Xiao-Diao Chen

Computing the minimum distance between a point and a NURBS curve

A new method is presented for computing the minimum distance between a point and a NURBS curve. It utilizes a circular clipping technique to eliminate the curve parts outside a circle with the test point as its center point. The radius of the elimination circle becomes smaller and smaller during the subdivision process. A simple condition for terminating the subdivision process is provided, which leads to very few subdivision steps in the new method. Examples are shown to illustrate the efficiency and robustness of the new method.

Computing the Hausdorff distance between two B-spline curves

This paper presents a geometric pruning method for computing the Hausdorff distance between two B-spline curves. It presents a heuristic method for obtaining the one-sided Hausdorff distance in some interval as a lower bound of the Hausdorff distance, which is also possibly the exact Hausdorff distance. Then, an estimation of the upper bound of the Hausdorff distance in an sub-interval is given, which is used to eliminate the sub-intervals whose upper bounds are smaller than the present lower bound. The conditions whether the Hausdorff distance occurs at an end point of the two curves are also provided. These conditions are used to turn the Hausdorff distance computation problem between two curves into a minimum or maximum distance computation problem between a point and a curve, which can be solved well. A pruning technique based on several other elimination criteria is utilized to improve the efficiency of the new method. Numerical examples illustrate the efficiency and the robustness of the new method.

Cubic B-spline curve approximation by curve unclamping

A new approach for cubic B-spline curve approximation is presented. The method produces an approximation cubic B-spline curve tangent to a given curve at a set of selected positions, called tangent points, in a piecewise manner starting from a seed segment. A heuristic method is provided to select the tangent points. The first segment of the approximation cubic B-spline curve can be obtained using an inner point interpolation method, least-squares method or geometric Hermite method as a seed segment. The approximation curve is further extended to other tangent points one by one by curve unclamping. New tangent points can also be added, if necessary, by using the concept of the minimum shape deformation angle of an inner point for better approximation. Numerical examples show that the new method is effective in approximating a given curve and is efficient in computation.

Improved Algebraic Algorithm on Point projection for B´eziercurves

This paper presents an improved algebraic pruning method for point projection for Bezier curves. It first turns the point projection into a root finding problem, and provides a simple but easily overlooked method to avoid finding invalid roots which is obviously irrelative to the closest point. The continued fraction method and its expansion are utilized to strengthen its robustness. Since NURBS curves can be easily turned into Bezier form, the new method also works with NURBS curves. Examples are presented to illustrate the efficiency and robustness of the new method.

Computing minimum distance between two implicit algebraic surfaces

The minimum distance computation problem between two surfaces is very important in many applications such as robotics, CAD/CAM and computer graphics. Given two implicit algebraic surfaces, a new method based on the offset technique is presented to compute the minimum distance and a pair of points where the minimum distance occurs. The new method also works where there are an implicit algebraic surface and a parametric surface. Quadric surfaces, tori and canal surfaces are used to demonstrate our new method. When the two surfaces are a general quadric surface and a surface which is a cylinder, a cone or an elliptic paraboloid, the new method can produce two bivariate equations where the degrees are lower than those of any existing method.

Free-Form Deformation with Rational DMS-Spline Volumes

In this paper, we propose a novel free-form deformation (FFD) technique, RDMS-FFD (Rational DMS-FFD), based on rational DMS-spline volumes. RDMS-FFD inherits some good properties of rational DMS-spline volumes and combines more deformation techniques than previous FFD methods in a consistent framework, such as local deformation, control lattice of arbitrary topology, smooth deformation, multiresolution deformation and direct manipulation of deformation. We first introduce the rational DMS-spline volume by directly generalizing the previous results related to DMS-splines. How to generate a tetrahedral domain that approximates the shape of the object to be deformed is also introduced in this paper. Unlike the traditional FFD techniques, we manipulate the vertices of the tetrahedral domain to achieve deformation results. Our system demonstrates that RDMS-FFD is powerful and intuitive in geometric modeling.

Computing the minimum distance between a point and a clamped B-spline surface

The computation of the minimum distance between a point and a surface is important for the applications such as CAD/CAM, NC verification, robotics and computer graphics. This paper presents a spherical clipping method to compute the minimum distance between a point and a clamped B-spline surface. The surface patches outside the clipping sphere which do not contain the nearest point are eliminated. Another exclusion criterion whether the nearest point is on the boundary curves of the surface is employed, which is proved to be superior to previous comparable criteria. Examples are also shown to illustrate efficiency and correctness of the new method.

Automatic G1 arc spline interpolation for closed point set

A method for generating an interpolation closed G1 arc spline on a given closed point set is presented. For the odd case, i.e. when the number of the given points is odd, this paper disproves the traditional opinion that there is only one closed G1 arc spline interpolating the given points. In fact, the number of the resultant closed G1 arc splines fulfilling the interpolation condition for the odd case is exactly two. We provide an evaluation method based on the arc length as well such that the choice between those two arc splines is made automatically. For the even case, i.e. when the number of the given points is even, the points are automatically moved based on weight functions such that the interpolation condition for generating closed G1 arc splines is satisfied, and that the adjustment is small. And then, the G1 arc spline is constructed such that the radii of the arcs in the spline are close to each other. Examples are given to illustrate the method.

Coincidence condition of two Bézier curves of an arbitrary degree

This paper discusses the coincidence condition of two Bezier curves of the same degree, which is necessary for a curve/curve intersection algorithm. It proves that two Bezier curves of the same degree are coincident with each other if and only if they have a coincident control polygon or they can be reparameterized into the same non-reparameterizable Bezier curve of a lower degree. A simple method is also provided for detecting the reparameterizable case where a Bezier curve can be reparameterized into a non-reparameterizable one of a lower degree. Graphical abstractDisplay Omitted HighlightsPresent coincidence conditions of two Bezier curves of the same degree.The resulting coincidence check is different from the point inversion technique.Our method is done by using equality conditions of explicit expressions.Coincidence check of two Bezier curves with partial overlapping is also discussed.

Conditions for the coincidence of two quartic Bézier curves

This paper presents efficient and necessary coincidence conditions for two quartic Bezier curves, i.e., either their two control polygons are coincident or the two curves can be reparameterized or degenerated into the same quadratic Bezier curve.