Global Self-intersections Research Articles

Function Representation Based Analytic Shape Hollowing Optimization

We present a function representation based analytic optimization framework for shape hollowing problem to achieve various functionalities. A compact description of model with cavities is constructed by introducing a field function, which could handle geometry and topology changes of interior shapes effortlessly without both local and global self-intersections. The variation of cavities is permitted in the entire interior volume without restrictions of skeletons and topology changes. Moreover, the function representation inherently defines smooth and accurate shapes that are independent of mesh resolution. Last but not least, the analytic derivatives of variables in gradient based optimization can be directly executed on the function representation without remeshing. Benefiting from the proposed function representation, our method provides an efficient and controllable framework to represent, analyze and optimize shape with cavities directly on functions. Various experiments have shown that our approach can obtain smooth and effective cavities that well predict the physical properties of fabricated models.

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 G1-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 Collision and Interference Detection for 3D Geometric Models

This paper describes a new approach to perform continuous collision and interference detection between a pair of arbitrarily complex objects moving according to general three-dimensional affine motions. Our approach, which does not require any envelope computations, recasts the problem of detecting collisions and computing the interfering subsets in terms of inherently parallel set membership classification tests of specific curves against the original (static) geometric representations. We show that our approach can compute the subsets of the moving objects that collide and interfere, as well as the times of collision, which has important applications in mechanical design and manufacturing. Our approach can be implemented for any geometric representation that supports curve-solid intersections, such as implicit and parametric representations. We describe an implementation of the proposed technique for solids given as a boundary representation (B-rep), and illustrate its effectiveness for several rigid and deformable moving objects bounded by tesselated and freeform surfaces of various complexities. Furthermore, we show that our approach can be extended to also identify the local and global self-intersections of the envelopes of the moving objects without requiring to compute these envelopes explicitly. The paper concludes by summarizing the proposed approach as well as reviewing relevant computational improvements that can decrease the computational cost of the prototype implementation by orders of magnitude.

Generation of a planar offset curve based on minimum distance

A robust algorithm for removing both local and global self intersections in a planar offset curve is presented. Our scheme is based on the minimum distance from a point to an original parametric curve. The method consists of the following two steps: 1) in the 2D plane, generation of the field of minimum distance from a point to an original planar parametric curve, 2) in the field, extraction of the iso-curve, regarding the offset distance as the minimum distance. The iso-curve is the non-self-intersection offset curve. Therefore, the advantage of our method is that the planar offset curve does not need to be approximated by a rational curve; however, both local and global self-intersections in the offset curve can be directly and automatically eliminated.

Trimming self-intersections in swept volume solid modeling

Swept volume solid modeling has been applied to many areas such as NC machining simulation and verification, robot workspace analysis, collision detection, and CAD. But self-intersections continue to be a challenging problem in the boundary representation of swept volume solids. A novel algorithm is presented in this paper to trim self-intersection regions in swept volume solids modeling. This trimming algorithm consists of two major steps: (1) roughly detecting self-intersection regions by checking intersections or overlapping of the envelop profiles; (2) splitting the whole envelop surfaces of the swept volume solid into separate non-self-intersecting patches to trim global self-intersections, and to trim local self-intersections, dividing local self-intersecting regions into patches and replacing self-intersecting patches with non-self-intersecting ones. Examples show that our algorithm is efficient and robust.

Self-intersection detection and elimination in freeform curves and surfaces

We present several algorithms for self-intersection detection, and possible elimination, in freeform planar curves and surfaces. Both local and global self-intersections are eliminated using a binormal-line criterion and a simple direct algebraic elimination procedure that enables the direct solution of the algebraic (self-)intersection constraints. All algorithms have been fully implemented and tested. Examples are presented for applications in self-intersection detection, self-intersection-free metamorphosis of curves, and proper trimming of self-intersections in offset curves.

Minkowski sum boundary surfaces of 3D-objects

Given two objects A and B with piecewise smooth boundary we discuss the computation of the boundary Γ of the Minkowski sum A + B. This boundary surface Γ is part of the envelope when B is moved by translations defined by vectors a ∈ A, or vice versa. We present an efficient algorithm working for dense point clouds or for triangular meshes. Besides this the global self-intersections of the boundary Γ are detected and resolved. Additionally we point to some relations between Minkowski sums and kinematics, and compute local quadratic approximations of the envelope.

Trimming local and global self-intersections in offset curves/surfaces using distance maps

A robust and efficient algorithm for trimming both local and global self-intersections in offset curves and surfaces is presented. Our scheme is based on the derivation of a rational distance map between the original curve or surface and its offset. By solving a bivariate polynomial equation for an offset curve or a system of three polynomial equations for an offset surface, all local and global self-intersection regions in offset curves or surfaces can be detected. The zero-set of polynomial equation(s) corresponds to the self-intersection regions. These regions are trimmed by projecting the zero-set into an appropriate parameter space. The projection operation simplifies the analysis of the zero-set, which makes the proposed algorithm numerically stable and efficient. Furthermore, in a post-processing step, a numerical marching method is employed, which provides a highly precise scheme for self-intersection elimination in both offset curves and surfaces. The effectiveness of our approach is demonstrated using several experimental results.

Offset of curves on tessellated surfaces

Geodesic offset of curves on surfaces is an important and useful tool of computer aided design for applications such as generation of tool paths for NC machining and simulation of fibre path on tool surfaces in composites manufacturing. For many industrial and graphic applications, tessellation representation is used for curves and surfaces because of its simplicity in representation and for simpler and faster geometric operations. The paper presents an algorithm for computing offset of curves on tessellated surfaces. A curve on tessellation (COT) is represented as a sequence of 3D points, with each line segment of every two consecutive points lying exactly on the tessellation. With an incremental approach of the algorithm to compute offset COT, the final offset curve position is obtained through several intermediate offset curve positions. Each offset curve position is obtained by offsetting all the points of COT along the tessellation in such a way that all the line segments gets offset exactly along the faces of tessellation in which the line segments are contained. The algorithm, based entirely on tessellation representation, completely eliminates the formation of local self-intersections. Global self-intersections if any, are detected and corrected explicitly. Offset of both open and closed tessellated curves, either in a plane or on a tessellated surface, can be generated using the proposed approach. The computation of offset COT is very accurate within the tessellation tolerance.