Intersections with Validated Error Bounds for Building Interval Solid Models

Interval arithmetic has been considered as a step forward to counter numerical robustness problem in geometric and solid modeling. The interval arithmetic boundary representation (Brep) scheme was developed to tackle this problem. In constructing an interval B-rep solid, robust and efficient computation of intersections between the bounding surfaces of the solid is a critical issue. To address this problem, a marching method based on a validated interval ordinary differential equation (ODE) solver was proposed, motivated by its potential for the interval B-rep model construction. In this paper, we concentrate on the issue of error control in model space using the validated ODE solver, and further explain that the validated ODE solver can be used in the construction of an interval B-rep solid model using such an error control. INTRODUCTION Solid modeling is one of the key computational tools aiding design and manufacturing, and has been successfully used in computer graphics, CAD/CAM and CAE. In pursuit of a stable solid modeling environment, which allows us to represent the deddress all correspondence to this author. Email: [email protected]; Tel.: 53-4555; MIT Room 5-428, Cambridge, MA 02139-4307, USA 1 sign shape in a topologically valid and consistent manner, several approaches have been proposed. Especially, the Boundary Representation (B-rep) is a frequently used technique to represent such a model. The mathematical theory behind B-rep is well established [1, 2] but in practice having an ideal B-rep solid model still remains beyond the reach of current computational technology primarily due to the inherent limitations present in computer arithmetic and curve approximations. These limitations, such as finite precision (round-off error), arise due to the use of finite set of floating point numbers to represent the infinite set of real numbers with limited number of bits. This may cause pathological behavior of geometric modeling algorithms resulting in significant computational errors, leading to break down of geometric computations and adversely affecting productivity [3]. To mitigate the effects of such limitations, one might use geometric modeling systems based on non-conventional arithmetic techniques. Typical examples are integer and rational arithmetic, and interval and lazy arithmetic [4–12]. Among them, computation techniques based on interval arithmetic have shown the potential to remedy robustness issues in numerical computation [13]. Interval methods can compute bounds in which the correct answer is guaranteed to be enclosed. For example, solution methods of an ordinary differential equation (ODE) based on interval arithmetic can take into account three sources of errors in Copyright c © 2005 by ASME the numerical computation of the solution; propagation of error in initial data, truncation error caused by truncating infinite series after a finite number of terms and round-off errors inherent to computation in a floating point environment [14]. As a first attempt to make the most use of interval arithmetic in the robust B-rep method, Hu et al. [9, 10] developed a data structure and Boolean operations for manifold and non-manifold interval boundary models. Later, the use of interval boundary models in boundary representation was topologically justified by Sakkalis et al. [15]. Interval arithmetic B-rep relies on efficient evaluation of intersection of surfaces and generation of boxes (validated error bounds in 3D model space). Let us assume the existence of a B-rep solid model M, as a well-defined conceptual object. The difficulty then is capturing this in an approximated numerical computational instantiation. An interval boundary representation model is generated from the solid M with face, edge and vertex boxes [15]. The face boxes cover the interior of the faces of M, whereas the edge boxes cover the boundary curves and the vertex boxes corner points of the faces of M. Note that face boxes do not intersect edges and vertices [15]. Please see Figure 1. Face boxes can be constructed by evaluating interval points Pre-image of Face Boxes Pre-image of Edge Boxes Pre-image of