Hyperbolic plane geometry revisited

... We prove first those cases when all sides of the triangle lie on real lines, respectively. We assume that the right angle is at C and that it is a real point because of our definition. • If A is an infinite point B and C are real ones then sinh c · sin α = ∞ · 0 is indeterminable and we can consider ...

... We prove first those cases when all sides of the triangle lie on real lines, respectively. We assume that the right angle is at C and that it is a real point because of our definition. • If A is an infinite point B and C are real ones then sinh c · sin α = ∞ · 0 is indeterminable and we can consider ...

Practice Workbook

... 18. Matt made the conjecture that the sum of two numbers is always greater than either number. Find a counterexample for his conjecture. 19. Find a counterexample for the statement “All numbers are less than zero.” 20. Find a counterexample for the statement “All bears are brown.” ...

... 18. Matt made the conjecture that the sum of two numbers is always greater than either number. Find a counterexample for his conjecture. 19. Find a counterexample for the statement “All numbers are less than zero.” 20. Find a counterexample for the statement “All bears are brown.” ...

Practice - McGraw Hill Higher Education

... of the decimal point. 1-3. Sample answers given. 1. a rational number less than 0 with a 3-digit repeating pattern 2.0120120120. . . 2. an irrational number between 7 and 8 ...

... of the decimal point. 1-3. Sample answers given. 1. a rational number less than 0 with a 3-digit repeating pattern 2.0120120120. . . 2. an irrational number between 7 and 8 ...

Contemporary Arguments For A Geometry of Visual Experience

... This paper addresses an argument that has emerged in recent years from the work of James Van Cleve, Gideon Yaffe and Gordon Belot for a spherical geometry of visual experience. The argument in question consists of a demonstration that the geometry of what is visible and spherical geometry are equiv ...

... This paper addresses an argument that has emerged in recent years from the work of James Van Cleve, Gideon Yaffe and Gordon Belot for a spherical geometry of visual experience. The argument in question consists of a demonstration that the geometry of what is visible and spherical geometry are equiv ...

Lengths of simple loops on surfaces with hyperbolic metrics Geometry & Topology G

... 2.4 We will recall the results obtained in [10] concerning the change of xi coordinates by 2. Suppose (p1 ∪. . .∪p3g+r−3 , b, col) is a marking on an oriented surface F , and DT is the associated Dehn–Thurston coordinate. Let [a] and [b] be two isotopy classes of curve systems so that their twisting ...

... 2.4 We will recall the results obtained in [10] concerning the change of xi coordinates by 2. Suppose (p1 ∪. . .∪p3g+r−3 , b, col) is a marking on an oriented surface F , and DT is the associated Dehn–Thurston coordinate. Let [a] and [b] be two isotopy classes of curve systems so that their twisting ...

Word - The Open University

... (a) Through what angle do the students turn at A? (b) When they arrive at D, what is the total angle that they have turned through relative to their starting direction? (c) When they return to S, through what angle must they turn in order to face in the direction in which they started? (d) When ...

... (a) Through what angle do the students turn at A? (b) When they arrive at D, what is the total angle that they have turned through relative to their starting direction? (c) When they return to S, through what angle must they turn in order to face in the direction in which they started? (d) When ...

Lesson 3-1 and Lesson 3-2

... The fifth axiom of Euclidean Geometry states that given any straight line and a point not on it, there exists one and only one straight line that passes through that point and never intersects the first line. The fifth axiom is also known as the Parallel Postulate. ...

... The fifth axiom of Euclidean Geometry states that given any straight line and a point not on it, there exists one and only one straight line that passes through that point and never intersects the first line. The fifth axiom is also known as the Parallel Postulate. ...

non-euclidean geometry

... The Euclidean Model ..............................................................… ..........................44 The Klein Disk Model .................................................................… .....................45 The metric of KDM ......................................................... ...

... The Euclidean Model ..............................................................… ..........................44 The Klein Disk Model .................................................................… .....................45 The metric of KDM ......................................................... ...

Non-Euclidean Geometry - Digital Commons @ UMaine

... The Euclidean Model ..............................................................… ..........................44 The Klein Disk Model .................................................................… .....................45 The metric of KDM ......................................................... ...

... The Euclidean Model ..............................................................… ..........................44 The Klein Disk Model .................................................................… .....................45 The metric of KDM ......................................................... ...

Geometry Q1 - Rocky Ford School District

... preserve distance and angle to those that do not.2 (CCSS: G-CO.2) vi.Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (CCSS: G-CO.4) vii.Given a geometric figure and a ...

... preserve distance and angle to those that do not.2 (CCSS: G-CO.2) vi.Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (CCSS: G-CO.4) vii.Given a geometric figure and a ...

Curriculum Cluster 2

... Parallel and Perpendicular Lines a.1 Builds on knowledge of number, operation, and quantitative reasoning; patterns, and algebraic thinking, geometry, measurement, and probability and statistics, solve meaningful problems by representing and transforming figures and analyzing relationships. a.6 Make ...

... Parallel and Perpendicular Lines a.1 Builds on knowledge of number, operation, and quantitative reasoning; patterns, and algebraic thinking, geometry, measurement, and probability and statistics, solve meaningful problems by representing and transforming figures and analyzing relationships. a.6 Make ...

