Surface–Surface-Intersection Computation Using a Bounding Volume Hierarchy with Osculating Toroidal Patches in the Leaf Nodes

Highlights

• A new BVH-based algorithm for computing the SSI curves for freeform surfaces

• Highly efficient and robust in handling the degenerate case of tangential intersections.

• Improvement based on the high approximation order and simplicity of osculating torus.

• Compact data structures for bounding the positions and normals of freeform surfaces.

Abstract

We present an efficient and robust algorithm for computing the intersection curve of two freeform surfaces using a Bounding Volume Hierarchy (BVH), where the leaf nodes contain osculating toroidal patches. The covering of each surface by a union of tightly fitting toroidal patches greatly simplifies the geometric operations involved in the surface–surface-intersection computation, i.e., the bounding of surface normals, the detection of surface binormals, the point projection from one surface to the other surface, and the intersection of local surface patches. Moreover, the hierarchy of simple bounding volumes (such as rectangle-swept spheres) accelerates the geometric search for the potential pairs of surface patches that may generate some curve segments in the surface–surface-intersection. We demonstrate the effectiveness of our approach by using test examples of intersecting two freeform surfaces, including some highly non-trivial examples with tangential intersections. In particular, we test the intersection of two almost identical surfaces, where one surface is obtained from the same surface, using a rotation around a normal line by a smaller and smaller angle $\theta =10^{-k}$ degree, $k=0,\dots ,5$. The intersection results are often given as surface subpatches in some highly tangential areas, and even as the whole surface itself, when $\theta =0.00001^{\circ }$.

Introduction

The problem of intersecting two freeform curves and surfaces in a reliable way is one of the main technical challenges in geometric and solid modeling [1], [2], [3], [4]. Unfortunately, in the last decade, there have been only a few publications in this fundamental research [5], [6]. On the other hand, the problem of intersecting two surfaces appears in many different forms as the solution of non-linear geometric constraints [7], [8], [9], where the constraints are often represented or automatically generated as rational freeform curves, surfaces, and multivariate volumes. Consequently, it is still extremely important to develop new techniques that can handle the surface–surface-intersection (SSI) problem in an efficient and reliable way.

Now, a natural question arises: what makes the revisiting of this problem worthwhile for some meaningful technical advancements? To answer this question in a positive perspective, we would like to mention the recent developments of spatial data structures (such as bounding volume hierarchy) for freeform curves and surfaces, which have made possible the acceleration of many important geometric algorithms, including collision detection, the minimum and Hausdorff distance computations, offset curve trimming, Minkowski sum computation, medial axis and Voronoi diagram construction, convex hull computation, etc [10], [11], [12], [13].

Planar freeform curves can be approximated by a hierarchy of $G^1$-biarcs with cubic convergence, which can be used for the construction of arc trees (i.e., BVH structures where the internal nodes are fat arcs [14]). The arc trees are a very powerful tool for the acceleration of algorithms for planar freeform curves. A natural dimension–extension of arc trees seems to be torus trees. Nevertheless, different from the planar case, it is extremely difficult to approximate freeform surfaces with $G^1$-continuous toroidal patches. As a compromise, we consider a union of osculating toroidal patches that tightly fits the given freeform surface. (We use the second order osculating torus patch of Liu et al. [15].) Thanks to the higher approximation order than triangular and bilinear approximations, the osculating toroidal patches provide good approximate solutions to the SSI problem even though they are not connected even with $C^0$-continuity.

In this work, we consider the acceleration of the SSI problem using a BVH for freeform surfaces. There are certainly some costs we have to pay for the memory space and the preprocessing time for the BVH construction. Nevertheless, the majority of recent acceleration algorithms are based on spatial data structures (often precomputed) at some additional costs. For example, an alternative approach (to the SSI problem) using triangulated meshes of the freeform surfaces would also require the construction of BVH trees in a preprocessing stage, often taking considerably more time and space than our approach. As the memory cost becomes cheaper, we may even assume the pre-constructed BVH structure as a part of the surface representation by encoding the structure compactly as a 2D array, much like an image, which is the approach we take in the current work.

For the sake of simplicity in the presentation, we consider the intersection of two bicubic Bézier surfaces, though the basic approach can easily be extended to higher degree trimmed B-spline surfaces or even to triangular Bézier surfaces. Taking uniform samples along each parameter direction, $S\left(\frac{i}{2^H},\frac{j}{2^H}\right)$, where $i,j=0,\dots ,2^H$, the BVH structure for a bicubic Bézier surface $S(u,v)$, ($0\le u,v\le 1$), can be represented in about the same way as that for the quadtree (of height $H$) for regular quad meshes (Fig. 1). The main difference is in the internal nodes which contain bounding volumes slightly thicker than those for a regular quad mesh. The extra thickness is needed for bounding the difference between a surface $S(u,v)$ and the quad mesh approximation (interpreted as a rectangular array of bilinear surface patches or pairs of triangles).

