Paths of algebraic hyperbolic curves

Cubic algebraic hyperbolic (AH) Bézier curves and AH spline curves are defined with a positive parameter α in the space spanned by {1, t, sinht, cosht}. Modifying the value of α yields a family of AH Bézier or spline curves with the family parameter α. For a fixed point on the original curve, it will move on a defined curve called "path of AH curve" (AH Bézier and AH spline curves) when α changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.

Non-uniform B-spline curves with multiple shape parameters

We introduce a kind of shape-adjustable spline curves defined over a non-uniform knot sequence.These curves not only have the many valued properties of the usual non-uniform B-spline curves,but also are shape adjustable under fixed control polygons.Our method is based on the degree elevation of B-spline curves,where maximum degrees of freedom are added to a curve parameterized in terms of a non-uniform B-spline.We also discuss the geometric effect of the adjustment of shape parameters and propose practical shape modification algorithms,which are indispensable from the user's perspective.

Adaptive triangular mesh coarsening with centroidal Voronoi tessellations

We present a novel algorithm for adaptive triangular mesh coarsening. The algorithm has two stages. First, the input triangular mesh is refined by iteratively applying the adaptive subdivision operator that performs a so-called red-green split. Second, the refined mesh is simplified by a clustering algorithm based on centroidal Voronoi tessellations (CVTs). The accuracy and good quality of the output triangular mesh are achieved by combining adaptive subdivision and the CVTs technique. Test results showed the mesh coarsening scheme to be robust and effective. Examples are shown that validate the method.

Degree elevation of unified and extended spline curves

Unified and extended splines(UE-splines), which unify and extend polynomial, trigonometric, and hyperbolic B-splines, inherit most properties of B-splines and have some advantages over B-splines. The interest of this paper is the degree elevation algorithm of UE-spline curves and its geometric meaning. Our main idea is to elevate the degree of UE-spline curves one knot interval by one knot interval. First, we construct a new class of basis functions, called bi-order UE-spline basis functions which are defined by the integral definition of splines.Then some important properties of bi-order UE-splines are given, especially for the transformation formulae of the basis functions before and after inserting a knot into the knot vector. Finally, we prove that the degree elevation of UE-spline curves can be interpreted as a process of corner cutting on the control polygons, just as in the manner of B-splines. This degree elevation algorithm possesses strong geometric intuition.

Easy cross parameterization for articulated shapes

This paper presents a novel interactive system for establishing compatible meshes for articulated shapes.Given two mesh surfaces,our system automatically generates both the global level component correspondence and the local level feature correspondence.Users can use some sketch-based tools to specify the correspondence in an intuitive and easy way.Then all the other vertex correspondences could be generated automatically.The cross parameterization preserves both high level and low level features of the shapes.The technique showed in the system benefits various applications in graphics including mesh inter-polation,deformation transfer,and texture transfer.

Quasi-angle-preserving mesh deformation using the least-squares approach

We propose an angle-based mesh representation, which is invariant under translation, rotation, and uniform scaling, to encode the geometric details of a triangular mesh. Angle-based mesh representation consists of angle quantities defined on the mesh, from which the mesh can be reconstructed uniquely up to translation, rotation,and uniform scaling. The reconstruction process requires solving three sparse linear systems: the first system encodes the length of edges between vertices on the mesh, the second system encodes the relationship of local frames between two adjacent vertices on the mesh, and the third system defines the position of the vertices via the edge length and the local frames. From this angle-based mesh representation, we propose a quasi-angle-preserving mesh deformation system with the least-squares approach via handle translation, rotation, and uniform scaling. Several detail-preserving mesh editing examples are presented to demonstrate the effectiveness of the proposed method.