We use octree spatial subdivision to generate point clouds on complex nonmanifold implicit surfaces in order to visualize them. The new spatial subdivision scheme only uses point sampling and an interval exclusion test. The algorithm includes a test for pruning the resulting plotting nodes so that only points in the closest nodes to the surface are used in rendering. This algorithm results in improved image quality compared to the naive use of intervals or affine arithmetic when rendering implicit surfaces, particularly in regions of high curvature. We discuss and compare CPU and GPU versions of the algorithm. We can now render nonmanifold features such as rays, ray-like tubes, cusps, ridges, thin sections that are at arbitrary angles to the octree node edges, and singular points located within plot nodes, all without artifacts. Our previous algorithm could not render these without severe aliasing. The algorithm can render the self-intersection curves of implicit surfaces by exploiting the fact that surfaces are singular where they self-intersect. It can also render the intersection curves of two implicit surfaces. We present new image space and object space algorithms for rendering these intersection curves as contours on one of the surfaces. These algorithms are better at rendering high curvature contours than our previous algorithms. To demonstrate the robustness of the node pruning algorithm we render a number of complex implicit surfaces such as high order polynomial surfaces and Gaussian curvature surfaces. We also compare the algorithm with ray casting interms of speed and image quality. For the surfaces presented here, the point clouds can be computed in seconds to minutes on atypical Intel based PC. Once this is done, the surfaces can be rendered at much higher frame rates to allow some degree of interactive visualization.

Abstract We are concerned with producing high‐quality images of parametric and implicit surfaces, in particular those with non‐manifold features. We present a point‐based technique for rendering implicit surfaces that uses octree spatial subdivision with a natural interval exclusion test that guarantees that no parts of the surface are missed. This allows us to render non‐manifold implicit surfaces at speeds comparable to parametric surfaces. We also derive criteria that guarantee complete pixel coverage of the surface. The point‐based method allows for hidden surface elimination using a z‐buffer, and shadow casting using a shadow buffer. We illustrate the technique with a number of surfaces, and discuss its advantages and disadvantages.

Abstract ACM CSS: I.3.7 3D Graphics and Realism, I.3.3 Picture/Image Generation

Two grid-based algorithms are presented for contouring functions of two variables over square plotting areas. The algorithms are simple to code, and achieve considerable efficiency through the use of quadtree techniques to produce non-uniform subdivisions of the plotting areas. One algorithm uses curvature properties of the functions being plotted to produce plotting cells of optimal size, and their relative speeds depend on the complexity of the functions. Both algorithms can be adapted for plotting over rectangular area.