Robert J. Cripps

Identification of inflection points and cusps on rational curves

Abstract Using homogeneous coordinates, a rational curve can be represented in a nonrational form. Based on such a nonrational representation of a curve, a simple method to identify inflection points and cusps on 2-D and 3-D rational curves is proposed.

Algorithms to support point-based cadcam

This paper presents a system that supports a point-based approach to CADCAM that was first suggested by McLaughlin. The core to the approach is the ability to interpolate a grid of points in a geometrically meaningful way that does not suffer from the disadvantages of the standard interpolation schemes based on NURBS. The essential features of the approach are based on recursive subdivision between existing points to generate intermediate points that are consistent with a smooth curvature profile. There are many advantages to this approach. Time-consuming surface fitting is not necessary; there are no problems with ensuring continuity between surfaces; data exchange can be carried out exactly. By chosing to work entirely with points, all of the standard interrogations available in CAE systems, including visualisations, FE analysis and NC machining, which are dependent on NURBS curves and surfaces have to be replicated. The feasibility of the point-based approach is illustrated by showing how geometric interrogations and machine tool paths can be accurately generated using just points.

Estimation of end curvatures from planar point data

Given a string of discrete planar points, the estimation of principal curvature vectors using circle fitting and Richardsons extrapolation principle has been considered by several authors. However, these methods can not be directly applied to end points, due to symmetry. This article extends these methods to cope with end points. The method is based on the construction of interpolating circles using the first (or last) four data points. Error analysis suggests that the accuracy of curvature estimation using circle fitting is determined by arc-lengths and derivatives of curvature with respect to arc-length. A comparison is made between the proposed four-point method and the well established threepoint method.

Shape preserving data reduction for 3D surface points

Abstract In reverse engineering, vast amounts of point data captured by certain types of scanners significantly increase the complexity and reduce the efficiency in downstream activities, such as surface modelling and manufacturing. Data reduction has therefore become an important issue. A new data reduction algorithm for surface points is developed to preserve the original shape using an error metric based on a Hausdorff distance of principal curvature vectors. The performance of the proposed method is illustrated by its application to a test surface and an industrial surface.

Smooth polynomial approximation of spiral arcs

Constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials. Even NURBS curves can exhibit unsatisfactory curvature profiles. Curve segments with monotonic curvature profiles, for example spiral arcs, exist but are intrinsically non-polynomial in nature and thus difficult to integrate into existing CAD systems. A method of constructing an approximation to a generalised Cornu spiral (GCS) arc using non-rational quintic Bezier curves matching end points, end slopes and end curvatures is presented. By defining an objective function based on the relative error between the curvature profiles of the GCS and its Bezier approximation, a curve segment is constructed that has a monotonic curvature profile within a specified tolerance.

G3 quintic polynomial approximation for Generalised Cornu Spiral segments

Within Computer Aided Design (CAD) there is a need to construct fair curves. The Generalised Cornu Spirals (GCSs) are a set of curves with a monotonic curvature profile and are hence considered fair but implementation in current CAD systems is not straightforward, partly due to not being in the usual polynomial form. A method to approximate a GCS using a quintic polynomial curve is presented. The method seeks to interpolate the GCS to satisfy the G^3 constraints at the end points with a quintic Bezier, leaving two degrees of freedom. An initial approximation is shown to be effective for the majority of GCS curves. Moreover, it is possible to determine when an initial approximation is likely to be poor. If this approximation does not meet the tolerance required, a search involving two parameters is performed. Characteristics of the search domain are used to establish a suitable starting value.

C1 monotone cubic Hermite interpolant

Abstract Constraining an interpolation to be shape preserving is a well established technique for modelling scientific data. Many techniques express the constraint variables in terms of abstract quantities that are difficult to relate to either physical values or the geometric properties of the interpolant. In this paper, we construct a piecewise monotonic interpolant where the degrees of freedom are expressed in terms of the weights of the rational Bezier cubic interpolant.

Pattern shape design for ceramic ware: an integrated solution

This paper presents an integrated approach to producing a two-dimensional decorative pattern for ceramic bone china cast ware components. The novelty of the approach is to integrate the approximate unwrapping of a generally non-developable surface with the constraints of the manufacturing process. The key components of the solution are an unwrapping algorithm to determine the shape of the pattern from the CAD model and an estimation of the scaling factors for the mould to compensate for shrinkage in drying and firing. The approach produces patterns that are at least as good as the traditional manual approach, which distort less when wrapped onto the final ceramic ware. Furthermore, because the unwrapping is produced from the CAD definition, the pattern design can be completed much earlier in the manufacturing process with a reduction in the lead time.

Improved surface bounds based on derivatives

Abstract The surface bounds of Filip et al. (1986) and Sheng and Hirsch (1992) are improved by considering the geometric, rather than the parametric, displacement of the surface from its linear interpolant (Barley, 1992).

Executive-centred system design for CAD applications

Abstract The paper discusses the development of a system architecture that is suitable for CAD applications. A generic structure is described that encompasses the interactions of the major components of the system. An object-oriented data structure and database-management system are also considered, and, finally, the application of the overall structure to a practical CAD problem is described with the software implementation details.