Projective Geometry

Projective Gometry is a branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface. Common examples are the shadows cast by opaque objects and motion pictures displayed on a screen.

In relation to perspective there are different kinds of projection as explained below.

Perspective and Projection

Perspective Category Theory identifies 3 basic classes of perspective, corresponding to the goals of perspective: to view, match, represent, create an illusion of, or immerse in the visual appearance of a spatial object/scene. Accordingly, optical perspective <IMAGING CLASS> is any perspective that uses, or purports to use, light, or EM radiation (ref. real, imaginary, or simulated light-rays, etc), to capture images/views of a spatial reality.

We also have the optical perspective <PROJECTING CLASS>, which projects images/shadows forward into spatial reality. Finally, we have the optical perspective <ILLUSIVE / IMMERSIVE CLASS>, which creates the illusion of, or immersion into, a real, simulated, or virtual optical world overlaid, displayed, or inserted into the human visual field.

It is important to realise that just as with composite perspective (category chaining) and category overloading, we can have examples of class chaining or composite perspective classes, as well as class overloading.

Projective Geometry (Mathematical)

In the context of projective geometry as explored by H.S.M. Coxeter, perspective transformations (often referred to as collineations or homologies) are classified based on the relationship between their centre (O) and their axis (s).

Coxeter’s work, particularly in his seminal text Projective Geometry, identifies five general types of collineations based on their fixed elements. However, when considering the broader categorisation of projective transformations of the plane into itself, there are seven basic types distinguished by the configuration of their invariant points and lines:

  1. General Homology: An axis a, a centre O (not on a), and another pair of corresponding points.
  2. Special Homology: The axis and centre are the same as Type 1, but with special characteristics.
  3. Elation (Type 2): An axis a and a centre that lies on a.
  4. A more restricted for of perspectivity.
  5. A specific case of projective collinearity.
  6. A specific case of projective collinearity.
  7. General collineation: No specific perspective centre or axis (sometimes classified as a generallinear transformation).

In Introduction to Geometry, Coxeter also classifies affinities (a type of projective transformation), which can include translations, rotations, reflections, and dilations. The exact numbering 1-7 for perspective transformations can vary between Coxeter’s primary text and textbooks that summarise his work. The most fundamental perspective transformations in Coxeter’s framework are the Homology and the Elation.

Projective Geometry of Perspective

The classes and/or types of perspective projection system are listed as follows

  • Orthographic Projection
  • Horizontal Oblique Projection
  • Vertical Oblique Projection
  • Oblique Projection
  • Orthogonal Converging
  • Isometric Projection
  • Perspective Projection (see: foreshortening)

Desargues Perspective Theorem

Desargues’s Perspective Theorem, published in 1648, links 3-D and 2-D geometry, stating that two triangles are in perspective axially if and only if they are in perspective centrally. Desargues’s theorem is a cornerstone of projective geometry, influencing both classical mathematics and computer graphics.

The contribution of Desargues to perspective consists of the definition of points and lines to infinity, found in his Brouillon Project. The theorem states that if two triangles ABC and A′B′C′, situated in three-dimensional space, are related to each other in such a way that they can be seen perspectively from one point (i.e., the lines AA′, BB′, and CC′ all intersect in one point), then the points of intersection of corresponding sides all lie on one line, provided that no two corresponding sides are parallel.

This concept is not only the fundamental principle of projective geometry, but is also an essential moment in the evolution of theory and practice of perspective, since it gives a general meaning to the punctum concursus (vanishing point).

Put simply, Desargues brought understanding to infinity, and optical/technical perspective became an instrument capable of treating the infinite in finite terms.