The Geometric Continuity of Rational Bezler Triangular Surfaces

The problems of geometric continuity for rational Bezter surfaces are discussed. Concise conditions of first order and second order geometric continuity for rational triangular Bézier surfaces are given. Meanwhile,a geometric condition for smoothness between adjacent rational Bézier surfaces and the transformation formulae between rational triangular patches and rational rectangular patches are obtained.

The Effects of Shape Parameters for Surface Joining with High Order Geometric Continuity

The necessary and sufficient conditions and an algorithm to reach continuity between adjacent Bézier patches are presented. The effects of shape parameters for surface connection of G4 geometric continuity are investigated in detail. The algorithm can be generalized directly to the case of surface joining with higher order geometric continuity. It has important applications in surface modeling and surface joining.

Global Continuity Adjustment and Local Shape Optimization Technique for Complex Trimmed Surface Model

Smoothly stitching multiple surfaces mainly represented by B-spline or NURBS together is an extremely important issue in complex surfaces modeling and reverse engineering. In recent years, a lot of progress has been made in smooth join of non-trimmed surface patches, while there has been seldom research on smoothly stitching trimmed surface patches together. This paper studies the problem of global continuity adjustment, damaged hole repair and local shape optimization for complex trimmed surface model, and presents a uniform scheme to deal with continuity adjustment of trimmed surfaces and geometric repair of local broken region. Constrained B-spline surface refitting technique and trim calculation are first utilized to achieve global G^1 continuity, and then local shape optimization functional is adopted to reduce fitting error and improve local quality of refitted surface patch. The proposed approach is applied to a discontinuity ship hull surface model with an irregular hole, and the result demonstrates the validation of our method. Furthermore, on the premise of global continuity, the proposed locally repairing damaged surface model provides a better foundation for following research work, such as topology recovery technique for complex surface model after geometric repair.

G^2 Continuity Algorithms between Adjacent NURBS Patches along Common Cubic Boundary Curve

Non-uniform rational B-spline (NURBS) curves and surfaces are becoming increasingly widespread. The author have explored G^1 continuity condition between adjacent NURBS surface patches along common cubic boundary curve. On the basis of the research performed, this paper presents a G^2 continuity condition between adjacent NURBS patches along common cubic boundary curve and deduces a specific algorithm for contro1 points and weights of NURBS patch. For making another NURBS patch and one given NURBS patch to attain G^2, according to algorithm condition, one can adjust another patch control points and weights. It is much more convenient for engineers to apply.

PRACTICAL G^2 CONTINUITY ALGORITHMS BETWEEN ADJACENT NURBS PATCHES

Non-uniform rationa1 B-spline (NURBS) curves and sdrices are becomingincreasingly widespread. 'The authors have explored G1 continuity condition between adja-cent NURBS surface patches along common boundary curve. This paper presents a G2continuity condition between adjacent NURBS patches along common quadratic boundarycurve and deduces a specific algorithm for control Points and weights of NURBS patch.For making another NURBS patch and one given NURBS patch to attain G2, according toalgorithms condition, one can adjust another patch control ponts and weights. It is muchmore convenient for engineers to apply.

Bézier representation of geometrically continuous splines

As an intrinsic measure of smoothness, geometric continuity is an important problem in the fields of computer aided geo- metric design. It can afford more degrees of freedom for manipulating the shape of curve. However, piecewise polynomial functions of geometrically continuous splines are difficult to be constructed. In this paper, the conversion matrix between geometrically con- tinuous spline basis functions and Bezier representation is analyzed. Based on this, construction of arbitrary degree geometrically continuous spline basis functions can be translated into a solution of linear system of equations. The original construction of geomet- rically continuous spline is simplified.

Cubic Algebraic Spline Curves Design

In this paper we propose a construction method of the planar cubic algebraic splinecurve with endpoint interpolation conditions and a specific analysis of its properties. Thepiecewise cubic algebraic curve has G2 continuous contact with the control polygon at twoendpoints and is G2 continuous between each segments of itself. The process of this method issimple and clear, and provides a new way of thinking to design implicit curves.

Wu Wen-tsün Formulae for the Blending of Pipe Surfaces

One of the central questions in CAGD is blending of pipe surfaces. Wu Wen-tsun[1]studied the problem by using the characteristic set method and derived a sufficient andnecessary condition for the existence of a GC1 blending cubic surface of two cylinders whose

G^0 and G^1 connection between two adjacent B-spline surfaces

In this paper, the smooth connection between two B-spline surfaces is discussed. First, a brief proof of some simple sufficient conditions of Go and G1 continuity is given. On this basis, a novel method for Go or G1 connection between two adjacent B-spline surfaces is presented. A reparameterization step is firstly taken for one of the surfaces such that they have the same parameterization in v direction, then, adjust their boundary control vertices to make them Go or Gl connected. The GI connection parameter is determined by an optimization problem. Compared with the existed methods, our method is simple and easy to be used in practice.

