Recent advances in shape–memory polymers: Structure, mechanism, functionality, modeling and applications

Computational geometry.

1 What is a liquid crystal 2 1.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Cholesterics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Columnar phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Lyotropic liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Introduction to Liquid Crystals

Precise Hausdorff distance computation between polygonal meshes

A local start search algorithm to compute exact Hausdorff Distance for arbitrary point sets

We present a real-time algorithm for computing the precise Hausdorff Distance (HD) between two planar freeform curves. The algorithm is based on an effective technique that approximates each curve with a sequence of G 1 biarcs within an arbitrary error bound. The distance map for the union of arcs is then given as the lower envelope of trimmed truncated circular cones, which can be rendered efficiently to the graphics hardware depth buffer. By sampling the distance map along the other curve, we can estimate a lower bound for the HD and eliminate many redundant curve segments using the lower bound. For the remaining curve segments, we read the distance map and detect the pixel(s) with the maximum distance. Checking a small neighborhood of the maximum-distance pixel, we can reduce the computation to considerably smaller subproblems, where we employ a multivariate equation solver for an accurate solution to the original problem. We demonstrate the effectiveness of the proposed approach using several experimental results.

Precise Hausdorff distance computation for planar freeform curves using biarcs and depth buffer

Efficient point-projection to freeform curves and surfaces

We present a compact representation for the bounding volume hierarchy (BVH) of freeform NURBS surfaces using Coons patches. Following the Coons construction, each subpatch can be bounded very efficiently using the bilinear surface determined by the four corners. The BVH of freeform surfaces is represented as a hierarchy of Coons patch approximation until the difference is reduced to within a given error bound. Each leaf node contains a single Coons patch, where a detailed BVH for the patch can be represented very compactly using two lists (containing curve approximation errors) of length proportional only to the height of the BVH. We demonstrate the effectiveness of our compact BVH representation using several experimental results from real-time applications in collision detection and minimum distance computation for freeform models.

Coons BVH for freeform geometric models

We present an efficient algorithm for projecting a given point to its closest point on a family of freeform C1-continuous curves and surfaces. The algorithm is based on an efficient culling technique that eliminates redundant curves and surfaces which obviously contain no projection from the given point. Based on this scheme, we can reduce the whole computation to considerably smaller subproblems, which are then solved using a numerical method. In several experimental results, we demonstrate the effectiveness of the proposed approach.

Efficient point projection to freeform curves and surfaces

We present an efficient algorithm for projecting a continuously moving query point to a family of planar freeform curves. The algorithm is based on the one-sided Hausdorff distance from the trajectory curve (of the query point) to the planar curves. Using a bounding volume hierarchy (BVH) of the planar curves, we estimate an upper bound $\overline{h}$ of the one-sided Hausdorff distance and eliminate redundant curve segments when they are more than distance $\overline{h}$ away from the trajectory curve. Recursively subdividing the trajectory curve and repeating the same elimination procedure to the BVH of the remaining curves, we can efficiently determine where to project the moving query point. The explicit continuous point projection is then interpreted as a curve reparameterization problem, for which we propose a few simple approximation techniques. Using several experimental results, we demonstrate the effectiveness of the proposed approach.

Young-Taek Oh

Papers

Precise Hausdorff distance computation for planar freeform curves using biarcs and depth buffer

Efficient point-projection to freeform curves and surfaces

Coons BVH for freeform geometric models

Efficient point projection to freeform curves and surfaces

Continuous point projection to planar freeform curves using spiral curves