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1.
G2 continuity of free-form surfaces is sometimes very important in engineering applications. The conditions for G2 continuity to connect two Bézier patches were studied and methods have been developed to ensure it. However, they have some restrictions on the shapes of patches of the Bézierpatch formulation. Gregory patch is a kind of free-form surface patch which is extended from Bézier patch so that four first derivatives on its boundary curves can be specified without restrictions of the compatibility condition. Several types of Gregory patches have been developed for intergral, rational, and NURBS boundary curves. In this paper, we propose some intergral boundary Gregorytype patches bounded by cubic and quartic curves for G2 continuity.  相似文献   

2.
A surface interpolation method for meshes of cubic curves is described. A mesh of cubic curve is constructed between the given vertices. This mesh is filled with Bézier patches, so that the surface is represented as a union of geometrically continuous bicubic quadrilateral and/or quartic triangular Bézier patches. The method is local and uses Farin's [Farin '83] conditions of G1 continuity between patches. The procedure for finding the needed control points of the Bézier patches is simple and efficient.  相似文献   

3.
The problem of interpolating a free form curve network with irregular topology is investigated in order to create a curvature continuous surface. The spanning curve segments are parametric quintic polynomials, the interpolating surface elements are biquintic Gregory patches. A necessary compatibility condition is formulated and proved which need to be satisfied at each node of the curve network. Constraints are derived for assuring G2 continuity between biquintic Gregory patches, which share a common side or a common corner point. The above conditions still leave certain geometric freedom for defining the entire G2 surface, so following some analysis a particular construction is presented, by which after computing the principle curvatures at each node the free parameters are locally set for each interpolating Gregory patch.  相似文献   

4.
High accuracy geometric Hermite interpolation   总被引:22,自引:0,他引:22  
We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C1 bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.  相似文献   

5.
The Hermite interpolation problem in the plane considered here is to join two points and to match given unit tangent vectors and signed curvatures at the two points with various G2 curves consisting of a pair of spirals. The rotation of the tangent vector of the interpolating curve from one point to the other is restricted to being less than π. The necessary and sufficient conditions for the existence of each of the various curves are given.  相似文献   

6.
Local control of interval tension using weighted splines   总被引:20,自引:0,他引:20  
Cubic spline interpolation and B-spline sums are useful and powerful tools in computer aided design. These are extended by weighted cubic splines which have tension controls that allow the user to tighten or loosen the curve on intervals between interpolation points. The weighted spline is a C1 piecewise cubic that minimizes a variational problem similar to one that a C2 cubic spline minimizes. A B-spline like basis is constructed for weighted splines where each basis function is nonnegative and nonzero only on four intervals. The basis functions sum up identically to one, thus curves generated by summing control points multiplied by the basis functions have the convex hull property. Different weights are built into the basis functions so that the control point curves are piecewise cubics with local control of interval tension. If all weights are equal, then the weighted spline is the C2 cubic spline and the basis functions are B-splines.  相似文献   

7.
Consumer products such as ping-pong paddles, can be designed by blending circles. To be visually pleasing it is desirable that the blend be curvature continuous without extraneous curvature extrema. Transition curves of gradually increasing or decreasing curvature between circles also play an important role in the design of highways and railways. Recently planar cubic and Pythagorean hodograph quintic spiral segments were developed and it was demonstrated how these segments can be composed pairwise to form transition curves that are suitable for G2 blending. It is now shown that a single cubic curve can be used for blending or as a transition curve with the guarantee of curvature continuity and fairness. Use of a single curve rather than two segments has the benefit that designers and implementers have fewer entities to be concerned with.  相似文献   

8.
We begin by characterizing notions of geometric continuity represented by connection matrices. Next we present a set of geometric properties that must be satisfied by all reasonable notions of geometric continuity. These geometric requirements are then reinterpreted as an equivalent collection of algebraic constraints on corresponding sets of connection matrices. We provide a general technique for constructing sets of connection matrices satisfying these criteria and apply this technique to generate many examples of novel notions of geometric continuity. Using these constraints and construction techniques, we show that there is no notion of geometric continuity between reparametrization continuity of order 3, (G3), and Frenet frame continuity of order 3, (F3); that there are several notions of geometric continuity between G4 and F4; and that the number of different notions of geometric continuity between Gn and Fn grows at least exponentially with n.  相似文献   

9.
We present an efficient algorithm for computing the Bézier points of a generalized cubic β-spline curve and show the connection with multiple knot insertion. We also consider the inverse problem of determining the β-spline vertices of a composite G2 Bézier curve. Finally, we briefly discuss how to construct the Bézier net of a tensor product β-spline surface.  相似文献   

10.
In this paper new methods of discretization (integer approximation) of algebraic spatial curves in the form of intersecting surfaces P(x, y, z) = 0 and Q(x, y, z) = 0 are analyzed.

The use of homogeneous cubical grids G(h3) to discretize a curve is the essence of the method. Two new algorithms of discretization (on 6-connected grid G6c(h3) and 26-connected grid G26(h3)) are presented based on the method above. Implementation of the algorithms for algebraic spatial curves is suggested. The elaborated algorithms are adjusted for application in computer graphics and numerical control of machine tools.  相似文献   


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