The Three Reflections Theorem


 Samson Casey
 1 years ago
 Views:
Transcription
1 The Three Reflections Theorem Christopher Tuffley Institute of Fundamental Sciences Massey University, Palmerston North 24th June 2009 Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
2 Outline 1 The Three Twodimensional Geometries Euclidean Spherical Hyperbolic 2 The Three Reflections Theorem Statement Proof 3 Orientation preserving isometries Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
3 The Three Twodimensional Geometries The Euclidean plane Euclidean The Euclidean plane is with the Euclidean distance d ( (x 1, y 1 ),(x 2, y 2 ) ) = E 2 = {(x, y) x, y R}, (x 1 x 2 ) 2 + (y 1 y 2 ) 2. d (x 2, y 2 ) (x 1, y 1 ) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
4 Arc length The Three Twodimensional Geometries Euclidean If γ : [a, b] E 2 is a smooth curve then length(γ) = b a ds, where ds 2 = dx 2 + dy 2 is the infinitesimal metric. γ(b) γ(a) γ (t) = ( dx dt ) 2 ( ) 2 + dy dt The distance from P to Q is the infimum of {length(γ) γ a curve from P to Q}. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
5 The Three Twodimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Translations Rotations Orientation reversing: Reflections Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
6 The Three Twodimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Translations Rotations Orientation reversing: Reflections Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
7 The Three Twodimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Translations Rotations Orientation reversing: Reflections Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
8 The Three Twodimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Translations Rotations Orientation reversing: Reflections Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
9 The Three Twodimensional Geometries Euclidean Euclidean isometries Definition An isometry is a distance preserving map. Euclidean examples Orientation preserving: Translations Rotations Orientation reversing: Reflections Glide reflections Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
10 The Three Twodimensional Geometries Spherical geometry Spherical Restrict the 3dimensional Euclidean metric to the unit sphere S 2 in R 3. ds 2 = dx 2 + dy 2 + dz 2 Arc length on S 2 is given by (3d) Euclidean arc length. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
11 The Three Twodimensional Geometries Lines in spherical geometry Spherical Lines in spherical geometry are great circles: the intersection of a plane through the origin with S 2. Great circles are geodesics: locally length minimising curves. Any two lines (great circles) intersect in a pair of antipodal points. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
12 The Three Twodimensional Geometries Lines in spherical geometry Spherical Lines in spherical geometry are great circles: the intersection of a plane through the origin with S 2. Great circles are geodesics: locally length minimising curves. Any two lines (great circles) intersect in a pair of antipodal points. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
13 The Three Twodimensional Geometries Spherical Spherical isometries Spherical isometries include rotations about a diameter reflections in a plane through the origin. A reflection in a plane through the origin may be regarded as a reflection in the corresponding great circle, i.e. spherical line. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
14 The Three Twodimensional Geometries Hyperbolic Hyperbolic geometry: the upper half plane model Hyperbolic geometry may be modelled by the upper half plane H 2 = {(x, y) R 2 y > 0}, with metric ds 2 = dx 2 + dy 2 y 2. The vectors shown all have the same hyperbolic length. Hyperbolic angle in H 2 coincides with Euclidean angle. Other models exist, including the conformal disc model. y x Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
15 The Three Twodimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 is a vertical ray x = constant, or a semicircle with centre on the xaxis. There is a unique line through any pair of distinct points Disjoint lines may be asymptotic or ultraparallel. The xaxis together with forms the circle at infinity. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
16 The Three Twodimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 is a vertical ray x = constant, or a semicircle with centre on the xaxis. There is a unique line through any pair of distinct points Disjoint lines may be asymptotic or ultraparallel. The xaxis together with forms the circle at infinity. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
17 The Three Twodimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 is a vertical ray x = constant, or a semicircle with centre on the xaxis. There is a unique line through any pair of distinct points Disjoint lines may be asymptotic or ultraparallel. The xaxis together with forms the circle at infinity. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
18 The Three Twodimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 is preserved by Horizontal translations z z + c, c real Euclidean dilations z ρz, ρ > 0 Reflections in vertical rays e.g. z z Inversions in semicircular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
19 The Three Twodimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 is preserved by Horizontal translations z z + c, c real Euclidean dilations z ρz, ρ > 0 Reflections in vertical rays e.g. z z Inversions in semicircular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
20 The Three Twodimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 is preserved by Horizontal translations z z + c, c real Euclidean dilations z ρz, ρ > 0 Reflections in vertical rays e.g. z z Inversions in semicircular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
