Library/Geometry/Projective geometry/Projective geometry (other)

Projective geometry (other)

Overview
Important

Projective geometry studies properties of figures that remain unchanged under projective transformations, such as perspective projections. Beyond the basics (points, lines, and the projective plane), projective geometry includes concepts like duality, cross ratio, conic sections, and harmonic division.

Important properties

  • Duality: Points and lines can be interchanged in many theorems.

  • Cross ratio: A special value associated with four collinear points, preserved by projective transformations.

  • Harmonic division: A special case of cross ratio where the value is -1.

  • Conics: Circles, ellipses, parabolas, and hyperbolas can all be seen as projective conics.

  • Desargues' and Pappus' Theorems: Fundamental results unique to projective geometry.