Offset Approximation Research Articles

Curvature continuous offset approximation based on circle approximation using quadratic Bézier biarcs

We present a method for G 2 end-point interpolation of offset curves using rational Bézier curves. The method is based on a G 2 end-point interpolation of circular arcs using quadratic Bézier biarcs. We also prove the invariance of the Hausdorff distance between two compatible curves under convolution. Using this result, we obtain the exact Hausdorff distance between an offset curve and its approximation by our method. We present the approximation algorithm and give numerical examples.

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C 1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.

One-sided offset approximation of freeform curves for interference-free NURBS machining

Generating valid tool path curves in NURBS form is important in realizing an efficient NURBS machining. In this paper, a method for computing one-sided offset approximations of freeform curves with NURBS format as tool paths is presented. The approach first uses line segments to approximate the progenitor curve with one-sided deviations. Based on the obtained line approximating curve and its offsets, a unilateral tolerance zone (UTZ) is constructed subsequently. Finally, a C 1 -continuous and completely interference-free NURBS offset curve is generated within the UTZ to satisfy the required tolerance globally. Since all of the geometric computations involved are linear, the proposed method is efficient and robust. Interference-free tool path generation thus can be achieved in NURBS based NC machining.

Rational offset approximation of rational Bézier curves

The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve the same polynomial or rational polynomial representations, it arouses difficulty in applications. Thus approximation methods have been introduced to solve this problem. In this paper, it has been pointed out that the crux of offset curve approximation lies in the approximation of parametric speed. Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented.

A novel approach for computing C2-continuous offset of NURBS curves

Computing offset curves is an important geometric operation in areas of CAD/CAM, robotics, cam design and many industrial applications. In this paper, an algorithm for computing offsets of NURBS curves using C2-continuous B-spline curves is presented. The progenitor curve in database is initially approximated by a line-fitting curve, and then the exact offset of this line-fitting curve is introduced as an initial offset. Based on the initial offset and a set of selected knots, an intended C2-continuous B-spline curve is subsequently constructed. The method uses a new error-measuring scheme, which is based on the convex hull property of Bezier curves and the idea of cumulative errors, to calculate the global error bound of offset approximation. The method obtains offset curves with C2 continuity and guarantees that the actual error bound is precisely within the prescribed tolerance. In addition, it also allows one to selectively parametrize the offset curve.

Qualitative and quantitative comparisons of B-spline offset surface approximation methods

Surface offset is one of the most useful operations in Computer Aided Geometric Design (CAGD). However, an implementation of this operator is not trivial primarily because the offset surface, in general, does not have the same representation as the original surface. Hence, it is difficult or impossible to represent an exact offset in a system with limited surface forms. For this reason, some CAGD surface offset operators produce results that are, at times, unsatisfactory. In this article, we discuss surface offset approximation methods in B-spline environment: both the original surfaces and their offsets are B-spline surfaces. This article summarizes research contained in the first named author's PhD thesis.

Computing offsets of NURBS curves and surfaces

This paper presents algorithms for computing offsets of NURBS curves and surfaces. The basic approach consists of four steps: (1) recognition of special curves and surfaces; (2) sampling the offset curve or surface based on bounds on second derivatives; (3) interpolating these points; and (4) removing all unwanted knots using the offset tolerance. The method provides a good handle on error control and results in the fewest number of control points compared to all published work. It also allows one to control the degree and the parametrization of the offset approximation.

On: “Suppression of sea‐floor‐scattered energy using a dip‐moveout approach—Application to the mid‐ocean ridge environment” by G. M. Kent, I. I. Kim, A. J. Harding, R. S. Detrick, and J. A. Orcutt (May‐June 1996 GEOPHYSICS, 61, 821–834)

In their paper, Kent et al. (1996) present an excellent case history of the use of dip moveout (DMO) and velocity‐filtering in the common midpoint (CMP) domain for the suppression of out‐of‐plane arrivals scattered from a deep sea‐floor. However, they imply that as a result of a “small offset approximation” the use of DMO in this way is limited to surveys recorded in water depths of at least a few kilometers with conventional streamer offsets. This is incorrect. I argue here that the application of DMO will reduce to water velocity the stacking velocity of arrivals scattered from the upper surface of the seafloor without any restriction on water depth. Furthermore, I argue that this use of DMO is simply an example of the equivalence between 2-D and 3-D DMO for marine surveys where all source‐receiver azimuths are equal, and that no “small offset approximation” is required. I first present a counter‐example to the claim of Kent et al. (1996) that seafloor scattering cannot be suppressed using DMO in shallow water, and then consider in more detail their argument for the application of DMO to out‐of‐plane scattering. In the discussion that follows, I only consider DMO in the context of a constant velocity medium.

A Symbolic Approach to Freeform Parametric Surface Blends

This paper presents a symbolic approach to the computation of blend surfaces across piecewise polynomial and rational surfaces. Curves in the parameter space of the primary surfaces can be specified as the rail curves, also known as trimlines, of the blend. Several techniques which require various levels of user interaction are presented for defining the cross-boundary tangent curves along the rail curves. The resulting blend is represented as a polynomial surface having tangent plane continuity with the primary surfaces to an accuracy bounded only by the machine precision. Also presented is a normalization method that approximates unit vector fields, an approach that might benefit other applications such as offset approximation and animation curve construction. © 1997 by John Wiley & Sons, Ltd.

