Hyperbolic geometry: sum of angles and Poincaré model
... 1.1.1. Theorem. The angle of parallelism varies inversely with the segment. (1) AB < A′ B ′ ⇐⇒ α(AB) > α(A′ B ′ ); (2) AB ∼ = A′ B ′ ⇐⇒ α(AB) = α(A′ B ′ ). Proof. We shall just prove that AB ∼ = A′ B ′ =⇒ α(AB) = α(A′ B ′ ) and AB < ...
... 1.1.1. Theorem. The angle of parallelism varies inversely with the segment. (1) AB < A′ B ′ ⇐⇒ α(AB) > α(A′ B ′ ); (2) AB ∼ = A′ B ′ ⇐⇒ α(AB) = α(A′ B ′ ). Proof. We shall just prove that AB ∼ = A′ B ′ =⇒ α(AB) = α(A′ B ′ ) and AB < ...
Non-Euclidean Geometry
... In doing so, he introduced the idea of deductive proof, employing the axiomatic method. The idea here was to choose a list of axioms, that one would take as being self-evidently true, and then to deduce from these more complicated theorems. ...
... In doing so, he introduced the idea of deductive proof, employing the axiomatic method. The idea here was to choose a list of axioms, that one would take as being self-evidently true, and then to deduce from these more complicated theorems. ...
Geometry 1: Intro to Geometry Introduction to Geometry
... G-CO.9 I can prove theorems about lines and angles. 15. Draw a diagram that fits the following criteria: Draw two lines and a transversal such that 1 andA 2 are corresponding angles, 2 and 3 are vertical angles, and 3 and 4 are corresponding angles. What type of angle pair is 1 and ...
... G-CO.9 I can prove theorems about lines and angles. 15. Draw a diagram that fits the following criteria: Draw two lines and a transversal such that 1 andA 2 are corresponding angles, 2 and 3 are vertical angles, and 3 and 4 are corresponding angles. What type of angle pair is 1 and ...
Geometry 1 - Phoenix Union High School District
... are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 15. Draw a diagram that fits the following criteria: Draw two lines and a transversal such that 1 and 2 are corresponding angles, ...
... are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. 15. Draw a diagram that fits the following criteria: Draw two lines and a transversal such that 1 and 2 are corresponding angles, ...
If the lines are parallel, then
... into many varied towers, mathematicians assemble a few definitions and assumptions into many varied theorems. The blocks are assembled with Hands, the axioms are assembled with Reason. All of Euclidean Geometry (the thousands of theorems) were all put together with a few different kinds of blocks. T ...
... into many varied towers, mathematicians assemble a few definitions and assumptions into many varied theorems. The blocks are assembled with Hands, the axioms are assembled with Reason. All of Euclidean Geometry (the thousands of theorems) were all put together with a few different kinds of blocks. T ...
Test #1 Review
... Compute various numbers, figures, correspondences, and equations using the concepts we have studied (e.g., congruence of triangles, and the Poincare Half-plane model of Hyperbolic Geometry). Understand the proofs presented in class, but, except for a proof that the EPP is equivalent to a particular ...
... Compute various numbers, figures, correspondences, and equations using the concepts we have studied (e.g., congruence of triangles, and the Poincare Half-plane model of Hyperbolic Geometry). Understand the proofs presented in class, but, except for a proof that the EPP is equivalent to a particular ...
Slide 1
... parallel postulate Given a line L and a point P not on L, there exists a unique line parallel to L through the point P. P ...
... parallel postulate Given a line L and a point P not on L, there exists a unique line parallel to L through the point P. P ...
8. Hyperbolic triangles
... geometry), the length of the hypotenuse is not substantially shorter than the sum of the lengths of the other two sides. Proof. Let ∆ be a triangle satisfying the hypotheses of the theorem. By applying a Möbius transformation, we may assume that the vertex with internal angle π/2 is at i and that t ...
... geometry), the length of the hypotenuse is not substantially shorter than the sum of the lengths of the other two sides. Proof. Let ∆ be a triangle satisfying the hypotheses of the theorem. By applying a Möbius transformation, we may assume that the vertex with internal angle π/2 is at i and that t ...
Absolute geometry
... Under the correspondence ABC ↔ XY Z, let two sides and the included angle of ABC be congruent, respectively, to the corresponding two sides and included angle of XY Z. That is, for example, AB ∼ ...
