Kevin G. Suffern

Interval methods in computer graphics

Abstract The methods of interval arithmetic are applied to graphics algorithms for contouring functions of two variables and rendering implicit surfaces. Interval methods result in algorithms that are guaranteed not to miss parts of the contours or surfaces down to a specified size in the viewing region. Thus, they provide a degree of robustness to the algorithms which is difficult to achieve when point sampling alone is used to detect the contours and surfaces.

Visualisation of implicit surfaces

Abstract We discuss the polygonisation and rendering of implicit surfaces through the adaptive subdivision of octree nodes. Three new algorithms which form part of this process, are discussed. The first algorithm removes the cracks that form on the surfaces between adjacent nodes that are subdivided to different depths in the octree. This algorithm fixes the cracks for arbitrary differences in depth between the adjacent nodes. The second algorithm allows user specified selected polygon outlines to be drawn on the surfaces, instead of drawing all the outlines, or no outlines. The third algorithm fixes the gaps that can appear in the polygon outlines when only selected outlines are drawn. We also investigated the use of an exact calculation of the curvature of the implicit surfaces to drive the adaptive subdivision. We conclude that this is not worthwhile doing, and give reasons.

Eigenvalues of the Chandrasekhar–Page angular functions

The Chandrasekhar–Page angular functions for the Dirac equation in the Kerr–Newman background are expanded as series of hypergeometric polynomials, and a three‐term recurrence relation is derived for the coefficients in these series. This leads to a transcendental equation for the determination of the separation constant which is obtained initially as a power series and is then iterated by the method of Blanch and Bouwkamp.

Adaptive polygonisation of non-manifold implicit surfaces

We discuss the polygonisation and rendering of non-manifold implicit surfaces using adaptive octree subdivision and interval arithmetic for surface exclusion in octree nodes. We present a new algorithm that polygonises some surfaces that self intersect, or have other non-manifold features such as separate sections that meet at points. Gradient information is used to resolve ambiguous polygonisations in plotting nodes. A line-stitching algorithm is discussed that allows for multiple polygons to be in a plotting node. We illustrate the algorithm with a number of surfaces that demonstrate its capabilities and limitations.

Point-Based Rendering of Non-Manifold Surfaces

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.

Point based rendering of non-manifold surfaces with contours

We present point based rendering techniques that render various types of contours as constant width slabs on surfaces. The techniques requires evaluations of the surface functions and gradients to render shaded images. We use slabs parallel to the principle planes, slabs located along a principal axis and rotated by arbitrary steps, slabs consisting of concentric spheres and slabs of constant Gaussian and mean curvatures. We also use the technique to render curvature maps of surfaces. We illustrate the techniques with a number of parametric and implicit surfaces, and discuss their advantages and disadvantages compared to other rendering techniques.

Effective instructional animation in 3D computer graphics education

A number of instructional animation are discussed for teaching 3D computer graphics. They are effective in their assigned task because they focus on concepts and processes that are difficult or impossible to see without animation. They also use a minimalist approach making the animations as simple as possible, focussed on what is relevant, and interactive. One animation is a program that allows students to explore viewing systems, two are concerned with ray tracing transparent objects, and one visualises the formation of marble.

Perspective views of polar coordinate functions

Abstract A special purpose hidden line algorithm is presented for plotting perspective views of grid surfaces for functions defined in spherical polar coordinates. By exploiting the geometric properties of grid surfaces the algorithm is linear time in the number of facets processed.

Ray Tracing Surfaces with Contours

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

Visualizing Nonmanifold and Singular Implicit Surfaces with Point Clouds

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 in terms of speed and image quality. For the surfaces presented here, the point clouds can be computed in seconds to minutes on a typical Intel based PC. Once this is done, the surfaces can be rendered at much higher frame rates to allow some degree of interactive visualization.