Fujio Yamaguchi

Some basic geometric test conditions in terms of Plücker coordinates and Plücker coefficients

We propose some basic geometric tests that are unified, projectively invariant, and do not require any auxiliary points. We include (1) linearly independent conditions of homogeneous flats and simplices; (2) coincidence, coplanarity, concurrency, and collinearity conditions of homogeneous flats; and (3) interference conditions between homogeneous simplices, defined both in projective space and in two-sided space. All the conditions are represented in terms of Plücker coordinates and Plücker coefficients.

Applications of the 4×4 determinant method and the POLYGON ENGINE

In the first half of the paper, various types of processing pertaining to a polygon, using the 4×4 determinant theories are explained along with a new containment test algorithm of a point in a polygon. In the latter half of the paper, a general-purpose geometric processor, the POLYGON ENGINE, is presented which can deal with various types of interference problems, such as Boolean operations in solid modelling, hidden line and surface eliminations, ray tracing and so on. It is, a successor of the TRIANGLE PROCESSOR and is also based upon the 4×4 determinant theories [4–6]. While the TRIANGLE PROCESSOR processes a triangulated polygon on a triangle-by-triangle basis, the POLYGON ENGINE can treat a polygon without triangulation. The latter is expected to be more functional, more efficient and easier to use.

Homogeneous bounding boxes as tools for intersection algorithms of rational bézier curves and surfaces

In the divide-and-conquer algorithm for detecting intersections of parametric rational Bézier curves (surfaces), we use bounding boxes in recursive rough checks. In this paper, we replace the conventional bounding box with a homogeneous bounding box, which is projectively defined. We propose a new rough check algorithm based on it. One characteristic of the homogeneous bounding box is that it contains a rational Bézier curve (surface) with weights of mixed signs. This replacement of the conventional bounding box by the homogeneous one does not increase the computation time.

Solid Modeling Based on a New Paradigm

The technique of solid modeling is essential in CAD/CAM applications, and is currently well established. However, problems remain, such as the lack of uniformity in geometric computations and the lack of stability of Boolean operations of two solids. In this paper, we introduce a theoretical solid modeling system that operates on boundary representations of polyhedral objects and is based on a new paradigm. The characteristics of the system are the following: (I) in Boolean Operations and modeling transformations, all geometric computations are performed by the 4 × 4 determinant method or the 4 × 4 matrix method in homogeneous space, which allows the system to avoid division operations, (2) all geometric computations are performed by the exact integer arithmetic, which makes the geometric algorithms stable and simple, and (3) primitive solids are constructed consistently in the integer domain, and the consistency is assured throughout Boolean operations and transformations.

Projectively invariant intersection detections for solid modeling

An intersection detection method for solid modeling which is invariant under projective transformations is presented. We redefine the fundamental geometric figures necessary to describe solid models and their dual figures in a homogeneous coordinate representation. Then we derive conditions, which are projectively invariant, for intersections between these primitives. We will show that a geometric processor based on the 4 x 4 determinant method is applicable to a wide range of problems with little modification. This method has applications in intersection detections of rational parametric curves and surfaces and hidden-line/surface removal algorithms.

A New Paradigm for Geometric Processing

A new paradigm for geometric processing is proposed. The paradigm can be featured by the four key phrases: (1) Totally Four Dimensional Homogeneous Processing, (2) 4 × 4 Matrix Method and 4 × 4 Determinant Method, (3) Integer Arithmetic with Adaptive Data‐Length Control Technique, (4) Systematic Paradigm Structures Based on the Duality.

Theoretical foundations for the 4×4 determinant method in computer graphics and geometric modelling

Theories of the 4×4 determinant method to resolve interference problems are described, in detail, in succession to the former paper [1]. First, various cases of the 4×4 determinant are discussed including the geometric implications by deriving a few fundamental relations. Secondly, normalization of the determinant is proposed. Thirdly, an intersection formula in homogeneous coordinates is verified which makes it possible to do consistent homogenous coordinate processing from the very beginning of geometric modelling to the very last of the objects displayed. Lastly an outline of how the 4×4 determinant method is applied to basic geometric problems is described.This article will provide, theoretical foundations for the 4×4 determinant method in computer graphics and geometric modelling.

A shift of playground for geometric processing from euclidean to homogeneous

The author argues that as long as euclidean processing provides the basis for CAD, there will always be a limit to the level of reliability achievable in a system. In other words, a CAD system contains within it elements that inevitably lead to the systems inaccuracy, instability, and complexity because aa division operation is the root cause of all evilo. The author proposes a geometric processing playground in which there is no need of performing a division operation; that is, atotally four-dimensional homogeneous processingo. He discusses the geometric processing from the points of its theoretical backgrounds, geometric definitions, geometric computations, topological definitions, and topological operations.

An adaptive, error-free computation based on the 4 × 4 determinant method

The 4×4 determinant method makes it possible to unify geometric processing by the computations of 4×4 determinants composed of homogeneous coordinants vectors of four points or coefficient vectors of four plane equations. Because the method needs not require a division operation, error-free geometric computation is not difficult to realize by means of integer arithemtic of appropriate data length. The present paper proposes an error-free, efficient computing method, which computes the 4×4 determinants by adaptively selecting integer arithmetic of variable data length. This technique is discussed theoretically and experimentally.

Applications of the 4 x 4 determinant method and the TRIANGLE PROCESSOR to various interference problems

In the beginning the paper reviews the 4×4 determinant method and the TRIANGLE PROCESSOR proposed by one of the present authors. The 4X4 determinant method makes it possible to unify various types of interference problems by computing some 4×4 determinants. The TRIANGLE PROCESSOR is a hardware processor based on the 4×4 determinant theories, which accelerates the processing speeds of these programs.