... Under the correspondence ABC ↔ XY Z, let two sides and the included angle of ABC be congruent, respectively, to the corresponding two sides and included angle of XY Z. That is, for example, AB ∼ ...
Geometry High Honors - Montclair Public Schools
... Recognize and apply basic properties of parallel and perpendicular lines. Understand and use the relationships between seven major types of quadrilaterals to solve problems. Recognize convex polygons and solve ...
... Recognize and apply basic properties of parallel and perpendicular lines. Understand and use the relationships between seven major types of quadrilaterals to solve problems. Recognize convex polygons and solve ...
Euclid axioms.
... there are infinitely many lines parallel to the given line through the point. All the other axioms were the same. This new geometry was shown to be consistent, and so another geometry could stand alongside Euclid’s as a possible way of modelling the universe. ...
... there are infinitely many lines parallel to the given line through the point. All the other axioms were the same. This new geometry was shown to be consistent, and so another geometry could stand alongside Euclid’s as a possible way of modelling the universe. ...
Geometry Unit 18: Euclidean vs Non-Euclidean Geometry 2009-2010
... Geometry Unit 18: Euclidean vs Non-Euclidean Geometry 2009-2010 ...
... Geometry Unit 18: Euclidean vs Non-Euclidean Geometry 2009-2010 ...
Theorem 6.3.1 Angle Sum Theorem for Hyperbolic Geometry
... This sketch depic ts the hyperbolic plane H2 us in g the Poincaré disk model. In this model, a line through tw o poin ts is def ined as the Euc lidean arc pas sing through the points and perpendic ular to the c irc le . Us e this document's custom tools to perform c onstructions on the hyperbolic pl ...
... This sketch depic ts the hyperbolic plane H2 us in g the Poincaré disk model. In this model, a line through tw o poin ts is def ined as the Euc lidean arc pas sing through the points and perpendic ular to the c irc le . Us e this document's custom tools to perform c onstructions on the hyperbolic pl ...
Euclidean Geometry and History of Non
... parallel line postulate with the postulate that no line may be drawn through a given point, parallel to a given line. Also known as Riemannian geometry. Read more: http://www.answers.com/topic/elliptic-geometry-1#ixzz1n8msBzGE Elliptic geometry (sometimes known as Riemannian geometry) is a non-Eucli ...
... parallel line postulate with the postulate that no line may be drawn through a given point, parallel to a given line. Also known as Riemannian geometry. Read more: http://www.answers.com/topic/elliptic-geometry-1#ixzz1n8msBzGE Elliptic geometry (sometimes known as Riemannian geometry) is a non-Eucli ...
QUESTIONS for latest set of presentations
... will never meet on either side. d. If two straight lines are cut by a transversal and the sum of the measure of the interior angles equals 180, then the two lines will never intersect, thus making them parallel. True or False: Saccheri was able to create a very convincing proof that showed if the ne ...
... will never meet on either side. d. If two straight lines are cut by a transversal and the sum of the measure of the interior angles equals 180, then the two lines will never intersect, thus making them parallel. True or False: Saccheri was able to create a very convincing proof that showed if the ne ...
Euclid`s Fifth Postulate
... (a) \If a straight line intersects one of two parallels (i.e, lines which do not intersect however far they are extended), it will intersect the other also." (b) \There is one and only one line that passes through any given point and is parallel to a given line." (c) \Given any gure there exists a ...
... (a) \If a straight line intersects one of two parallels (i.e, lines which do not intersect however far they are extended), it will intersect the other also." (b) \There is one and only one line that passes through any given point and is parallel to a given line." (c) \Given any gure there exists a ...
pdf of Non-Euclidean Presentation
... Beltrami and Klein made a model of nonEuclidean geometry in a disk, with chords being the lines. But angles are measured in a complicated way. Poincaré discovered a model made from points in a disk and arcs of circles orthogonal to the boundary of the disk. Angles are measured in the usual way. ...
... Beltrami and Klein made a model of nonEuclidean geometry in a disk, with chords being the lines. But angles are measured in a complicated way. Poincaré discovered a model made from points in a disk and arcs of circles orthogonal to the boundary of the disk. Angles are measured in the usual way. ...
