Blending Surfaces Research Articles

ELIMINATING EXTRANEOUS SOLUTIONS IN CURVE AND SURFACE OPERATIONS

We study exact representations for offset curves and surfaces, for equal-distance curves and surfaces, and for fixed- and variable-radius blending surfaces. The representations are systems of nonlinear equations that define the curves and surfaces as natural projections from a higher-dimensional space into 3-space. We show that the systems derived by naively translating the geometric constraints defining the curves and surfaces can entail degeneracies that result in additional solutions that have no geometric significance. We characterize these extraneous solution points geometrically, and then augment the systems with auxiliary equations of a uniform structure that exclude all extraneous solutions. Thereby, we arrive at representations that capture the geometric intent of the curve and surface definitions precisely.

Discrete bilinear blending and its application in rendering curved surfaces

Bilinear blended patches (BBPs), though somewhat limited in their application to geometric modelling, can prove to be very useful as drawing primitives when rendering curved surfaces. An algorithm is described to generate discrete bilinear blending surfaces using integer arithmetic only. The algorithm is proven to be correct with respect to its formally stated specifications.

Spherical and Circular Blending of Functional Surfaces

Abstract A method is presented for designing smooth blending between structural surfaces when only local alterations of the original surfaces are allowed. A simple kinematic analogy to the blending process is to represent the blending surface as the envelope of a sphere, possibly of variable radius, moving while in contact with the original surfaces. This scheme is then generalized to blending surfaces which can be represented by sweeping a (possibly variable) arc of circle. By design, this method guarantees continuity of the unit surface normal and of the normal radius of curvature along the common boundary with the blending surface, called the linkage curve.

The displacement method for implicit blending surfaces in solid models

To date, methods that blend solids, that is, B-rep or CSG models, with implicit functions require successive composition of the blending functions to handle an arbitrary solid model. The shape of the resulting surfaces depends upon the algebraic distances defined by these functions. To achieve meaningful shapes, previous methods have relied on blending functions that have a pseudo-Euclidean distance measure. These methods are abstracted, resulting in some general observations. Unfortunately, the functions used can exhibit unwanted discontinuities. A new method, the displacement form of blending, embeds the zero surface of the blending functions in a form for which algebraic distance is C 1 continuous in the entire domain of definition. Characteristics of the displacement form are demonstrated using the superelliptic blending functions. Intuitive and mathematical underpinnings are provided.

Blending parametric surfaces

A blending surface is a surface that smoothly connects two given surfaces along two arbitrary curves, one on each surface. This is particularly useful in the modeling operations of filleting a sharp edge between joining surfaces or connecting disjoint surfaces. In this paper we derive a new surface formulation for representing surfaces which are blends of parametric surfaces. The formulation has the advantage over the traditional rational polynomial approach in that point and normal values have no gaps between the blending surface and the base surfaces. Shape control parameters that control the cross-sectional shape of the blending surface are also available. In addition, the base surfaces are not restricted to any particular type of surface representation as long as they are parametrically defined and have a well-defined and continuous normal vector at each point. The scheme is extensible to higher orders of geometric continuity, although we concentrate on G 1 .

Quadratic blending surfaces for complex corners

The potential method for blending implicitly defined surfaces is extended in several ways. First, the method of blending an arbitrary number of surfaces to form a convex corner is shown. Second, blending surfaces are constructed for all cases where three surfaces meet transversely. In all cases only quadratic blendings are needed. These two extensions constitute the first part of the paper. In the second part a construction is given for a very general class of corners, which contains the corners studied in the first part. The general method is not a straight-forward extension of the constructions in the first part. In particular, the blendings produced by the special-purpose methods of the first part are more symmetric. All constructions rely on some simple properties of quadratic hypersurfaces and can easily be automated. The resulting blendings enjoy some highly desirable properties: they are easily computed, have simple descriptions and are well suited for the subsequent computations which may need to be done — for instance, for rendering purposes. Detailed numerical examples are included in both parts of the paper.

