Kan-Le Shi

Anisotropic Delaunay Meshes of Surfaces

Anisotropic simplicial meshes are triangulations with elements elongated along prescribed directions. Anisotropic meshes have been shown well suited for interpolation of functions or solving PDEs. They can also significantly enhance the accuracy of a surface representation. Given a surface S endowed with a metric tensor field, we propose a new approach to generate an anisotropic mesh that approximates S with elements shaped according to the metric field. The algorithm relies on the well-established concepts of restricted Delaunay triangulation and Delaunay refinement and comes with theoretical guarantees. The star of each vertex in the output mesh is Delaunay for the metric attached to this vertex. Each facet has a good aspect ratio with respect to the metric specified at any of its vertices. The algorithm is easy to implement. It can mesh various types of surfaces like implicit surfaces, polyhedra, or isosurfaces in 3D images. It can handle complicated geometries and topologies, and very anisotropic metric fields.

Gn blending multiple surfaces in polar coordinates

This paper proposes a method of G^n blending multiple parametric surfaces in polar coordinates. It models the geometric continuity conditions of parametric surfaces in polar coordinates and presents a mechanism of converting a Cartesian parametric surface into its polar coordinate form. The basic idea is first to reparameterize the parametric blendees into the form of polar coordinates. Then they are blended simultaneously by a basis function in the complex domain. To extend its compatibility, we also propose a method of converting polar coordinate blending surface into N NURBS patches. One application of this technique is to fill N-sided holes. Examples are presented to show its feasibility and practicability.

Filling n -sided regions with G 1 triangular Coons B-spline patches

Filling n-sided regions is an essential operation in shape and surface modeling. Positional and tangential continuities are highly required in designing and manufacturing. We propose a method for filling n-sided regions with untrimmed triangular Coons B-spline patches, preserving G1 continuity exactly. The algorithm first computes a central point, a central normal, the central, and the corner derivative vectors. Then the region is split into n triangular areas by connecting the central point to each corner of the boundary. These inner curves and all cross-boundary derivatives are computed fulfilling G1 compatibility conditions. And finally, the triangular patches are generated in the Coons B-spline form, one boundary of which is regressed to the central vertex. Neither positional nor tangential error is introduced by this method. And only one degree elevation is needed.

Special Section on CAD/Graphics 2013: Projecting points onto planar parametric curves by local biarc approximation

This paper proposes a geometric iteration algorithm for computing point projection and inversion on planar parametric curves based on local biarc approximation. The iteration begins with initial estimation of the projection of the prescribed test point. For each iteration, we construct a biarc that locally approximates a segment on the original curve starting from the current projective point. Then we compute the projective point for the next iteration, as well as the parameter corresponding to it, by projecting the test point onto this biarc. The iterative process terminates when the projective point satisfies the required precision. Examples demonstrate that our algorithm converges faster and is less dependent on the choice of the initial value compared to the traditional geometric iteration algorithms based on single-point approximation.

The Transition Between Sharp and Rounded Features and the Manipulation of Incompatible Boundary in Filling n-sided Holes

N-sided hole filling plays an important role in vertex blending. Piegl and Tiller presented an algorithm to interpolate the given boundary and cross-boundary derivatives in B-spline form. To deal with the incompatible cases that their algorithm cannot handle, we propose an extension method to manipulate the transition between sharp and rounded features. The algorithm first patches n crescent-shaped extended surfaces to the boundary with G2 continuity to handle incompatibility problem in the corners. Then, we compute the inner curves and the corresponding cross-boundary derivatives fulfilling tangent and twist compatibilities. The generated B-spline Coons patches are G1-continuously connected exactly, and have epsilon-G1 continuity with the extended surfaces. Our method improves the continuity-quality of the shape and reduces the count of the inserted knots. It can be applied to all G0-continuous boundary conditions without any restrictions imposed on the boundary or cross-boundary derivatives. It generates better shapes than some popular industrial modeling systems on these incompatible occasions. Some examples underline its feasibility.

