Dereck S. Meek

On surface normal and Gaussian curvature approximations given data sampled from a smooth surface

Approximations to the surface normal and to the Gaussian curvature of a smooth surface are often required when the surface is defined by a set of discrete points. The accuracy of an approximation can be measured using asymptotic analysis. The errors of several approximations to the surface normal and to the Gaussian curvature are compared.

Approximating smooth planar curves by arc splines

When a smooth curve is used to describe the path of a computer-controlled cutting machine, the path is usually approximated by many straight line segments. It is preferable to describe the cutting path as an arc spline, a tangent continuous piecewise curve made of circular arcs and straight line segments. This paper presents an algorithm for finding an arbitrarily close arc spline approximation of a smooth curve.

Clothoid spline transition spirals

Highway and railway designers use clothoid splines (planar G2 curves consisting of straight line segments, circular arcs, and clothoid segments) as center lines in route location. This paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures and unit tangents at the two points. Conditions are given for the existence and uniqueness of the clothoid spline transition spirals, and algorithms for finding them are outlined.

Approximation of discrete data by G1 arc splines

Abstract Arc splines, i.e. G1 curves made of circular arcs and straight-line segments, are important, as they are the paths that are used by automatically controlled cutting machinery. Many algorithms for approximately representing discrete data by a polygon have been published. In the paper, simple modifications of two of those algorithms are applied to the problem of the approximate representation of discrete data by an arc spline.

The use of Cornu spirals in drawing planar curves of controlled curvature

Abstract Cornu spirals or clothoids have been used in highway design for many years. In the past the spirals have been found manually by draftsmen. The purpose of this paper is to show that it is practical to find these spirals with a micro-computer. The design curve will be made up of arcs of circles and segments of Cornu spirals joined in such a way that the curvature is continuous throughout, and takes its largest values on the arcs of circles. Thus, the radii of the circles used will limit, and control the curvature of the whole design curve.

Technical section: A controlled clothoid spline

A clothoid has the property that its curvature varies linearly with arclength. This is a useful feature for the path of a vehicle whose turning radius is controlled as a linear function of the distance travelled. Highways, railways and the paths of car-like robots may be composed of straight line segments, clothoid segments and circular arcs. Control polylines are used in computer aided design and computer aided geometric design applications to guide composite curves during the design phase. This article examines the use of a control polyline to guide a curve composed of segments of clothoids, straight lines, and circular arcs.

A Pythagorean hodograph quintic spiral

Abstract A polynomial curve with a Pythagorean hodograph has the properties that its arc-length is a polynomial of its parameter, and its offset is a rational algebraic expression. A quintic is the lowest degree Pythagorean hodograph curve that may have an inflection point and that inflection point allows a segment of it to be joined to a straight line segment while preserving continuity of curvature, continuity of position, and continuity of tangential direction. The curvature of a spiral varies monotonically with arc-length. Spiral segments are useful in the design of fair curves. A Pythagorean hodograph quintic spiral is presented which allows the design of fair curves in a nurbs based cad system. It is also suitable for applications such as highway design in which the clothoid has traditionally been used.

Geometric Hermite interpolation with Tschirnhausen cubics

Explicit formulae are found that give the unique Tschirnhausen cubic that solves a geometric Hermite interpolation problem. That solution is used to create a planar G1 spline by joining segments of Tschirnhausen cubics. If the geometric Hermite data is from a smooth function, the Tschirnhausen cubic approximates the smooth function. The error in the approximation of a short segment of length h can be expressed as a power series in h. The error is O(h4) and the coefficient of the leading term is found.

A triangular G1 patch from boundary curves

Abstract For some applications it is necessary to fit an irregular surface to given data, e.g. to develop a geometric model of a human skeletal bone from computerized tomography scans. Such a surface does not always have easily distinguishable isoparametric lines. It is thus not convenient to use standard global curve fitting techniques such as those based on B-splines. A global method may also smooth away essential features. A reasonable approach is to use a composite surface where individual surface patches are locally determined. To obtain some visual smoothness it is desirable that these patches join their neighbours in a manner that preserves positional as well as tangent plane continuity. Several methods have been presented for constructing surfaces in such a manner. A common initial stage in developing the patches is to determine a network of boundary curves. This article reports on some results using boundary curves based on a recent technique for point normal interpolation.

Approximating quadratic NURBS curves by arc splines

Abstract When a quadratic NURBS curve is used to describe the path of a computer-controlled cutting machine, the NURBS curve is usually approximated by many straight-line segments. It is preferable to describe the cutting path as an arc spline, a tangent-continuous, piecewise curve made of circular arcs and straight-line segments. The paper presents an algorithm for finding an arbitrarily close arc-spline approximation to a quadratic NURBS curve.