21 The Three Twodimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 is preserved by Horizontal translations z z + c, c real Euclidean dilations z ρz, ρ > 0 Reflections in vertical rays e.g. z z Inversions in semicircular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
22 The Three Twodimensional Geometries Hyperbolic Hyperbolic isometries The metric ds 2 = dx 2 + dy 2 y 2 is preserved by Horizontal translations z z + c, c real Euclidean dilations z ρz, ρ > 0 Reflections in vertical rays e.g. z z Inversions in semicircular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
23 The Three Reflections Theorem Statement The Three Reflections Theorem The following hold in each of the three geometries E 2, S 2 and H 2. Theorem (Characterisation of lines) P The set of points equidistant from a pair of distinct points P and Q is a line. Reflection in this line exchanges P and Q. Q Conversely, every line is the set of points equidistant from a suitably chosen pair of points P, Q. Corollary (The Three Reflections Theorem) Any isometry is a product of at most three reflections. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
24 The Three Reflections Theorem Proof Step 1: three points determine an isometry Lemma Any point P is uniquely determined by its distances to three noncollinear points A, B, C. Consequently, any isometry is completely determined by the images of any three noncollinear points. A B P C Proof. Suppose Q has the same distances to A, B, C. Then A, B, C must lie on the line equidistant from P and Q, contradicting the fact they are not collinear. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
25 The Three Reflections Theorem Proof Step 1: three points determine an isometry Lemma Any point P is uniquely determined by its distances to three noncollinear points A, B, C. A P Consequently, any isometry is completely determined by the images of any three noncollinear points. Q B C Proof. Suppose Q has the same distances to A, B, C. Then A, B, C must lie on the line equidistant from P and Q, contradicting the fact they are not collinear. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
26 The Three Reflections Theorem Proof Step 1: three points determine an isometry Lemma Any point P is uniquely determined by its distances to three noncollinear points A, B, C. A P Consequently, any isometry is completely determined by the images of any three noncollinear points. Q B C Proof. Suppose Q has the same distances to A, B, C. Then A, B, C must lie on the line equidistant from P and Q, contradicting the fact they are not collinear. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
27 The Three Reflections Theorem Proof Step 2: decompose isometries into reflections. Given an isometry φ, let A, B, C be noncollinear. 1 If A φ(a), reflect in the line equidistant from A and φ(a). 2 If B φ(b), reflect in the line equidistant from B and φ(b). 3 If C φ(c), reflect in the line equidistant from C and φ(c). A φ(a) φ(c) φ(b) C B The product of these reflections must be φ, because it coincides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
28 The Three Reflections Theorem Proof Step 2: decompose isometries into reflections. Given an isometry φ, let A, B, C be noncollinear. 1 If A φ(a), reflect in the line equidistant from A and φ(a). 2 If B φ(b), reflect in the line equidistant from B and φ(b). 3 If C φ(c), reflect in the line equidistant from C and φ(c). A φ(a) φ(c) φ(b) C B B C The product of these reflections must be φ, because it coincides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
29 The Three Reflections Theorem Proof Step 2: decompose isometries into reflections. Given an isometry φ, let A, B, C be noncollinear. 1 If A φ(a), reflect in the line equidistant from A and φ(a). 2 If B φ(b), reflect in the line equidistant from B and φ(b). 3 If C φ(c), reflect in the line equidistant from C and φ(c). A φ(a) φ(c) φ(b) C C B B C The product of these reflections must be φ, because it coincides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
30 The Three Reflections Theorem Proof Step 2: decompose isometries into reflections. Given an isometry φ, let A, B, C be noncollinear. 1 If A φ(a), reflect in the line equidistant from A and φ(a). 2 If B φ(b), reflect in the line equidistant from B and φ(b). 3 If C φ(c), reflect in the line equidistant from C and φ(c). A φ(a) φ(c) φ(b) C C B B C The product of these reflections must be φ, because it coincides on A, B, C. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
31 Orientation preserving isometries Orientation preserving isometries Orientation preserving isometries are products of two reflections. In Euclidean geometry, there are two cases: intersecting mirror lines: rotations parallel mirror lines: translations Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
32 Orientation preserving isometries Orientation preserving isometries Orientation preserving isometries are products of two reflections. In Euclidean geometry, there are two cases: intersecting mirror lines: rotations parallel mirror lines: translations Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
33 Orientation preserving isometries Orientation preserving isometries Orientation preserving isometries are products of two reflections. In Euclidean geometry, there are two cases: intersecting mirror lines: rotations parallel mirror lines: translations Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
34 Orientation preserving isometries Orientation preserving isometries Orientation preserving isometries are products of two reflections. In Euclidean geometry, there are two cases: intersecting mirror lines: rotations parallel mirror lines: translations Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
