Study Guide - U.I.U.C. Math
... Poincare and Klein models, including distance formulae Parallel Axiom in hyperbolic geometry Angle measure in Poincare model; perpendicularity in Klein model Limiting parallels Ideal points and ideal triangles Angle defect Saccheri and Lambert quadrilaterals Area in hyperbolic geomet ...
... Poincare and Klein models, including distance formulae Parallel Axiom in hyperbolic geometry Angle measure in Poincare model; perpendicularity in Klein model Limiting parallels Ideal points and ideal triangles Angle defect Saccheri and Lambert quadrilaterals Area in hyperbolic geomet ...
Document
... )EBB’ is )DBB’ , and let us call )DB’B’s supplement )X, hence )X ≅ )DBB’. Angle )X and )EBB’ both share a side, and they are congruent, so by Axiom C4 they have to be equal (that is their remaining sides have to coincide). Since )X = )EBB’ is a supplement to )DB’B, we conclude that B’E and B’D are o ...
... )EBB’ is )DBB’ , and let us call )DB’B’s supplement )X, hence )X ≅ )DBB’. Angle )X and )EBB’ both share a side, and they are congruent, so by Axiom C4 they have to be equal (that is their remaining sides have to coincide). Since )X = )EBB’ is a supplement to )DB’B, we conclude that B’E and B’D are o ...
transformationunit
... What to Know in Chapter 16: Transformational Geometry To complete this chapter successfully student will: 1. Recognize (see and name) transformations including rigid and size transformations. 2. Define (both conceptually and abstractly) and use the four rigid motions: translation, rotation, reflecti ...
... What to Know in Chapter 16: Transformational Geometry To complete this chapter successfully student will: 1. Recognize (see and name) transformations including rigid and size transformations. 2. Define (both conceptually and abstractly) and use the four rigid motions: translation, rotation, reflecti ...
Program for ``Topology and Applications``
... Boris Doubrov: The classi ication of three-dimensional homogeneous spaces with non-solvable transformation groups Abstract: Sophus Lie classi ied all 1- and 2-dimensional homogeneous spaces and outlined the ideas of classifying 3-dimensional spaces in volume 3 of “Transformation groups” by him and F ...
... Boris Doubrov: The classi ication of three-dimensional homogeneous spaces with non-solvable transformation groups Abstract: Sophus Lie classi ied all 1- and 2-dimensional homogeneous spaces and outlined the ideas of classifying 3-dimensional spaces in volume 3 of “Transformation groups” by him and F ...
the group exercise in class on Monday March 28
... endpoints? Does this number depend on which points you choose? How does this compare to Euclidean geometry? Lines and Angle Measure 4. Construct two lines (great circles) on the sphere. It is important to remember that lines must divide the sphere into two identical pieces. In how many points do the ...
... endpoints? Does this number depend on which points you choose? How does this compare to Euclidean geometry? Lines and Angle Measure 4. Construct two lines (great circles) on the sphere. It is important to remember that lines must divide the sphere into two identical pieces. In how many points do the ...
Geometry (PDF 145KB)
... The term geometry is derived from the two Greek words geo and metron. It means "to measure the Earth." The great irony is that the most basic building block in geometry, The Point, has no measurement at all. Let’s begin by looking at points that must exist in order to have lines and planes, as lines ...
... The term geometry is derived from the two Greek words geo and metron. It means "to measure the Earth." The great irony is that the most basic building block in geometry, The Point, has no measurement at all. Let’s begin by looking at points that must exist in order to have lines and planes, as lines ...
Lie sphere geometry
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous ""line-sphere correspondence"" between the space of lines and the space of spheres in 3-dimensional space.