A hybrid polygons-points algorithm based on a hierarchical decomposition of the bounding volume using an octree spatial data structure, and testing cell occupation with interval arithmetic is given. Polygons are produced where curvature in occupied cells is low and points are used where curvature is high. As a result, points are rendered around extreme features and polygons around regular features. Using this approach implicit surfaces of high complexity, such as recursively defined curvature surfaces, are rendered. A discussion of the implementation is given for a GPU version of the algorithm and a comparison with GPU ray-casting is made. As the approach uses subdivision and it can be used for surface modelling. An advantage of the approach is the few artefacts that occur with singular points, cusps, rays and curves of self intersection. Production of antialiased animations and multiple views of the surface can be faster than in ray-casting. The algorithm is robust, renders singular and non-manifold surface and the algorithm works for algebraic and non-algebraic surfaces.

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.