स्नातक (प्रारंभिक)Geometry & Topologyपाठ उपलब्ध
Non-Euclidean geometry: spherical and hyperbolic
What happens when the parallel postulate fails: geometry on the sphere, the Poincaré disc and half-plane models of the hyperbolic plane, angle sums and area, curvature as the measure of the difference, and the independence of the parallel postulate.
math.geom.non-euclidean-geometry
इस पर आधारित
25 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Work with great circles, spherical triangles and the spherical law of cosines, and prove that the angle sum of a spherical triangle exceeds π by its area divided by R².
- Describe geodesics and angles in the Poincaré disc and upper half-plane, and explain why the angle sum of a hyperbolic triangle is less than π.
- Explain what it means for the parallel postulate to be independent of Euclid's other axioms, and how models establish this.
परिभाषा
- A great circle is the intersection of a sphere with a plane through its centre; arcs of great circles are the shortest paths on the sphere.
- In the Poincaré disc model, points are the points of the open unit disc and lines are diameters and arcs of circles meeting the boundary circle at right angles.
- The angle excess of a spherical triangle with angles α, β, γ is α + β + γ − π; the defect of a hyperbolic triangle is π − (α + β + γ).
प्रमेय
- Girard's theorem: a spherical triangle on a sphere of radius R has area R²(α + β + γ − π).
- In the hyperbolic plane of curvature −1 a triangle has area π − (α + β + γ); in particular every triangle has area less than π.
- In hyperbolic geometry, through a point not on a line there are infinitely many lines that do not meet it.
- The parallel postulate is independent of the other axioms of Euclidean geometry (Beltrami, Klein, Poincaré).
आम भ्रांतियाँ
- Thinking non-Euclidean geometry is a contradiction of Euclid; it is a different consistent geometry obtained by changing one axiom.
- Believing the 'lines' of the Poincaré disc are curved in the hyperbolic plane itself; they are curved only in the Euclidean picture of the model.
- Assuming spherical geometry satisfies all of Euclid's axioms except the parallel postulate; two antipodal points lie on infinitely many great circles, so other axioms also change.
आगे ले जाता है
वैकल्पिक खोजें
- Riemannian geometry Spherical, Euclidean and hyperbolic geometry are the model spaces of constant curvature +1, 0 and −1; Riemannian geometry studies spaces whose curvature varies from point to point.
- Model theory The hyperbolic models show that the parallel postulate cannot be proved from Euclid's other axioms: a consistent model of the other axioms in which it fails exists (relative to the consistency of Euclidean geometry).
- ← The Pythagorean theorem
अधिगम पथ
आगे पढ़ने के लिए
- Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries (2008)
- Robin Hartshorne, Geometry: Euclid and Beyond (2000)
- James W. Anderson, Hyperbolic Geometry (2005)