The surface approximation error can be bounded using a simple formula developed by Filip et al. [16]. (Krishnamurthy et al. [13], [17] also use this formula for the construction of various BVH trees for freeform surfaces.) Using a 2D array of size $(1+2^H)\times (1+2^H)$, where the $x,y,z$ coordinates of $S\left(\frac{i}{2^H},\frac{j}{2^H}\right)$ are stored in each element, we can encode the BVH structure of the surface $S(u,v)$ in a compact way, where the thickness of the internal nodes at a level $h$ $(\le H)$ can be estimated by the formula of Filip et al. [16]. (This is because the higher level $h$ approximates the surface $S(u,v)$ using a coarser quad mesh with vertices taken at $S\left(\frac{i}{2^h},\frac{j}{2^h}\right)$, where $i,j=0,\dots ,2^h$.

The BVH-based intersection algorithm for freeform surfaces is about the same as other conventional algorithms for meshes. The main difference is in the intersection of two primitives stored in the leaf nodes. Instead of intersecting two triangles, we intersect two osculating toroidal patches, which may produce intersection curve segments of more complex shapes such as those with multiple branches and even X-junctions. The first row of Fig. 2 shows two examples of the tangential intersection, each with an X-junction. As we move the blue patch up and down along the normal direction, the X-junction disappears and the intersection curve splits to two branches as shown in the second and third rows of the figure.

When two surfaces intersect tangentially at a point, they share the same normal line (i.e., their binormal line) at the point. The location of an X-junction or a near X-junction can be detected by computing the binormal lines to the two surfaces and checking the signed distance between the corresponding binormal-surface intersection points. Depending on the sign, we can decide which type of (almost) tangential intersection curve the two surfaces have. For example, in Fig. 2, when measured along the normal direction of the blue torus, the binormal-surface intersection point of the red torus is located at a signed distance of zero, negative, and positive, respectively, in each of the three rows. Osculating toroidal patches greatly simplify this test by converting the distance computation to a much simpler problem of solving a polynomial equation of degree 8 [18], [19]. The polynomial equation can be solved considerably faster than the general case of computing binormal lines for two bivariate surfaces, where we need to deal with a system of four equations in four variables.

For toroidal patches, their Gauss maps are easy to compute. When there is no overlap in the Gauss maps of the toroidal patches, the intersection curve may have at most one connected branch, which can be traced very efficiently using numerical techniques. Otherwise, we take further steps (such as subdivision or resampling of osculating tori) for detecting the correct topology of SSI curves. Though this is quite a complex task in general due to tangential surface intersections, the binormal construction algorithm for toroidal patches greatly simplifies this process. We subdivide the surfaces at the binormal-surface intersection points, which can remove an X-junction or reduce the number of branches in the intersection curve. In other words, we can make (almost) X-junctions appear only at (or around) the corner points of surface patches, after the subdivision.

The numerical curve tracing can be carried out either on the original surfaces or on the osculating toroidal patches. According to the experimental results reported in Section 5, the difference is insignificant as we cover the entire surface using a union of tightly fitting osculating toroidal patches within a small Hausdorff distance bound, such as $ϵ=10^{-5}$.

The main contributions of this work can be summarized as follows:

• We propose a new BVH-based algorithm for computing the SSI curves for freeform surfaces, which outperforms other conventional algorithms in computing speed and robustness by computing the BVH structure in a preprocessing time.

• The improvement is based on the high approximation order of osculating toroidal patches to the freeform surfaces and the geometric simplicity of toroidal patches in supporting primitive operations (needed for the SSI computation) such as the Gauss map computation, the overlap test between two Gauss maps, the binormal line construction, the projection of points onto a surface, etc.

• The BVH of freeform surfaces can be represented in a compact way using a rectangular array of uniform sampling points and other simple spatial data structures which resemble the mipmap structure for texture mapping [20].

• The main advantage of our SSI algorithm is in handling the degenerate case of (almost) tangential surface intersections, where the conventional subdivision methods have difficulty pruning a large number of potentially overlapping pairs of small surface patches.

• The osculating toroidal patches in the leaf nodes not only bound the small surface patches corresponding to the leaf nodes, but they also bound their surface normals. This will be discussed in Section 3.3.

• Further trimming the toroidal patches into a shape similar to the surface patches (using a surface matching technique to be discussed briefly at the end of Section 3.3), we can bound their Hausdorff distance within a small tolerance $ϵ=10^{-5}$ (by repeating the local subdivisions recursively if necessary).

• The 1–1 matching between each local surface patch and the corresponding trimmed toroidal patch (within a Hausdorff distance $ϵ=10^{-5}$) makes the SSI curve constructed in the $uv$-parameter space of one surface $S_1(u,v)$ also 1–1 matched with its counterpart SSI curve in the $st$-space of the other surface $S_2(s,t)$. (The Hausdorff distance between the two matching SSI curves is then bounded by $kϵ=k\cdot 10^{-5}$, where $k$ is a small constant.) More details will be discussed in Section 4