A Symbolic Approach to Freeform Parametric Surface Blends

This paper presents a symbolic approach to the computation of blend surfaces across piecewise polynomial and rational surfaces. Curves in the parameter space of the primary surfaces can be specified as the rail curves, also known as trimlines, of the blend. Several techniques which require various levels of user interaction are presented for defining the cross-boundary tangent curves along the rail curves. The resulting blend is represented as a polynomial surface having tangent plane continuity with the primary surfaces to an accuracy bounded only by the machine precision. Also presented is a normalization method that approximates unit vector fields, an approach that might benefit other applications such as offset approximation and animation curve construction. © 1997 by John Wiley & Sons, Ltd.

Comparing offset curve approximation methods

Offset curves have diverse engineering applications, spurring extensive research on various offset techniques. In a paper on offset curve approximation (Lee et al., 1996), we suggested a new approach based on approximating the offset circle instead of the offset curve itself. To demonstrate the effectiveness of this approach, we compared it extensively with previous methods. To our surprise, Tiller and Hanson's (1984) simple method outperforms other methods for offsetting (piecewise) quadratic curves, even though its performance is not as good for high degree curves. The experimental results revealed other interesting facts, too. Had these details been reported several years ago, we believe offset approximation research might have developed somewhat differently. This article is intended to fill an important gap in the literature. We conducted qualitative as well as quantitative comparisons employing various contemporary offset approximation methods for freeform curves in the plane. We measured the efficiency of the offset approximation in terms of the number of control points generated while making the approximations within a prescribed tolerance.

ERROR BOUNDED VARIABLE DISTANCE OFFSET OPERATOR FOR FREE FORM CURVES AND SURFACES

Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the control polygon/mesh and then attempts to estimate the error of the approximated offset over a region. Most of the current offset algorithms estimate the error using a finite set of samples taken from the region and therefore can not guarantee the offset approximation is within a given tolerance over the whole curve or surface. This paper presents new methods to globally bound the error of the approximated offset of freeform curves and surfaces and then automatically derive new approximations with improved accuracy. These tools can also be used to develop a global error bound for a variable distance offset operation and to detect and trim out loops in the offset.

Expressive brush strokes

Expressive brush strokes are modeled using a technique based on variable offset approximation of uniform cubic B-splines. The trajectory of a brush stroke is represented as a 3D cubic B-spline and each bristle as a 3D offset cubic B-spline of this trajectory. The first two coordinates are the spatial position coordinates and the third coordinate represents the shade of each bristle. This technique facilitates the process of inputting data, simplifies the computation task, and provides some advantages in the animation process of brush strokes.

Self-intersection of an offset surface

A method is described for obtaining the self-intersection curves of a collapsed offset surface. The method comprises two essential processes. The first is to detect the self-intersection points of a collapsed offset surface by successive unidimensional searches. The second is to calculate self-intersection curves by numerical integration by using some of the points obtained in the first process as initial points. All offset surface data needed is directly generated from the original surface without offset approximation. This method is applied to some collapsed offset surfaces whose original surfaces are defined by a B-spline.

Offset approximation of uniform B-splines

A simple and efficient method is developed for generating offset curves interactively using a set of control knots for a uniform cubic B-spline.

LARGE‐OFFSET APPROXIMATIONS IN THE MODIFIED CAGNIARD METHOD FOR COMPUTING SYNTHETIC SEISMOGRAMS: A SURVEY1

ABSTRACTLarge horizontal offset and large vertical offset approximations in the modified Cagniard method for computing synthetic seismograms in a horizontally layered model of the earth are derived. They apply to each of the generalized‐ray constituents into which the seismic wave motion is decomposed. For the results applying to large horizontal offset, which are known in the seismological literature, a simplified derivation is given. The results for large vertical offset, which are of particular interest to vertical seismic profiling, are new. The use of the large‐offset approximations, both horizontal and vertical, leads to a considerable reduction in computation time for synthetic seismograms compared with the full three‐dimensional version of the modified Cagniard method.

The approximation of non-degenerate offset surfaces

This paper describes algorithms for approximating the offsets to general piecewise parametric surfaces by networks of bicubic patches. The surface to be offset may be composed of arbitrary parametric patches, provided precise tangent continuity obtains across all patch boundaries and the surface metric is everywhere non-singular (i.e., a unique surface normal is defined). The approximation scheme consists of three stages: (1) A differential surface analysis is performed to ascertain the extremum principal radii of curvature for each patch — this constrains the offset magnitude that may be specified without degeneration of the offset. (2) The parametric domain of each patch is sub-divided, and a bicubic approximant to the offset for each sub-domain is computed. (3) A tolerance analysis is performed on each patch of the offset approximation to evaluate its extremum deviations from the true offset. The accuracy of the offset approximation increases with the degree of subdivision of the parametric domain of each patch. Fractional errors in the range 10 −2 to 10 −3 are typical for two- or three- fold sub-division of each parametric variable (four- and nine- fold area division respectively), provided the surface is smooth and its smallest concave radius of curvature considerably exceeds the specified offset magnitude. A fully automatic tolerance-based surface offset capability may be developed by providing feedback between stages (3) and (2), successive degrees of parametric sub-division being determined by the errors from prior approximations until the desired accuracy is achieved.