Yong-Joon Kim

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

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 G1 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.

Efficient point-projection to freeform curves and surfaces

We present an efficient algorithm for projecting a given point to its closest point on a family of freeform 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. For monotone spiral planar curves with no inflection, we show that a few simple geometric tests are sufficient to guarantee the convergence of numerical methods to the closest point. In several experimental results, we demonstrate the effectiveness of the proposed approach.

Precise gouging-free tool orientations for 5-axis CNC machining

We present a precise approach to the generation of optimized collision-free and gouging-free tool paths for 5-axis CNC machining of freeform NURBS surfaces using flat-end and rounded-end (bull nose) tools having cylindrical shank. To achieve high approximation quality, we employ analysis of hyper-osculating circles (HOCs) (Wang et?al., 1993a,b), that have third order contact with the target surface, and lead to a locally collision-free configuration between the tool and the target surface. At locations where an HOC is not possible, we aim at a double tangential contact among the tool and the target surface, and use it as a bridge between the feasible HOC tool paths. We formulate all such possible two-contact configurations as systems of algebraic constraints and solve them. For all feasible HOCs and two-contact configurations, we perform a global optimization to find the tool path that maximizes the approximation quality of the machining, while being gouge-free and possibly satisfying constraints on the tool tilt and the tool acceleration. We demonstrate the effectiveness of our approach via several experimental results. We present an algorithm for generating optimized gouging-free tool path for 5-Axis CNC machining.We employ analysis of hyper-osculating circles that provides third order approximation of the surface.Double tangential contact between the tool and the target surface is employed to connect feasible hyper-osculating tool paths.A robust collision and gouging detection algorithm is provided.We introduce a global optimization algorithm that maximizes the geometric matching between the tool and the target surface.

Coons BVH for freeform geometric models

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.

Short Communication to SMI 2011: Efficient convex hull computation for planar freeform curves

We present an efficient real-time algorithm for computing the precise convex hulls of planar freeform curves. For this purpose, the planar curves are initially approximated with G^1-biarcs within a given error bound @e in a preprocessing step. The algorithm is based on an efficient construction of approximate convex hulls using circular arcs. The majority of redundant curve segments can be eliminated using simple geometric tests on circular arcs. In several experimental results, we demonstrate the effectiveness of the proposed approach, which shows the performance improvement in the range of 200-300 times speed up compared with the previous results (Elber et al., 2001) [8].

Efficient Hausdorff Distance computation for freeform geometric models in close proximity

We present an interactive-speed algorithm for computing the Hausdorff Distance (HD) between two freeform geometric models represented with NURBS surfaces. The algorithm is based on an effective technique for matching a surface patch from one model to the corresponding nearby surface patch on the other model. To facilitate the matching procedure, we employ a bounding volume hierarchy (BVH) for freeform NURBS surfaces, which provides a hierarchy of Coons patches and bilinear surfaces approximating the NURBS surfaces (Kim et al., 2011 [1]). Comparing the local HD upper bound against a global HD lower bound, we can eliminate the majority of redundant surface patches from further consideration. The resulting algorithm and the associated data structures are considerably simpler than the previous BVH-based HD algorithms. As a result, we can compute the HD of two freeform geometric models efficiently and robustly even when the two models are in close proximity. We demonstrate the effectiveness of our approach using several experimental results.

Efficient offset trimming for planar rational curves using biarc trees

We present an efficient algorithm for trimming both local and global self-intersections in planar offset curves. The algorithm is based on a G^1-continuous biarc approximation of the given planar curves. We first consider an implementation that employs a distance map which can be stored in the graphics hardware depth buffer. The depth-buffer approach is easier to implement than a different approach that is based on a biarc-tree, a hierarchical data structure for the biarc approximation of the given planar curves. Though more involved technically, the biarc-tree algorithm is more efficient both in computing time and in memory space needed for storing the data structure. We demonstrate the real-time performance of our algorithm using several experimental results.

Continuous point projection to planar freeform curves using spiral curves

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

Volumetric Boolean sum

\overline{h}

Interactive tree modeling and deformation with collision detection and avoidance

of the one-sided Hausdorff distance and eliminate redundant curve segments when they are more than distance