Parametric Selection and Continual Combination of Building Surface

A method to reparametrize G retional curve to obtain a C^1 curve is given. A practical G^1 continual connective between adjacent NURUS patches along common guadratic boundary curve is presented in this paper, and a specific algorithm for control points and weights of NURBS patches is discussed.

The G^3 spline basis functions

The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters＇ values. It makes the equation＇s solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.

Knot Insertion Algorithms for ECT B-spline Curves

Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, also the segmentation of the curve can be rea- lized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties .This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized P61ya poly- nomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear com- bination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.

THE APPLICATION OF G^1 JOIN OF RECTANGULAR AND TRIANGULAR BEZIER PATCHES IN SURFACE MODELING

An application of techniques is presented to construct G￣1 smooth surfaces by using acombination of the rectangular and triangular Bezier patches of degree as low as possible. TheG￣1 smooth surfaces have the local property and interpolate the given data and inherit thetopology imposed by the given space convex quadrilateral partition and triangulation. The papergeneralizes current approaches for assembling of rectangular and triangular patches.

B-SPLINE SURFACE SMOOTH JOINING

Optimization techniques are being applied to solve the problems of surface interpolation, approximation, smooth joining and fairing, aiming at corresponding objective functions. This paper focuses on the construction of fair surface interpolating the given mesh of curved boundaries with G 2 adjustment at comers and G 1, G 2 smoothness between adjacent patches. Many papers on surface blending have been presented, but almost all of them are restricted to the discussion of Bezier patches, there are no good results for B-spline surface. This paper gives a solution to the B-spline surface, allowing the surface to degenerate at comer in and have different parameterization along the common boundary of two patches.

CAGD AND NC MILLING OF THE CASTING MOULD OF HELICAL INTAKE PORT

A new method of the design and machining the casting moulds of helical intake ports of diesel engines is presented based on CAD / CAM technique. The problems about the smooth continuity between patches, the cutter interference and the determination of the mould split curve are discussed in details.The CAGD system of the intake port has been developed. and the casting mould is machined sucessfully on NC machine.

G^1 Continuous Condition of B-Spline Surfaces with Double Knots

The conditions for G1 continuity between two adjacent bicubic B-spline surfaces with double interior knots along their common boundary curve are obtained in this paper, which are directly represented by the control points of the two B-spline surfaces. As stated by Shi Xi-quan and Zhao Yan, a local scheme of constructing G1 continuous B-spline surface models with single interior knots does not exist; we may achieve a local scheme of (true) G1 continuity over an arbitrary B-spline surface network using these conditions.

Approximate Degree Reduction of Bezier Curves

Degree reduction of parametric curves and surfaces is an important process in data communication between CAD systems. The degenerate condition of Bezier curves and the constrained optimization method are used to develop a new degree reduction method for Bezier curves. An error analysis of the degree reduction is also given. The degree reduction scheme is combined with a subdivision algorithm to generate lower degree approximations which are within some preset error tolerance of the prescribed Bezier curve. Geometric continuity between adjacent curve segments is also considered in the subdivision/degree reduction process.

Storage-efficient reconstruction framework for planar contours

A storage-efficient reconstruction framework for cartographic planar contours is developed.With a smaller number of control points,we aim to calculate the area and perimeter as well as to reconstruct a smooth curve.The input data forms an oriented contour,each control point of which consists of three values:the Cartesian coordinates(x,y)and tangent angleθ.Two types of interpolation methods are developed,one of which is based on an arc spline while the other one is on a cubic Hermite spline.The arc spline-based method reconstructs a G1 continuous curve,with which the exact area and perimeter can be calculated.The benefit of using the Hermite spline-based method is that it can achieve G2 continuity on most control points and can obtain the exact area,whereas the resulting perimeter is approximate.In a numerical experiment for analytically defined curves,more accurate computation of the area and perimeter was achieved with a smaller number of control points.In another experiment using a digital elevation model data,the reconstructed contours were smoother than those by a conventional method.

Interpolation by G^(2) Quintic Pythagorean-Hodograph Curves

In this paper,the G^(2) interpolation by Pythagorean-hodograph(PH)quintic curves in R^(d),d≥2,is considered.The obtained results turn out as a useful tool in practical applications.Independently of the dimension d,they supply a G^(2) quintic PH spline that locally interpolates two points,two tangent directions and two curvature vectors at these points.The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns.Although several solutions might exist,the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case.The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method.Numerical examples confirm the efficiency of the proposed method.

CAN A CUBIC SPLINE CURVE BE G^(3)

This paper proposes a method to construct an G^(3)cubic spline curve from any given open control polygon.For any two inner Bezier points on each edge of a control polygon,we can de ne each Bezier junction point such that the spline curve is G2-continuous.Then by suitably choosing the inner Bezier points,we can construct a global G^(3)spline curve.The curvature combs and curvature plots show the advantage of the G^(3)cubic spline curve in contrast with the traditional C2 cubic spline curve.