Direct trimming of NURBS surfaces on the GPU

Abstract

This paper presents a highly efficient direct trimming technique for NURBS surfaces, which is applicable to tessellation-based rendering as well as ray tracing systems. The central idea is to split the trim curves into monotonic segments with respect to the two parameter dimensions of the surface patches. We use an optimized bisection method to classify a point with respect to each monotonic trim curve segment without performing an actual intersection test. Our hierarchical acceleration structure allows the use of a large number of such curve segments and performs the bisection method only for points contained in the bounding boxes of the curve segments.We have integrated our novel point classification scheme into a GPU-based NURBS ray casting system and implemented the entire trimmed NURBS rendering algorithm in a single OpenGL GLSL shader. The shader can handle surfaces and trim curves of arbitrary degrees, which allows the use of original CAD data without incorporating any approximations. Performance data confirms that our trimming approach can deal with hundreds of thousands of trim curves at interactive rates. Our point classification scheme can be applied to other application domains dealing with complex curved regions including flood fills, font rendering and vector graphics mapped on arbitrary surfaces.

Crack‐free rendering of dynamically tesselated B‐rep models

“…If it does, we look up the u, v coordinate resulting from the Newton iteration in the trim structure to determine if the intersection actually lies on the B-Rep face. This follows the fragment classification method of Schollmeyer and Fröhlich [SF09]. If the Newton iterations do not converge, or if the trim structure lookup reveals the fragment is off the face, execution continues on to the next neighboring fragment.…”

Calculation of the Physical Optics Scattering by Trimmed NURBS Surfaces

“…For the first step, the input NURBS trimming curve is transformed into the nonparametric explicit representation with respect to either -or -direction in parameter domain, as shown in [5], which, taking -direction for example, yields the explicit Bézier curve . The -monotonic points on each trimming curve are the extrema on the curve [6], which are obtained by solving , and the trimming curves are split at each monotonic point by the Bézier subdivision, yielding the -monotonic curve segments, as shown in Fig. 2(a).…”

“…At the third step, the -monotonic segments are stored in an acceleration data structure (ADS) based on the dual binary tree. Based on the ADS, an efficient PCA can be derived for the 2-D ray-curve intersection problem, which is available in [6].…”

“…Advantages of the adopted PCA can be summarized as follows: It is possible to reduce the unnecessary ray-curve intersection computation through the fast indexing of binary tree; besides, unlike the traditional ray-curve intersection solving approaches such as Newton iteration method or Bézier clipping algorithm [6], the -monotonic segment-based bisection ray-curve intersection solving algorithm gains in efficiency and numerical stability because it is free of the computation cost derivative operation. The numerical experiment that investigates the performance of the adopted PCA is given in Section III.…”

An Border-Stable Approach to NURBS Surface Rendering for Ray Tracing

“…Schollmeyer and Froehlich [9] describe an approach for NURBS surface ray tracing, where surface trimming is used to set of monotonic Bézier curves. For finding of intersection point the bisection method is used.…”