Cusps, triangle groups and hyperbolic 3-folds

... bounds for this volume are known, see [19, 9] for example. The latter estimates are off by at most one order of magnitude. In this paper we provide a number of interesting examples of small volume hyperbolic 3-orbifolds. For instance we provide an infinite family of co-finite volume discrete groups ...

... bounds for this volume are known, see [19, 9] for example. The latter estimates are off by at most one order of magnitude. In this paper we provide a number of interesting examples of small volume hyperbolic 3-orbifolds. For instance we provide an infinite family of co-finite volume discrete groups ...

Non-Euclidean Geometry and a Little on How We Got Here

... In the beginning geometry was a collection of rules for computing lengths, areas, and volumes. Many were crude approximations derived by trial and error. This body of knowledge, developed and used in construction, navigation, and surveying by the Babylonians and Egyptians, was passed to the Greeks. ...

... In the beginning geometry was a collection of rules for computing lengths, areas, and volumes. Many were crude approximations derived by trial and error. This body of knowledge, developed and used in construction, navigation, and surveying by the Babylonians and Egyptians, was passed to the Greeks. ...

X - Ms. Williams – Math

... the theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other (Theorem 3-6) using the Converse of the Alternate Interior Angles Theorem instead of the Converse of the Corresponding Angles Postulate? By the Vertical Angles Theorem, ...

... the theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other (Theorem 3-6) using the Converse of the Alternate Interior Angles Theorem instead of the Converse of the Corresponding Angles Postulate? By the Vertical Angles Theorem, ...

MasterJinHasvoldseter

... we are familiar with depends on the hypothesis that, given a line and a point not on that line, there exists one and only one line through the point parallel to the line. However, some of his proofs his books are less than perfect. Much work was done during the nineteenth century to attain the perfe ...

... we are familiar with depends on the hypothesis that, given a line and a point not on that line, there exists one and only one line through the point parallel to the line. However, some of his proofs his books are less than perfect. Much work was done during the nineteenth century to attain the perfe ...

Axiomatic Geometry: Euclid and Beyond

... To draw a straight-line from any point to any other point. To produce a finite straight-line continuously in a straight-line. To draw a circle with any center and radius. All right-angles are equal to one another. Parallel axiom: If a straight line ` cuts two other straight lines m and n such that t ...

... To draw a straight-line from any point to any other point. To produce a finite straight-line continuously in a straight-line. To draw a circle with any center and radius. All right-angles are equal to one another. Parallel axiom: If a straight line ` cuts two other straight lines m and n such that t ...

On Klein`s So-called Non

... that the use of the notion of transformation and of projective invariant by geometers like Poncelet8 had prepared the ground for Klein’s general idea that a geometry is a transformation group. The fact that the three constant curvature geometries (hyperbolic, Euclidean and spherical) can be develope ...

... that the use of the notion of transformation and of projective invariant by geometers like Poncelet8 had prepared the ground for Klein’s general idea that a geometry is a transformation group. The fact that the three constant curvature geometries (hyperbolic, Euclidean and spherical) can be develope ...

2205 Unit 1 NOTES - North Penn School District

... Theorem 2-2 Congruent Supplements Theorem: If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent ...

... Theorem 2-2 Congruent Supplements Theorem: If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent ...

Name - North Penn School District

... Theorem 2-5 If two angles are both congruent and supplementary, then each is a right angle. ...

... Theorem 2-5 If two angles are both congruent and supplementary, then each is a right angle. ...

Hyperbolic Geometry and 3-Manifold Topology

... one in Nη , the cover of N with fundamental group generated by η. If Nt (η) is a radius-t, tubular neighborhood of η in Nη , then orthogonal projection of Nη \ Nt (η) to Nt (η) inﬁnitesimally reduces arc length by at least cosh(d) where d is distance to Nt (η). Since γ is a uniformly bounded distanc ...

... one in Nη , the cover of N with fundamental group generated by η. If Nt (η) is a radius-t, tubular neighborhood of η in Nη , then orthogonal projection of Nη \ Nt (η) to Nt (η) inﬁnitesimally reduces arc length by at least cosh(d) where d is distance to Nt (η). Since γ is a uniformly bounded distanc ...

isosceles trapezoid

... An isosceles trapezoid is a trapezoid whose legs are congruent and that has two congruent angles such that their common side is a base of the trapezoid. Thus, in an isosceles trapezoid, any two angles whose common side is a base of the trapezoid are congruent. In Euclidean geometry, the convention i ...

... An isosceles trapezoid is a trapezoid whose legs are congruent and that has two congruent angles such that their common side is a base of the trapezoid. Thus, in an isosceles trapezoid, any two angles whose common side is a base of the trapezoid are congruent. In Euclidean geometry, the convention i ...

6.1 Warm Up The diagram includes a pair of congruent triangles

... Geometry 5.1 Perpendiculars and Bisectors ...

... Geometry 5.1 Perpendiculars and Bisectors ...

alternate interior angles

... Definition and illustration (if applicable): a geometric system based on the five postulates of Euclid 1. A straight line can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn havin ...

... Definition and illustration (if applicable): a geometric system based on the five postulates of Euclid 1. A straight line can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn havin ...

Geometry Vocabulary

... Definition and illustration (if applicable): a geometric system based on the five postulates of Euclid 1. A straight line can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn havin ...

... Definition and illustration (if applicable): a geometric system based on the five postulates of Euclid 1. A straight line can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn havin ...

# 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.