Basics of Hyperbolic Geometry
... The map z → −z is just reflection in the vertical geodesic coming out of the point 0. This map is a hyperbolic isometry, and also fixes every point on the vertical geodesic. As another example, the map Let R(z) = 1/z is a hyperbolic isometry that fixes every point on the geodesic connecting −1 to 1. ...
... The map z → −z is just reflection in the vertical geodesic coming out of the point 0. This map is a hyperbolic isometry, and also fixes every point on the vertical geodesic. As another example, the map Let R(z) = 1/z is a hyperbolic isometry that fixes every point on the geodesic connecting −1 to 1. ...
File
... cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic. Euclid's (400BC) important geometry book The Elements formed the basis for most of the geometry studied in schools ever since. The Fifth c. 400 B.C. - 1800 A. D. Postulate Controvers Ther ...
... cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic. Euclid's (400BC) important geometry book The Elements formed the basis for most of the geometry studied in schools ever since. The Fifth c. 400 B.C. - 1800 A. D. Postulate Controvers Ther ...
15 the geometry of whales and ants non
... Legendre (whom we have met already) believed that he had proved it. So, at ...
... Legendre (whom we have met already) believed that he had proved it. So, at ...
Non-Euclidean Geometry - Department of Mathematics | Illinois
... The fifth postulate is very different from the first four and Euclid was not even completely satisfied with it Being that it was so different it led people to wonder if it is possible to prove the fifth postulate using the first four Many mathematicians worked with the fifth postulate and it actuall ...
... The fifth postulate is very different from the first four and Euclid was not even completely satisfied with it Being that it was so different it led people to wonder if it is possible to prove the fifth postulate using the first four Many mathematicians worked with the fifth postulate and it actuall ...
MATH 301 Survey of Geometries Homework Problems – Week 5
... (a) The two sides of each interior angle of a triangle ∆ on a sphere determine two congruent lunes with lune angle the same as the interior angle. Show how the three pairs of lunes determined by the three interior angles α, β, γ cover the sphere, with some overlap. (See ...
... (a) The two sides of each interior angle of a triangle ∆ on a sphere determine two congruent lunes with lune angle the same as the interior angle. Show how the three pairs of lunes determined by the three interior angles α, β, γ cover the sphere, with some overlap. (See ...
Euclid`s Postulates - Homeschool Learning Network
... 6. Draw two points on your orange. Given your answer to (5), how many different lines can you draw through those two points? Remember the definition of a line: a line is defined by two POINTS, and two POINTS define one and only one line. This definition never changes, no matter what geometry you are ...
... 6. Draw two points on your orange. Given your answer to (5), how many different lines can you draw through those two points? Remember the definition of a line: a line is defined by two POINTS, and two POINTS define one and only one line. This definition never changes, no matter what geometry you are ...
Hyperbolic geometry quiz solutions
... Note that this does not implt that k = ℓ or k = 1/ℓ (in the same way that 1 + 4 = 2 + 3 does not imply that 1 = 2 or 1 = 3). Multiplying by kℓ gives k2 ℓ + ℓ = kℓ2 + k which can be factorised as (k − ℓ)(kℓ − 1) = 0. Hence k = ℓ or k = 1/ℓ. [6 marks] k+ ...
... Note that this does not implt that k = ℓ or k = 1/ℓ (in the same way that 1 + 4 = 2 + 3 does not imply that 1 = 2 or 1 = 3). Multiplying by kℓ gives k2 ℓ + ℓ = kℓ2 + k which can be factorised as (k − ℓ)(kℓ − 1) = 0. Hence k = ℓ or k = 1/ℓ. [6 marks] k+ ...
Introduction to Geometry
... Label all your points. What are the measures of the two smaller angles? ...
... Label all your points. What are the measures of the two smaller angles? ...
Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry. In hyperbolic geometry the parallel postulate of Euclidean geometry is replaced with:For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.(compare this with Playfair's axiom the modern version of Euclid's parallel postulate)Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space.When geometers first realised they worked with something else than the standard Euclidean geometry they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry. It was for putting it in the now rarely used sequence elliptic geometry (spherical geometry) , parabolic geometry (Euclidean geometry), and hyperbolic geometry.In Russia it is commonly called Lobachevskian geometry after one of its discoverers, the Russian geometer Nikolai Lobachevsky.This page is mainly about the 2 dimensional or plane hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry.Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases.