Generating soft shadows in real-time is difficult. Exact methods (such as ray tracing, and multiple light source simulation) are too slow, while approximate methods often over-estimate the umbra regions. In this paper, we introduce a new algorithm based on the shadow map method to quickly and accurately render soft shadows produced by a light source. Our method builds inner and outer translucent fins on objects to represent the penumbra area inside and outside hard shadows respectively. The fins are traced into multi-layered light space maps to store illuminance adjustment to shadows. The viewing space illuminance buffer is then calculated using those maps. Finally, by blending illuminance and shading, a scene with highly accurate soft shadow effects is produced. Our method does not suffer from umbra over-estimation. Physical relations between light, objects, and shadows demonstrate the soundness of our approach.

In this paper we analyze an algorithm which solves the point projection and the "inversion" problems for parametric curves and surfaces. It consists of a geometric second order iteration which converges faster than existing first order methods, and whose sensitivity to the choice of initial values is small. Applications include the ICP algorithm for shape registration.

We give a new, simple algorithm for simultaneous degree elevation and knot insertion for B-spline curves. The method is based on the simple approach of computing derivatives using the control points, resampling the knot vector, and then computing the new control points from the derivatives. We compare our approach with previous algorithms and illustrate it with examples.

We investigate the propagation of errors through geometric transfor-mations, such as reflections, rotations, similarity transformations, and projections, and also the scalar product of vectors. This means computing tolerance zones of points which undergo such transformations, if these points and the transforma-tions themselves are given by toleranced input data.

Strain fields provide a method of deformation measurement based on physics. Using these as a tool, we can analyze deformation of objects in a measurable way. We have developed a new morphing technique based on strain field interpolation. Shape shaking and squeezing, which often happen when using linear interpolation for morphing, do not arise in our approach. We have also developed a new method to create isomorphic meshes from corresponding objects in two images. Meshes generated by this method have much fewer triangles than other methods, which greatly decreases calculation loads in the morphing process.

Significant visual effects arise from surface mesostructure, such as fine-scale shadowing, occlusion and silhouettes. To efficiently render its detailed appearance, we introduce a technique called viewdependent displacement mapping (VDM) that models surface displacements along the viewing direction. Unlike traditional displacement mapping, VDM allows for efficient rendering of selfshadows, occlusions and silhouettes without increasing the complexity of the underlying surface mesh. VDM is based on per-pixel processing, and with hardware acceleration it can render mesostructure with rich visual appearance in real time.

This paper addresses the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. The basic idea of the approach is to find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying these conditions. To obtain a merged curve without superfluous knots, we present a new knot adjustment algorithm for adjusting the end k knots of a kth order B-spline curve without changing its shape. The more general problem of merging curves to pass through some target points is also discussed.

This paper presents an algorithm for extending B spline carves and surfaces. Based on the unclamping algorithm for B-spline curves, we propose a new algorithm for extending B spline curves that extrapolates using the recurrence property of the de Boor algorithm. This algorithm provides a nice extension, with maximum continuity, to the original curve segment. Moreover, it can be applied to the extension of B spline surfaces. Extension to both single and multiple target points/curves are considered in this paper.

Solving geometric constraint systems in 3-D is much more complicated than that in 2-D because the number of variables is larger and some of the results valid in 2-D cannot be extended for 3-D. In this paper, we propose a new DOF based graph constructive method to geometric constraint systems solving that can efficiently handle well-, over-and under-constrained systems based on the dependence analysis. Tile basic idea is that the solutions of some geometric elements depend on some others because of the constraints between them. if some geometric elements depend on each other, they must be solved together, in our approach, we first identify all structurally redundant constraints, then we add some constraints to well constrain the system. And we prove that the order of a constraint system after processing-under constrained cases is not more than that of the original system multiplied by 5. After that, we apply a recursive searching process to identify all the clusters, which is shown to be capable of getting the mimimum order-reduction result of a well constrained system. We also briefly describe tile constraint evaluation phase and show the implementation results of our method.

NURBS surfaces are among the most commonly used parametric surfaces in CAGD and Computer Graphics. This paper investigates shape modihcation of NURBS surfaces with geometric constraints, such as point, normal vector, and curve constraints. Two new metbods are presented by constrained optimization anl energy minimization. The former is based on minimizing changes in control net of surfaces, whereas the latter is based on strain energy minimization. By these two methods, we change control points and weights of an original surface, such that the modified surface satisfies the given constraints. Comparison results and practical examples are also given.