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Constructive projective extension of an incidence plane

Mark Mandelkern

Abstract

A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional pencils. Generalized definitions are adopted to construct a projective extension. The main axioms of projective geometry are verified. The methods used are in accord with Bishop-type modern constructivism.

Constructive projective extension of an incidence plane

Abstract

A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional pencils. Generalized definitions are adopted to construct a projective extension. The main axioms of projective geometry are verified. The methods used are in accord with Bishop-type modern constructivism.
Paper Structure (5 sections, 29 theorems, 9 equations)

This paper contains 5 sections, 29 theorems, 9 equations.

Table of Contents

  1. Preliminaries
  2. Pencils
  3. Virtual lines
  4. Extension points and extension lines
  5. Projective extension

Key Result

Lemma 2.2

A pencil may be contained in at most one regular pencil.

Theorems & Definitions (62)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • ...and 52 more