3D B-spline curve construction from orthogonal views with self-overlapping projection segments

Constructing B-spline curves from orthogonal views is a common operation in the process of car modelling. Existing reconstructing methods for 3D free-form curves cannot effectively handle the situation where the projection curves in the input views are self-overlapping, which appears frequently in practice. In this paper we focus on solving the problem above. We track the input views simultaneously and generate an ordered set of sampling space points. Then we reorganize the planar points in each view according to the sampling points. Weight values are assigned to all points, and the points around the overlapping segments are split into multiple ones. Then the resulting curve can be evaluated through minimizing the constructed energy function. Examples show the efficiency of our method. Graphical abstractDisplay Omitted HighlightsWe propose a stepwise tracking method to generate ordered 3D points from two input orthogonal views.We propose a method to construct 3D B-spline curves from planar views through minimizing the energy function with weight values.The method can efficiently detect the situation of self-overlapping for curve segments from input views.

G n filling orbicular N-sided holes using periodic B-spline surfaces

The orbicular N-sided hole filling problem is usually introduced by filleting an end-point of a part with large radius. The existing methods based on quadrilateral partition or constrained-optimization can rarely generate high-order continuous blending surfaces under these circumstances. This paper first reparameterizes the boundary of the specified orbicular N-sided hole to ensure the compatibility of neighboring cross-boundary derivatives on the connecting points, preserving their Gn continuity. Then we compute the control points of the periodic B-spline surface using the sufficient Gn continuity condition on the pole and the algorithm of extending parametric surfaces. This method generates single blending surface, which can be converted into standard Bspline surface by adding knots without introducing errors. It only elevates the degree of the boundary by n. The construction method is simple and efficient, without iteration nor large-scale matrix solving. It achieves Gn continuity under compatible conditions. The blending examples underline its feasibility and practicability.

G 2 B-spline interpolation to a closed mesh

This paper focuses on interpolating vertices and normal vectors of a closed quad-dominant mesh G^2-continuously using regular Coons B-spline surfaces, which are popular in industrial CAD/CAM systems. We first decompose all non-quadrangular facets into quadrilaterals. The tangential and second-order derivative vectors are then estimated on each vertex of the quads. A least-square adjustment algorithm based on the homogeneous form of G^2 continuity condition is applied to achieve curvature continuity. Afterwards, the boundary curves, the first- and the second-order cross-boundary derivative curves are constructed fulfilling G^2 continuity and compatibility conditions. Coons B-spline patches are finally generated using these curves as boundary conditions. In this paper, the upper bound of the rank of G^2 continuity condition matrices is also strictly proved to be 2n-3, and the method of tangent-vector estimation is improved to avoid petal-shaped patches in interpolating solids of revolution. Several examples demonstrate its feasibility.

Rendering chamfering structures of sharp edges

In most realtime applications such as 3D games, in order to reduce the complexity of the scene being rendered, objects are often made by simple and large primitives. Thus, the phenomenon of edge highlighting, which would require chamfering structures made by lots of small patches at the seaming, is absent and is often faked by “highlights” drawn on the texture. We proposed a realistic realtime rendering procedure for highlighting chamfering structures, or rounded edges, by considering specified edges as thin cylinders and obtained the intensity via integration. We derived a brief approximated formula generalized from Blinn’s shadow model, and used a precomputed integration table to accelerate the render speed and reduce resources needed. The algorithm is implemented with shader language, and can be considered as a post-process on original result. Evaluation shows that the effect on rendering speed is limited even for scenes with large scale of vertices.

Polar NURBS Surface with Curvature Continuity

Polar NURBS surface is a kind of periodic NURBS surface, one boundary of which shrinks to a degenerate polarpoint. The specific topology of its control-point mesh offers the ability to represent a cap-like surface, which iscommon in geometric modeling. However, there is a critical and challenging problem that hinders its application:curvature continuity at the extraordinary singular pole. We first propose a sufficient and necessary condition ofcurvature continuity at the pole. Then, we present constructive methods for the two key problems respectively:how to construct a polar NURBS surface with curvature continuity and how to reform an ordinary polar NURBSsurface to curvature continuous. The algorithms only depend on the symbolic representation and operations ofNURBS, and they introduce no restrictions on the degree or the knot vectors. Examples and comparisons demonstratethe applications of the curvature-continuous polar NURBS surface in hole-filling and free-shape modeling.