Computing medial axes of generic 3D regions bounded by B-spline surfaces

A new approach is presented for computing the interior medial axes of generic regions in R³ bounded by C⁽⁴⁾-smooth parametric B-spline surfaces. The generic structure of the 3D medial axis is a set of smooth surfaces along with a singular set consisting of edge curves, branch curves, fin points and six junction points. In this work, the medial axis singular set is first computed directly from the B-spline representation using a collection of robust higher order techniques. Medial axis surfaces are computed as a time trace of the evolving self-intersection set of the boundary under the the eikonal (grassfire) flow, where the bounding surfaces are dynamically offset along the inward normal direction. The eikonal flow results in special transition points that create, modify or annihilate evolving curve fronts of the (self-) intersection set. The transition points are explicitly identified using the B-spline representation. Evolution of the (self-) intersection set is computed by adapting a method for tracking intersection curves of two different surfaces deforming over generalized offset vector fields. The proposed algorithm accurately computes connected surfaces of the medial axis as well its singular set. This presents a complete solution along with accurate topological structure.     

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.     

The method of resolvents: A technique for the implicitization, inversion, and intersection of non-planar, parametric, rational cubic curves

Let P(t) be a non-planar, parametric, rational cubic curve. The method of resolvents is applied to: (1) construct three quadric surfaces whose intersection is equal to P(t) (implicitization); (2) solve for the parameter t as the ratio of two linear expressions in the coordinates x, y, z (inversion). The results of these two operations are then applied to construct an optimal, robust, intersection algorithm for any two non-planar rational cubic curves, and it is shown that two such curves can intersect in at most five points. Specializations of these results for non-planar, integral, cubic curves are derived, and extensions of these techniques to non-planar, rational cubic, Bézier curves are also discussed.     

Quadratic approximation to plane parametric curves and its application in approximate implicitization

Expressing complex curves with simple parametric curve segments is widely used in computer graphics, CAD and so on. This paper applies rational quadratic B-spline curves to give a global C ¹ continuous approximation to a large class of plane parametric curves including rational parametric curves. Its application in approximate implicitization is also explored. The approximated parametric curve is first divided into intrinsic triangle convex segments which can be efficiently approximated with rational quadratic Bézier curves. With this approximation, we keep the convexity and the cusp (sharp) points of the approximated curve with simple computations. High accuracy approximation is achieved with a small number of quadratic segments. Experimental results are given to demonstrate the operation and efficiency of the algorithm.     

G 2 interpolation and blending on surfaces

We introduce a method for curvature-continuous (G ²) interpolation of an arbitrary sequence of points on a surface (implicit or parametric) with prescribed tangent and geodesic curvature at every point. The method can also be used forG ² blending of curves on surfaces. The interpolation/blending curve is the intersection curve of the given surface with a functional spline (implicit) surface. For the construction of blending curves, we derive the necessary formulas for the curvature of the surfaces. The intermediate results areG ² interpolation/blending methods in IR².     

Plus curves and surfaces

The formulations for parametric curves and surfaces that are based on control points are revised to use control lines and control planes instead. Curves defined by control lines are called control-line curves or plus curves, and surfaces defined by control planes are called control-plane surfaces or plus surfaces; the plus implies that in addition to the control points, gradient vectors at the control points are used to design curves and surfaces. The new curve and surface formulations provide more flexibility than traditional formulations in geometric design. Properties of plus curves and surfaces are investigated and an application of plus surfaces in smooth parametric representation of polygon meshes is introduced.     

The Intersection of Two Ringed Surfaces and Some Related Problems

We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪_uC^u generated by a moving circle. Given two ringed surfaces ∪_uC^u₁ and ∪_vC^v₂, we formulate the condition C^u₁ ∩ C^v₂ ≠ ∅ (i.e., that the intersection of the two circles C^u₁ and C^v₂ is nonempty) as a bivariate equation λ(u, v)=0 of relatively low degree. Except for redundant solutions and degenerate cases, there is a rational map from each solution of λ(u, v)=0 to the intersection point C^u₁ ∩ C^v₂. Thus it is trivial to construct the intersection curve once we have computed the zero-set of λ(u, v)=0. We also analyze exceptional cases and consider how to construct the corresponding intersection curves. A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the ray-tracing of ringed surfaces. Surfaces of linear extrusion and surfaces of revolution reduce their respective intersection algorithms to simpler forms than those for ringed surfaces and ruled surfaces. In particular, the bivariate equation λ(u, v)=0 is reduced to a decomposable form, f(u)=g(v) or 6f(u)−g(v)6=|r(u)|, which can be solved more efficiently than the general case.     

基于活动仿射标架反求Nurbs曲线/曲面参数

Nurbs曲线/曲面在反求参数上的数值不稳定性,是Nurbs曲线/曲面的致命缺点.该文介绍了用于参数曲线/曲面求交的活动仿射标架(moving affine frame,简称MAF)方法.基于MAF方法的原理,提出了反求Nurbs曲线/曲面参数的一种新方法.该方法在数值稳定性和效率上均高于各种传统的迭代法,并已应用于商品化三维CAD系统GEMS 5.0.     

Marching along surfacesurface intersection curves with an adaptive step length

A method for generating points on the intersection of two C² smooth parametric surfaces is presented. In each generated point the tangent and the curvature of the intersection curve are obtained from the surface positions, first and second derivatives. Initial approximation of the next point lies on a parabola approximating the intersection curve in a vicinity of the last point found. The length of the parabola between the two points is evaluated so that its maximal deviation from the chord joining the points is not greater than a given deviation tolerance. The new point is relaxed to the intersection curve.