Simple and Robust Boolean Operations for Triangulated Surfaces

Abstract

Boolean operations on geometric models are important in numerical simulation and serve as essential tools in the fields of computer-aided design and computer graphics. The accuracy of these operations is heavily influenced by finite precision arithmetic, a commonly employed technique in geometric calculations, which introduces numerical approximations. To ensure robustness in Boolean operations, numerical methods relying on rational numbers or geometric predicates have been developed. These methods circumvent the accumulation of rounding errors during computation, thus preserving accuracy. Nonetheless, it is worth noting that these approaches often entail more intricate operation rules and data structures, consequently leading to longer computation times. In this paper, we present a straightforward and robust method for performing Boolean operations on both closed and open triangulated surfaces. Our approach aims to eliminate errors caused by floating-point operations by relying solely on entity indexing operations, without the need for coordinate computation. By doing so, we ensure the robustness required for Boolean operations. Our method consists of two main stages: (1) Firstly, candidate triangle intersection pairs are identified using an octree data structure, and then parallel algorithms are employed to compute the intersection lines for all pairs of triangles. (2) Secondly, closed or open intersection rings, sub-surfaces, and sub-blocks are formed, which is achieved entirely by cleaning and updating the mesh topology without geometric solid coordinate computation. Furthermore, we propose a novel method based on entity indexing to differentiate between the union, subtraction, and intersection of Boolean operation results, rather than relying on inner and outer classification. We validate the effectiveness of our method through various types of Boolean operations on triangulated surfaces.

Accurately Computing the Interacted Volume of Molecules over Their 3D Mesh Models

“…After getting all the edges of the intersecting contour, we use the j endpoints (x 1 , y 1 ), (x 2 , y 2 ), ...(x j , y j ) of the intersecting contour to compute the intersection area At each slice of the common layer, the intersecting volume is the total area of intersecting contours multiplied by the slice interval h. The total volume of interacted parts between A 1 and B 1 can be expressed as (14) where S Kd P denotes the total area of intersecting contours in the K P -th common layer.…”

“…Determining interacted regions between molecular models is similar to performing Boolean intersection operations on 3D grid models in computer science, 14 which aligns with the molecular 3D isosurface grid models generated using the marching cube (MC) algorithm. 15,16 The essence of 3D Boolean intersection operations lies in efficiently executing spatial triangle intersection tests.…”

“…Determining interacted regions between molecular models is similar to performing Boolean intersection operations on 3D grid models in computer science, which aligns with the molecular 3D isosurface grid models generated using the marching cube (MC) algorithm. , The essence of 3D Boolean intersection operations lies in efficiently executing spatial triangle intersection tests. − However, processing based on 3D grid models tends to lead to complexity and unreliability . In contrast, planar line segments have fewer geometric features than spatial triangles, simplifying planar polygon Boolean intersection operation.…”