Projective geometry

इस पर आधारित

33 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)

आप सीखेंगे

• Use homogeneous coordinates to describe points and lines of the projective plane, including points at infinity.
• State and prove the theorems of Desargues and Pappus, and explain the principle of duality.
• Show that projective transformations preserve cross-ratio and that all non-degenerate real conics are projectively equivalent.

परिभाषा

• The real projective plane ℝP² is the set of lines through the origin in ℝ³; the point with homogeneous coordinates [x : y : z] is the line through (x, y, z) ≠ 0.
• A projective transformation of ℝP² is a map induced by an invertible 3 × 3 matrix; the group of these is PGL(3, ℝ).
• The cross-ratio of four distinct collinear points with coordinates a, b, c, d on a line is ((a − c)(b − d)) / ((a − d)(b − c)).

प्रमेय

• Any two distinct lines in the projective plane meet in exactly one point, and any two distinct points lie on exactly one line.
• Cross-ratio is invariant under projective transformations.
• Any four points of ℝP², no three collinear, can be mapped to any other such four by a unique projective transformation.
• Desargues' theorem holds in every projective plane coordinatised by a division ring; Pappus' theorem holds exactly when the division ring is commutative.

आम भ्रांतियाँ

• Thinking there is a separate point at infinity for each direction and orientation; opposite directions give the same point at infinity.
• Believing distances and angles are projective notions; projective transformations preserve incidence and cross-ratio, not lengths or angles.

वैकल्पिक खोजें

• Algebraic geometry Projective space is the natural home of algebraic geometry: Bézout's theorem counts intersections of plane curves correctly only in the complex projective plane.
• Surfaces, polyhedra and the Euler characteristic Topologically, the real projective plane is a non-orientable surface with Euler characteristic 1: a disc with antipodal boundary points identified.
• ← Euclidean geometry and proof
• ← Conic sections

आगे पढ़ने के लिए

• H. S. M. Coxeter, Projective Geometry (1974)
• Jürgen Richter-Gebert, Perspectives on Projective Geometry (2011)