35 Orientation preserving isometries Orientation preserving isometries Orientation preserving isometries are products of two reflections. In Euclidean geometry, there are two cases: intersecting mirror lines: rotations parallel mirror lines: translations Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
36 Orientation preserving isometries Orientation preserving isometries Orientation preserving isometries are products of two reflections. In Euclidean geometry, there are two cases: intersecting mirror lines: rotations parallel mirror lines: translations Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
37 Orientation preserving isometries Orientation preserving isometries Orientation preserving isometries are products of two reflections. In Euclidean geometry, there are two cases: intersecting mirror lines: rotations parallel mirror lines: translations Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
38 The sphere Orientation preserving isometries Any two distinct lines in S 2 intersect = every orientation preserving isometry of S 2 is a rotation. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
39 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
40 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
41 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
42 Orientation preserving isometries The hyperbolic plane Three cases: intersecting lines: rotations asymptotic lines: limit rotations (includes horizontal Euclidean translations) ultraparallel lines: hyperbolic translations (includes Euclidean dilations) Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
43 Orientation preserving isometries Orientation preserving isometries, classified by pairs of reflections Spherical intersecting lines rotation disjoint lines Euclidean rotation parallel lines: translation Hyperbolic rotation asymptotic lines: ultraparallel lines: limit rotation translation Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
44 Going further Going further In each geometry, an orientation reversing isometry is a glide reflection. Subgroups of the isometry group lead to quotient surfaces with the given geometry. Euclidean threespace has a Four Reflections Theorem. There are eight model geometries in three dimensions: E 3, S 3, H 3, S 2 E 1, H 2 E 1, Nil, SL 2 R, Solv. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19
halfline the set of all points on a line on a given side of a given point of the line
Geometry Week 3 Sec 2.1 to 2.4 Definition: section 2.1 halfline the set of all points on a line on a given side of a given point of the line notation: is the halfline that contains all points on the
More informationUNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE
UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE NINA LEUNG Abstract. This paper discusses 2dimensional locally Euclidean geometries and how these geometries can describe musical chords. Contents
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is NonEuclidean Geometry? Most geometries on the plane R 2 are noneuclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationalternate interior angles
alternate interior angles two nonadjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationLecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS
1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps
More informationGeometry Chapter 1 Vocabulary. coordinate  The real number that corresponds to a point on a line.
Chapter 1 Vocabulary coordinate  The real number that corresponds to a point on a line. point  Has no dimension. It is usually represented by a small dot. bisect  To divide into two congruent parts.
More informationGeometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
More informationPoincaré Models of Hyperbolic Geometry
Chapter 5 Poincaré Models of Hyperbolic Geometry 5.1 The Poincaré Upper Half Plane Model The first model of the hyperbolic plane that we will consider is due to the French mathematician Henri Poincaré.
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationSolutions to Practice Problems
Higher Geometry Final Exam Tues Dec 11, 57:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles
More informationLecture 26 181. area = πr 2 c 4 r4 + o(r 4 ).
Lecture 26 181 Figure 4.5. Relating curvature to the circumference of a circle. the plane with radius r (Figure 4.5). We will see that circumference = 2πr cr 3 + o(r 3 ) where c is a constant related to
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationTheorem 3.1. If two circles meet at P and Q, then the magnitude of the angles between the circles is the same at P and Q.
3 rthogonal circles Theorem 3.1. If two circles meet at and, then the magnitude of the angles between the circles is the same at and. roof. Referring to the figure on the right, we have A B AB (by SSS),
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More informationPROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin
PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationWhat is inversive geometry?
What is inversive geometry? Andrew Krieger July 18, 2013 Throughout, Greek letters (,,...) denote geometric objects like circles or lines; small Roman letters (a, b,... ) denote distances, and large Roman
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationDetermine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationExploring Spherical Geometry
Exploring Spherical Geometry Introduction The study of plane Euclidean geometry usually begins with segments and lines. In this investigation, you will explore analogous objects on the surface of a sphere,
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationLecture 2: Homogeneous Coordinates, Lines and Conics
Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4
More informationGeometry Notes Chapter 12. Name: Period:
Geometry Notes Chapter 1 Name: Period: Vocabulary Match each term on the left with a definition on the right. 1. image A. a mapping of a figure from its original position to a new position. preimage B.
More informationConjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)
Mathematical Sentence  a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement
More informationWHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?
WHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More informationTHE THURSTON METRIC ON HYPERBOLIC DOMAINS AND BOUNDARIES OF CONVEX HULLS
THE THURSTON METRIC ON HYPERBOLIC DOMAINS AND BOUNDARIES OF CONVEX HULLS MARTIN BRIDGEMAN AND RICHARD D. CANARY Abstract. We show that the nearest point retraction is a uniform quasiisometry from the
More informationHyperbolic Islamic Patterns A Beginning
Hyperbolic Islamic Patterns A Beginning Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 558122496, USA Email: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/
More informationFixed Point Theorems in Topology and Geometry
Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler
More informationFinal Review Geometry A Fall Semester
Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over
More information4.1 Euclidean Parallelism, Existence of Rectangles
Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles Definition 4.1 Two distinct
More informationCoordinate Plane Project
Coordinate Plane Project C. Sormani, MTTI, Lehman College, CUNY MAT631, Fall 2009, Project XI BACKGROUND: Euclidean Axioms, Half Planes, Unique Perpendicular Lines, Congruent and Similar Triangle Theorems,
More informationReading material on the limit set of a Fuchsian group
Reading material on the limit set of a Fuchsian group Recommended texts Many books on hyperbolic geometry and Kleinian and Fuchsian groups contain material about limit sets. The presentation given here
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationGeometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment
Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points
More informationSolutions to Vector Calculus Practice Problems
olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x
More informationInversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)
Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1
Student Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More informationTerminology: When one line intersects each of two given lines, we call that line a transversal.
Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal.
More informationThe measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures
8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 20132014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More information3.1 Triangles, Congruence Relations, SAS Hypothesis
Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)
More informationIf Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the RightHandRule.
Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions.
More informationWorking with Wireframe and Surface Design
Chapter 9 Working with Wireframe and Surface Design Learning Objectives After completing this chapter you will be able to: Create wireframe geometry. Create extruded surfaces. Create revolved surfaces.
More informationClassification and Structure of Periodic Fatou Components
Classification and Structure of Periodic Fatou Components Senior Honors Thesis in Mathematics, Harvard College By Benjamin Dozier Adviser: Sarah Koch 3/19/2012 Abstract For a given rational map f : Ĉ Ĉ,
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationChapter 6 Notes: Circles
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
More informationtr g φ hdvol M. 2 The EulerLagrange equation for the energy functional is called the harmonic map equation:
Notes prepared by Andy Huang (Rice University) In this note, we will discuss some motivating examples to guide us to seek holomorphic objects when dealing with harmonic maps. This will lead us to a brief
More informationClassical theorems on hyperbolic triangles from a projective point of view
tmcsszilasi 2012/3/1 0:14 page 175 #1 10/1 (2012), 175 181 Classical theorems on hyperbolic triangles from a projective point of view Zoltán Szilasi Abstract. Using the CayleyKlein model of hyperbolic
More informationMA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
More informationSolving Geometric Problems with the Rotating Calipers *
Solving Geometric Problems with the Rotating Calipers * Godfried Toussaint School of Computer Science McGill University Montreal, Quebec, Canada ABSTRACT Shamos [1] recently showed that the diameter of
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationLesson 18: Looking More Carefully at Parallel Lines
Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using
More information2. Length and distance in hyperbolic geometry
2. Length and distance in hyperbolic geometry 2.1 The upper halfplane There are several different ways of constructing hyperbolic geometry. These different constructions are called models. In this lecture
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationWeek 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test
Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan
More informationDESARGUES THEOREM DONALD ROBERTSON
DESARGUES THEOREM DONALD ROBERTSON Two triangles ABC and A B C are said to be in perspective axially when no two vertices are equal and when the three intersection points AC A C, AB A B and BC B C are
More informationChapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
More informationDECOMPOSING SL 2 (R)
DECOMPOSING SL 2 R KEITH CONRAD Introduction The group SL 2 R is not easy to visualize: it naturally lies in M 2 R, which is 4 dimensional the entries of a variable 2 2 real matrix are 4 free parameters
More informationcircumscribed circle Vocabulary Flash Cards Chapter 10 (p. 539) Chapter 10 (p. 530) Chapter 10 (p. 538) Chapter 10 (p. 530)
Vocabulary Flash ards adjacent arcs center of a circle hapter 10 (p. 539) hapter 10 (p. 530) central angle of a circle chord of a circle hapter 10 (p. 538) hapter 10 (p. 530) circle circumscribed angle
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationReview B: Coordinate Systems
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B2 B.1.1 Infinitesimal Line Element... B4 B.1.2 Infinitesimal Area Element...
More informationChapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
More informationvector calculus 2 Learning outcomes
29 ontents vector calculus 2 1. Line integrals involving vectors 2. Surface and volume integrals 3. Integral vector theorems Learning outcomes In this Workbook you will learn how to integrate functions
More informationTrainer/Instructor Notes: Transformations Terms and Definitions Geometry Module 12
Terms and Definitions A line is a straight, continuous arrangement of infinitely many points. It is infinitely long and extends in two directions but has no width or thickness. It is represented and named
More informationSHINPEI BABA AND SUBHOJOY GUPTA
HOLONOMY MAP FIBERS OF CP 1 STRUCTURES IN MODULI SPACE SHINPEI BABA AND SUBHOJOY GUPTA Abstract. Let S be a closed oriented surface of genus g 2. Fix an arbitrary nonelementary representation ρ: π 1
More informationMath 497C Sep 9, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More information61. Pascal s Hexagon Theorem.
. Pascal s Hexagon Theorem. Prove that the three points of intersection of the opposite sides of a hexagon inscribed in a conic section lie on a straight line. Hexagon has opposite sides,;, and,. Pascal
More informationChapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
More informationTilings of the sphere with right triangles III: the asymptotically obtuse families
Tilings of the sphere with right triangles III: the asymptotically obtuse families Robert J. MacG. Dawson Department of Mathematics and Computing Science Saint Mary s University Halifax, Nova Scotia, Canada
More informationDepartment of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY AND GROUPS
Department of Pure Mathematics and Mathematical Statistics University of Cambridge GEOMETRY AND GROUPS Albrecht Dürer s engraving Melencolia I (1514) Notes Michaelmas 2012 T. K. Carne. t.k.carne@dpmms.cam.ac.uk
More informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationTriangle Congruence and Similarity A CommonCoreCompatible Approach
Triangle Congruence and Similarity A CommonCoreCompatible Approach The Common Core State Standards for Mathematics (CCSSM) include a fundamental change in the geometry program in grades 8 to 10: geometric
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationDouble Tangent Circles and Focal Properties of SpheroConics
Double Tangent Circles and Focal Properties of SpheroConics HansPeter Schröcker University of Innsbruck Unit Geometry and CAD April 10, 008 Abstract We give two proofs for the characterization of a spheroconic
More informationVarious Ways of Representing Surfaces and Basic Examples
Chapter 1 Various Ways of Representing Surfaces and Basic Examples Lecture 1. a. First examples. For many people, one of the most basic images of a surface is the surface of the Earth. Although it looks
More informationCreating Repeating Patterns with Color Symmetry
Creating Repeating Patterns with Color Symmetry Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 558123036, USA Email: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/
More informationProjective Geometry: A Short Introduction. Lecture Notes Edmond Boyer
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Contents 1 Introduction 2 11 Objective 2 12 Historical Background 3 13 Bibliography 4 2 Projective Spaces 5 21 Definitions 5 22 Properties
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More information* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.
* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying
More information55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.
Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit
More informationWintersemester 2015/2016. University of Heidelberg. Geometric Structures on Manifolds. Geometric Manifolds. by Stephan Schmitt
Wintersemester 2015/2016 University of Heidelberg Geometric Structures on Manifolds Geometric Manifolds by Stephan Schmitt Contents Introduction, first Definitions and Results 1 Manifolds  The Group way....................................
More informationChapter 4 Circles, TangentChord Theorem, Intersecting Chord Theorem and Tangentsecant Theorem
Tampines Junior ollege H3 Mathematics (9810) Plane Geometry hapter 4 ircles, Tangenthord Theorem, Intersecting hord Theorem and Tangentsecant Theorem utline asic definitions and facts on circles The
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C2 Vertical Angles Conjecture If two angles are vertical
More informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School  Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationHonors Geometry Final Exam Study Guide
20112012 Honors Geometry Final Exam Study Guide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In each pair of triangles, parts are congruent as marked.
More informationExploring Geometric Transformations in a Dynamic Environment Cheryll E. Crowe, Ph.D. Eastern Kentucky University
Exploring Geometric Transformations in a Dynamic Environment Cheryll E. Crowe, Ph.D. Eastern Kentucky University Overview The GeoGebra documents allow exploration of four geometric transformations taught
More information