スウィープ補間曲面ソリッドのＣＳＧ表現法に関する研究

スウィープ補間曲面ソリッドのＣＳＧ表現法に関する研究

The geometry of projective blending surfaces

Blending surfaces smoothly join two or more primary surfaces that otherwise would intersect in edges. We outline the potential method for deriving blending surfaces, and explain why the method needs to be considered in projective parameter space, concentrating on the case of blending quadrics. Let W be the quadratic polynomial substituted for the homogenizing variable of parameter space. We show that a blending surface derived in projective parameter space is the projective image of a different blending surface derived in affine parameter space, provided that W = U 2 for some linear U. All blending surfaces may therefore by classified on basis of the projective classification of W.

Piecewise quadric blending of implicitly defined surfaces

A method for blending implicitly defined surfaces is proposed. The function defining the blending surface is constructed from the functions defining the surfaces to be blended with the aid of a piecewise quadric function.

Direct least-squares fitting of algebraic surfaces

In the course of developing a system for fitting smooth curves to camera input we have developed several direct (i.e. noniterative) methods for fitting a shape (line, circle, conic, cubic, plane, sphere, quadric, etc.) to a set of points, namely exact fit, simple fit, spherical fit, and blend fit. These methods are all dimension-independent, being just as suitable for 3D surfaces as for the 2D curves they were originally developed for.Exact fit generalizes to arbitrary shapes (in the sense of the term defined in this paper) the well-known determinant method for planar exact fit. Simple fit is a naive reduction of the general overconstrained case to the exact case. Spherical fit takes advantage of a special property of circles and spheres that permits robust fitting; no prior direct circle fitters have been as robust, and there have been no previous sphere fitters. Blend fit finds the best fit to a set of points of a useful generalization of Middleditch-Sears blending curves and surfaces, via a nonpolynomial generalization of planar fit.These methods all require ( am + bn ) n 2 operations for fitting a surface of order n to m points, with a = 2 and b = 1/3 typically, except for spherical fit where b is larger due to the need to extract eigenvectors. All these methods save simple fit achieve a robustness previously attained by direct algorithms only for fitting planes. All admit incremental batched addition and deletion of points at cost an 2 per point and bn 3 per batch.

Quadratic blending surfaces

A blending surface is a surface that smoothly connects two primary surfaces. Usually such surfaces are intended to smooth edges in which the primary surfaces would intersect otherwise. Blending surfaces are important to geometric modelling since virtually all solid objects designed and manufactured have them. The paper investigates the generality of a method for deriving blending surfaces automatically, given the primary surfaces are quadrics. The main result is the following theorem. Given two quadric surfaces g = 0 and h=0 in general position, and given a nondegenerate, quartic space curve on each surface: • • There exist quartic blending surfaces f = 0 tangent to the quadrics g and h in the respective space curves if and only if the curves of tangency lie on a common quadric k = 0. • • All such blending surfaces satisfy the equation g·h − μk 2 = 0 , where μ is a number. • • All such blending surfaces may be derived using the derivation method described elsewhere in greater detail. Applications of this theorem are also discussed.

Protection of reservoirs

The geometry of cutting flutes and the surfaces of end mills is one of the crucial parameters affecting the quality of the machining in the case of end milling. These are usually represented by two-dimensional models. This paper describes in detail the methodology to model the geometry of a flat end mill in terms of three-dimensional parameters. The geometric definition of the end mill is developed in terms of surface patches; flutes as helicoidal surfaces, the shank as a surface of revolution and the blending surfaces as bicubic Bezier and biparametric sweep surfaces. The proposed model defines the end mill in terms of three-dimensional rotational angles rather than the conventional two dimensional angles. To validate the methodology, the flat end milling cutter is directly rendered in OpenGL environment in terms of three-dimensional parameters. Further, an interface is developed that directly pulls the proposed three-dimensional model defined with the help of parametric equations into a commercial CAD modeling environment. This facilitates a wide range of downstream technological applications. The modeled tool is used for finite element simulations to study the cutting flutes under static and transient dynamic load conditions. The results of stress distribution (von mises stress), translational displacement and deformation are presented for static and transient dynamic analysis for the end mill cutter flute and its body. The method described in this paper offers a simple and intuitive way of generating high-quality end mill models for use in machining process simulations. It can be easily extended to generate other tools without relying on analytical or numerical formulations.