Clipping using homogeneous coordinates

ABSTRACT

Clipping is the process of determining how much of a given line segment lies within the boundaries of the display screen. Homogeneous coordinates are a convenient mathematical device for representing and transforming objects. The space represented by homogeneous coordinates is not, however, a simple Euclidean 3-space. It is, in fact, analagous to a topological shape called a “projective plane”. The clipping problem is usually solved without consideration for the differences between Euclidean space and the space represented by homogeneous coordinates. For some constructions, this leads to errors in picture generation which show up as lines marked invisible when they should be visible. This paper will examine these cases and present techniques for correctly clipping the line segments.

REFERENCES

• 1: Barr, S., Experiments in Topology, Thomas Y. Crowell Co., New York, 1964.
• 2: Riesenfeld, R. F., "Homogeneous Coordinates and Projective Planes in Computer Graphics", JACM, to appear.
• 3: Roberts, L. G., "Homogeneous Matrix Representation and Manipulation of N-Dimensional Constructs", MIT Lincoln Laboratory, MS 1405, May 1965.
• 4: Ivan E. Sutherland , Gary W. Hodgman, Reentrant polygon clipping, Communications of the ACM, v.17 n.1, p.32-42, Jan. 1974 [doi>10.1145/360767.360802]

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.3 Picture/Image Generation
          Subjects: Display algorithms

Additional Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Hierarchy and geometric transformations

General Terms:
Theory