Rational Offsets Research Articles

Offset-rational sinusoidal spirals in Bézier form

Mineur et al. [Mineur, Y., Lichah, T., Castelain, J.M., Giaume, H., 1998. A shape controlled fitting method for Bézier curves. Computer Aided Geometric Design 15 (9), 879–891] presented a remarkable family of Bézier curves, characterized by a control polygon with constant angle at each vertex and constant ratio between lengths of consecutive edges. We note that such curves belong to a classical family of polynomial sinusoidal spirals that encompasses Tschirnhausen's cubic, and can also be expressed as p-Bézier curves (Bézier curves in polar coordinates). These spirals have rational offsets, obtained simply by adding two p-Bézier curves, namely the spiral and a circle. Such a construction includes the offset to a parabola.

Hermite interpolation by Pythagorean hodograph curves of degree seven

Polynomial Pythagorean hodograph (PH) curves form a remarkable subclass of polynomial parametric curves; they are distinguished by having a polynomial arc length function and rational offsets (parallel curves). Many related references can be found in the article by Farouki and Neff on C 1 Hermite interpolation with PH quintics. We extend the C 1 Hermite interpolation scheme by taking additional curvature information at the segment boundaries into account. As a result we obtain a new construction of curvature continuous polynomial PH spline curves. We discuss Hermite interpolation of G 2 [C 1 ] boundary data (points, first derivatives, and curvatures) with PH curves of degree 7. It is shown that up to eight possible solutions can be found by computing the roots of two quartic polynomials, With the help of the canonical Taylor expansion of planar curves, we analyze the existence and shape of the solutions. More precisely, for Hermite data which are taken from an analytical curve, we study the behaviour of the solutions for decreasing stepsize Δ. It is shown that a regular solution is guaranteed to exist for sufficiently small stepsize Δ, provided that certain technical assumptions are satisfied. Moreover, this solution matches the shape of the original curve; the approximation order is 6. As a consequence, any given curve, which is assumed to be G 2 (curvature continuous) and to consist of analytical segments can approximately be converted into polynomial PH form. The latter assumption is automatically satisfied by the standard curve representations of Computer Aided Geometric Design, such as Bezier or B-spline curves. The conversion procedure acts locally, without any need for solving a global system of equations, It produces G 2 polynomial PH spline curves of degree 7.

Hermite interpolation by piecewise polynomial surfaces with rational offsets

We present a construction for polynomial spline surfaces with a piecewise linear field of normal vectors. As main advantageous feature these surfaces possess exact rational offsets. The spline surface is composed of quartic Clough–Tocher-type macro elements. Each element is capable of matching boundary data consisting of three points with associated normal vectors. The collection of the macro elements forms a G 1 continuous spline surface. With the help of a reparamaterization technique we obtain an exact rational representation of the offset surfaces by rational triangular spline surfaces of degree 10.

A Laguerre geometric approach to rational offsets

Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.

Interpolating G1-splines with rational offsets

In (Pottmann, 1995), a geometric characterization of rational PH-curves is presented. Using this result we developed an explicit Bézier representation for interpolating G 1-Hermite PH-splines referring to local coordinate systems. Furthermore a simple geometric criterion for avoiding singularities is proposed.

Pythagorean-hodograph quintic transition curves of monotone curvature

Walton and Meek [Walton, D.J. and Meek, D.S., A Pythagorean-hodograph quintic spiral. Computer Aided Design, 1996, 28, 943–950] have recently advocated the use of Pythagorean-hodograph quintics of monotone curvature, or “PH spirals” for short, as transitional elements that give G 2 connections of linear and circular arcs in applications such as layout of highways and railways—in which context PH curves provide the important advantage of rational offsets and exact rectifications. They construct a PH quintic, interpolating an initial point and tangent and a final tangent, with monotone curvature variation from zero to a given (extremum) final value. We show that using the complex representation for PH curves greatly simplifies this problem and also reveals that the method of Walton and Meek yields a special instance among a one-parameter family of solutions. The additional degree of freedom of PH spirals identified herein relaxes constraints otherwise required to guarantee their existence, and offers the designer precise control over their total length and/or the ability to fine-tune their curvature distributions.

Rational curves and surfaces with rational offsets

Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = ( s u × s v) t ́ | s u × s v t ́ | the offset surface at distance d is s d(u,v) = s(u,v) + d n(u,v) . This is in general not a rational representation, since t ́ | s u × s v is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets s d(u,v) . The analogous question is solved for curves, which is an extension of Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose arc length parameter s( t) is a rational function of t. Offsets arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property for practical use.

Curve design with rational Pythagorean-hodograph curves

The dual Bezier representation offers a simple and efficient constructive approach to rational curves with rational offsets (rational PH curves). Based on the dual form, we develop geometric algorithms for approximating a given curve with aG 2 piecewise rational PH curve. The basic components of the algorithms are an appropriate geometric segmentation andG 2 Hermite interpolation. The solution involves rational PH curves of algebraic class 4; these curves and important special cases are studied in detail.