Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constructive Projective Geometry

Mark Mandelkern

Abstract

Of the great theories of classical mathematics, projective geometry, with its powerful concepts of symmetry and duality, has been exceptional in continuing to intrigue investigators. The challenge put forth by Errett Bishop (1928-1983), "Every theorem proved with nonconstructive methods presents a challenge: to find a constructive version, and to give it a constructive proof", motivates a large portion of current constructive work. This challenge can be answered by discovering the hidden constructive content of classical projective geometry. Here we briefly outline, with few details, recent constructive work on the real projective plane, and projective extensions of affine planes. Special note is taken of a number of interesting open problems that remain; these show that constructive projective geometry is still a theory very much in need of further effort. The Bishop-type constructive mathematics discussed in the present paper proceeds from a viewpoint well-nigh opposite that of either formal logic or recursive function theory.

Constructive Projective Geometry

Abstract

Of the great theories of classical mathematics, projective geometry, with its powerful concepts of symmetry and duality, has been exceptional in continuing to intrigue investigators. The challenge put forth by Errett Bishop (1928-1983), "Every theorem proved with nonconstructive methods presents a challenge: to find a constructive version, and to give it a constructive proof", motivates a large portion of current constructive work. This challenge can be answered by discovering the hidden constructive content of classical projective geometry. Here we briefly outline, with few details, recent constructive work on the real projective plane, and projective extensions of affine planes. Special note is taken of a number of interesting open problems that remain; these show that constructive projective geometry is still a theory very much in need of further effort. The Bishop-type constructive mathematics discussed in the present paper proceeds from a viewpoint well-nigh opposite that of either formal logic or recursive function theory.
Paper Structure (25 sections, 25 theorems, 8 equations)

This paper contains 25 sections, 25 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Real projective plane
  3. Axioms
  4. Axiom Group C
  5. Desargues's Theorem
  6. Duality
  7. Harmonic conjugates
  8. Projectivities
  9. Fundamental Theorem
  10. Conics
  11. Construction of a conic
  12. Properties
  13. Pascal's Theorem
  14. Polarity
  15. Tangents and secants
  16. ...and 10 more sections

Key Result

Theorem 1

Converse of Desargues's Theorem. If two triangles are perspective from an axis, then they are perspective from a center. [M16, Thm. 3.2]

Theorems & Definitions (44)

  • Definition
  • Theorem
  • proof
  • Theorem
  • Theorem
  • Definition
  • Theorem
  • Corollary
  • Theorem
  • Theorem
  • ...and 34 more