* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

# Download Spherical Geometry

Survey

Document related concepts

Multilateration wikipedia , lookup

Cartan connection wikipedia , lookup

Algebraic geometry wikipedia , lookup

Projective plane wikipedia , lookup

Shape of the universe wikipedia , lookup

Analytic geometry wikipedia , lookup

Cartesian coordinate system wikipedia , lookup

Contour line wikipedia , lookup

Perspective (graphical) wikipedia , lookup

Tensor operator wikipedia , lookup

History of trigonometry wikipedia , lookup

Rational trigonometry wikipedia , lookup

Lie sphere geometry wikipedia , lookup

Duality (projective geometry) wikipedia , lookup

Geometrization conjecture wikipedia , lookup

Hyperbolic geometry wikipedia , lookup

History of geometry wikipedia , lookup

Transcript

Spherical Geometry Sep 282:31 PM 1 Euclidean Geometry Spherical Geometry 1. Lines extends indefinitely and has no thickness or width. 1. Line is a great circle that divides the sphere into two equal halfspheres 2. A line is the shortest path between two points. 2. There is a unique great circle passing through any pair of nonpolar points. 3. A straight line is infinite. 3. A great circle is finite and returns to its original starting point. 4. Given three collinear points, one point is always between the other two. 4. Given three collinear points, each point could be in the middle of the other two. 5. Perpendicular lines intersect at one point. 5. Perpendicular lines intersect at two points. 6. Perpendicular lines form four right angles. 6. Perpendicular lines form eight right angles. Sep 281:33 PM 2 Euclid's Fifth Postulate (Parallel Postulate) If there is a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line. Is this true in spherical geometry? Why? NO! There are no parallel lines in spherical geometry. Sep 282:13 PM 3