D.S. Meek

A planar cubic Be´zier spiral

A planar cubic Bezier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bezier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.

Planar G 2 transition curves composed of cubic Bézier spiral segments

In curve and surface design it is often desirable to have a planar transition curve, composed of at most two spiral segments, between two circles. The purpose may be practical, e.g., in highway design, or aesthetic. Cubic Bezier curves are commonly used in curve and surface design because they are of low degree, are easily evaluated, and allow inflection points. This paper generalizes earlier results on planar cubic Bezier spiral segments which were proposed as transition curve elements, and examines techniques for curve design using the new results.

Curvature extrema of planar parametric polynomial cubic curves

Parametric polynomial cubic curve segments are widely used in computer-aided design and computer-aided geometric design applications because their flexibility makes them suitable for use in the interactive design of curves and surfaces. Shape properties of these segments have been studied and results on the occurrence of cusps, and whether the segment is S- or C-shaped, are available in the literature. Their critical points, however, are not as well known. This paper presents results on the number and location of curvature extrema of these segments. All possible numbers of curvature extrema are found and it is demonstrated that all possible curvature extrema can be obtained numerically for any planar parametric cubic curve. These results are useful in the study of the fairness of curves designed with parametric polynomial cubic curve segments.

Planar G 2 transition with a fair Pythagorean hodograph quintic curve

Recently planar cubic and Pythagorean hodograph quintic transition curves that are suitable for G2 blending were developed. They are suitable for blending, e.g. rounding corners, or for smooth transition between two curves, e.g. two circular arcs. It was shown that a single cubic segment can be used as a transition curve with the guarantee that an S-shaped transition curve will have no curvature extrema, and a C-shaped transition curve will have a single curvature extremum. The results for the cubic curve are now extended to Pythagorean hodograph quintic curves. A Pythagorean hodograph curve has the attractive properties that its arclength is a polynomial of its parameter, and its offset is rational. A quintic is the lowest degree Pythagorean hodograph curve that may have an inflection point. Pythagorean hodograph curves with no curvature extrema for an S-shaped transition, and a single curvature extremum for a C-shaped transition are suitable for the design of fair curves, e.g. in highway design, or for blending in CAD applications.

Approximation of quadratic Be´zier curves by arc splines

Abstract The paths of cutting tools used in a computer-aided manufacturing environment are usually described by means of circular arcs and straight line segments. In a computer aided design environment, however, objects are often designed using B-splines or Bezier curves. This paper develops a simple technique to find an arbitrarily close approximation to a quadratic Bezier curve by a G1 curve consisting of circular arcs.

Spiral arc spline approximation to a planar spiral

A biarc is a one-parameter family of G 1 curves that can satisfy G 1 Hermite data at two points. An arc spline approximation to a smooth planar curve can be found by reading G 1 Hermite data from the curve and tting a biarc between each pair of data points. The resulting collection of biarcs forms a G 1 arc spline that interpolates the entire set of G 1 Hermite data. If the smooth curve is a spiral, it is desirable that the arc spline approximation also be a spiral. Several methods are described for choosing the free parameters of the biarcs so that the arc spline approximation to a smooth spiral is a spiral. c 1999 Elsevier Science B.V. All rights reserved.

The inverses of Toeplitz band matrices

Abstract The elements of the inverse of a Toeplitz band matrix are given in terms ofthe solution of a difference equation. The expression for these elements is a quotient of determinants whose orders depend the number of nonzero superdiagonals but not on the order of the matrix. Thus, the formulae are particularly simple for lower triangular and lower Hessenberg Toeplitz matrices. When the number of nonzero superdiagonals is small, sufficient conditions on the solution of the abovementioned difference equation can be given to ensure that the inverse matrix is positive. If the inverse is positive, the row sums can be expressed in terms of the solution of the difference equation.

Planar two-point G1 Hermite interpolating log-aesthetic spirals

Log-aesthetic spirals are currently being studied as fair curves that can be used in computer aided design. A family of planar log-aesthetic spirals that include a point of zero curvature is used in this paper. The two-point G^1 Hermite data that is considered has some restrictions on the angles. This paper proves that for any member of the family, a unique segment of that spiral can be found that matches given two-point G^1 Hermite data.

Planar G 2 Hermite interpolation with some fair, C-shaped curves

G2 Hermite data consists of two points, two unit tangent vectors at those points, and two signed curvatures at those points. The planar G2 Hermite interpolation problem is to find a planar curve matching planar G2 Hermite data. In this paper, a C-shaped interpolating curve made of one or two spirals is sought. Such a curve is considered fair because it comprises a small number of spirals. The C-shaped curve used here is made by joining a circular arc and a conic in a G2 manner. A curve of this type that matches given G2 Hermite data can be found by solving a quadratic equation. The new curve is compared to the cubic Bezier curve and to a curve made from a G2 join of a pair of quadratics. The new curve covers a much larger range of the G2 Hermite data that can be matched by a C-shaped curve of one or two spirals than those curves cover.

Curve design with more general planar Pythagorean-hodograph quintic spiral segments

Spiral segments are useful in the design of fair curves. They are important in CAD/CAM applications, the design of highway and railway routes, trajectories of mobile robots and other similar applications. The quintic Pythagorean-hodograph (PH) curve discussed in this article is polynomial; it has the attractive properties that its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. This paper generalises earlier results on planar PH quintic spiral segments and examines techniques for designing fair curves using the new results.