We solve the geometric Hermite interpolation problem with planar Pythagorean-hodograph cubics. For every Hermite data, we determine the exact number of the geometric Hermite interpolants and represent the interpolants explicitly. We also present a simple criterion for determining whether the interpolants have a loop or not.

Trends in Classical Analysis, Geometric Function Theory, and Geometry of Conformal Invariants

The present special issue of the Journal of Abstract and Applied Analysis is devoted to trends in classical analysis, geometric function theory, and geometry of conformal mappings. In the sense of the title of this journal, we wanted to present a spectrum of research themes reaching from applied analysis to pure analysis and from applications of analysis in geometry to applications in differential equations and integral equations. Further, our aim was to find articles on classical function theory as well as on its generalizations in several directions. One additional aspect, that was paid attention to, is the tendency of mathematics to use computers to solve problems by new and effective algorithms. In detail, we addressed to the following themes. That we did not forget classical pure analysis is proved by an article that considers harmonic functions on Riemann manifolds and by an article on geometry and topology of Banach spaces. The classical geometric function theory is represented by an article on univalence criterions associated with the nth derivative. The relationships between conformal mappings and integral equations are in the scope of two articles, where algorithms are proved to compute the mappings of unbounded multiply connected as well as bounded multiply connected regions onto slit regions. Other old themes of classical function theory are entire functions for which we publish a paper on uniqueness theorems for monomials of entire functions. Concerning the generalizations of classical analysis, we incorporated a paper on the stability of solutions of fractional differential equations and two papers on harmonic mappings, specially one on the general theory of log-harmonic mappings and one on certain classes of harmonic mappings defined by convolutions. The papers on the generalizations of holomorphic functions are completed by a longer article on the distribution of zeros and poles of the rational approximants of a nonholomorphic function on an interval. The aspect of new algorithms is addressed in an article on Hermite interpolation using Mobius transformations of planar Pythagorean-hodograph cubics and in a paper where algorithms to calculate inverse Z-transforms on the unit disc by number-theoretical methods are proved. We hope that in this broad variety of analytic themes many researchers can find something interesting and new.

PYTHAGOREAN-HODOGRAPH CUBICS AND GEOMETRIC HERMITE INTERPOLATION

In this paper, we present the geometric Hermite interpolation for planar Pythagorean-hodograph cubics for some general Hermite data. curve ro(t) = r(t) + dn(t) at a xed distance d from a given polynomial curve r(t), in the direction of its unit normal vector n(t), is an important problem in computer aided geometric design. The oset ro(t) is not, in general, a rational curve. Thus we need some approximation schemes to deal with osets of a polynomial curves. However, if the speed function of a given polynomial curve is a polynomial, then an oset curve can be expressed in a rational parametrization. For this reason, the Pythagorean-hodograph (PH) curves, whose speed function is a polynomial, were introduced by Farouki and Sakkalis ((6)). Since then, there have been vast researches on this class of curves by themselves and others ((1), (2), (3), (4), (5), and (11)). A PH curve r(t) = (x(t);y(t)) means a special polyno- mial curve which is characterized by the algebraic property that its hodograph r 0 (t) = (x 0 (t);y 0 (t)) satises the Pythagorean condition, that is, x 02 (t) +y 02 (t) = 2 (t) for some polynomial (t). So, the arc-length function and the osets

Solvability of [formula omitted] Hermite interpolation by spatial Pythagorean-hodograph cubics and its selection scheme

We provide necessary and sufficient conditions for the existence of solutions of G 1 Hermite interpolation problem by spatial PH cubics. To deal with the cases where multiple solutions exist, we also provide a selection scheme to choose one “good” interpolant curve based on the concept of analytic continuation.

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.

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

It is shown that, depending upon the orientation of the end tangents $\t_0, \t_1$ relative to the end point displacement vector $\Delta\p=\p_1-\p_0$, the problem of $G^1$ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points $\p_0,\ldots,\p_n$ in $\mathbb{R}^3$, compatible with a $G^1$ piecewise--PH--cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape--preservation properties of the resulting curves, is illustrated by a selection of computed examples.

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.

Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling

This article is devoted to cubic Pythagorean hodograph (PH) curves which enjoy a number of remarkable properties, such as polynomial arc-length function and existence of associated rational frames. First we derive a construction of such curves via interpolation of G 1 Hermite boundary data with Pythagorean hodograph cubics. Based on a thorough discussion of the existence of solutions we formulate an algorithm for approximately converting arbitrary space curves into cubic PH splines, with any desired accuracy. In the second part of the article we discuss applications to sweep surface modeling. With the help of the associated rational frames of PH cubics we construct rational representations of sweeping surfaces. We present sufficient criteria ensuring G 1 continuity of the sweeping surfaces. This article concludes with some remarks on offset surfaces and rotation minimizing frames.