The Three Reflections Theorem

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The Three Reflections Theorem"

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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional Geometries Spherical geometry Spherical Restrict the 3-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 Two-dimensional 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 co-incides 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 Two-dimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 is a vertical ray x = constant, or a semi-circle with centre on the x-axis. There is a unique line through any pair of distinct points Disjoint lines may be asymptotic or ultraparallel. The x-axis together with forms the circle at infinity. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

16 The Three Two-dimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 is a vertical ray x = constant, or a semi-circle with centre on the x-axis. There is a unique line through any pair of distinct points Disjoint lines may be asymptotic or ultraparallel. The x-axis together with forms the circle at infinity. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

17 The Three Two-dimensional Geometries Hyperbolic Lines in the upper half plane model A line in H 2 is a vertical ray x = constant, or a semi-circle with centre on the x-axis. There is a unique line through any pair of distinct points Disjoint lines may be asymptotic or ultraparallel. The x-axis together with forms the circle at infinity. Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

18 The Three Two-dimensional 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 semi-circular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

19 The Three Two-dimensional 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 semi-circular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

20 The Three Two-dimensional 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 semi-circular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

21 The Three Two-dimensional 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 semi-circular lines e.g. z 1/ z i Christopher Tuffley (Massey University) The Three Reflections Theorem 24th June / 19

22 The Three Two-dimensional 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 semi-circular 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 non-collinear points A, B, C. Consequently, any isometry is completely determined by the images of any three non-collinear 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 non-collinear points A, B, C. A P Consequently, any isometry is completely determined by the images of any three non-collinear 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 non-collinear points A, B, C. A P Consequently, any isometry is completely determined by the images of any three non-collinear 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 non-collinear. 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 co-incides 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 non-collinear. 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 co-incides 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 non-collinear. 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 co-incides 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 non-collinear. 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 co-incides 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 three-space 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

half-line the set of all points on a line on a given side of a given point of the line

half-line 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 half-line the set of all points on a line on a given side of a given point of the line notation: is the half-line that contains all points on the

More information

UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE

UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE NINA LEUNG Abstract. This paper discusses 2-dimensional locally Euclidean geometries and how these geometries can describe musical chords. Contents

More information

GEOMETRY CONCEPT MAP. Suggested Sequence:

GEOMETRY 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 information

The Math Circle, Spring 2004

The Math Circle, Spring 2004 The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

More information

alternate interior angles

alternate interior angles alternate interior angles two non-adjacent 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 information

4. Expanding dynamical systems

4. 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 information

Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS

Lecture 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 information

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Geometry 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 information

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: 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 information

Poincaré Models of Hyperbolic Geometry

Poincaré 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 information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: 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 information

Solutions to Practice Problems

Solutions to Practice Problems Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

More information

Lecture 26 181. area = πr 2 c 4 r4 + o(r 4 ).

Lecture 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 information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. 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 information

Theorem 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.

Theorem 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 information

Chapter 1: Essentials of Geometry

Chapter 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 information

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE 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 information

Geometry Course Summary Department: Math. Semester 1

Geometry 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 information

1.7 Cylindrical and Spherical Coordinates

1.7 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 two-dimensional coordinate system in which the

More information

What is inversive geometry?

What 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 information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES 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 information

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Name 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 information

1. 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. 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 information

Determine 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

Determine 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 information

Solutions to Homework 10

Solutions 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 information

Metrics on SO(3) and Inverse Kinematics

Metrics 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 information

Exploring Spherical Geometry

Exploring 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 information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 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 real-life problem. algebraic expression - An expression with variables.

More information

Lecture 2: Homogeneous Coordinates, Lines and Conics

Lecture 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 information

Geometry Notes Chapter 12. Name: Period:

Geometry 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 information

Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)

Conjunction 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 information

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? WHICH LINEAR-FRACTIONAL 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 information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles 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 information

THE THURSTON METRIC ON HYPERBOLIC DOMAINS AND BOUNDARIES OF CONVEX HULLS

THE 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 quasi-isometry from the

More information

Hyperbolic Islamic Patterns A Beginning

Hyperbolic Islamic Patterns A Beginning Hyperbolic Islamic Patterns A Beginning Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-2496, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

Fixed Point Theorems in Topology and Geometry

Fixed 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 information

Final Review Geometry A Fall Semester

Final 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 information

4.1 Euclidean Parallelism, Existence of Rectangles

4.1 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 information

Coordinate Plane Project

Coordinate 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 information

Reading material on the limit set of a Fuchsian group

Reading 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 information

Number Sense and Operations

Number 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 information

Geometry 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. 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 information

Solutions to Vector Calculus Practice Problems

Solutions 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 information

Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Inversion. 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 information

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1 Student Name: Teacher: Date: District: Miami-Dade 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 information

Geometric Transformations

Geometric 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 information

Section 1.8 Coordinate Geometry

Section 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 information

Connecting Transformational Geometry and Transformations of Functions

Connecting 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 information

Terminology: When one line intersects each of two given lines, we call that line a transversal.

Terminology: 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 information

The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures

The 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 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to

More information

Circle Name: Radius: Diameter: Chord: Secant:

Circle 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 information

3.1 Triangles, Congruence Relations, SAS Hypothesis

3.1 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 information

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule.

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule. 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 information

Working with Wireframe and Surface Design

Working 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 information

Classification and Structure of Periodic Fatou Components

Classification 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 information

Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected 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 information

Chapter 6 Notes: Circles

Chapter 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 information

tr g φ hdvol M. 2 The Euler-Lagrange equation for the energy functional is called the harmonic map equation:

tr g φ hdvol M. 2 The Euler-Lagrange 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 information

Classical theorems on hyperbolic triangles from a projective point of view

Classical theorems on hyperbolic triangles from a projective point of view tmcs-szilasi 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 Cayley-Klein model of hyperbolic

More information

MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem

MA 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 information

Solving Geometric Problems with the Rotating Calipers *

Solving 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 information

MAT 1341: REVIEW II SANGHOON BAEK

MAT 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 information

Lesson 18: Looking More Carefully at Parallel Lines

Lesson 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 information

2. Length and distance in hyperbolic geometry

2. Length and distance in hyperbolic geometry 2. Length and distance in hyperbolic geometry 2.1 The upper half-plane There are several different ways of constructing hyperbolic geometry. These different constructions are called models. In this lecture

More information

Solving Simultaneous Equations and Matrices

Solving 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 information

Week 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test

Week 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 information

DESARGUES THEOREM DONALD ROBERTSON

DESARGUES 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 information

Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 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 information

DECOMPOSING SL 2 (R)

DECOMPOSING 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 information

circumscribed circle Vocabulary Flash Cards Chapter 10 (p. 539) Chapter 10 (p. 530) Chapter 10 (p. 538) Chapter 10 (p. 530)

circumscribed 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 information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric 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 information

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean 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 information

Review B: Coordinate Systems

Review B: Coordinate Systems MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B-2 B.1.1 Infinitesimal Line Element... B-4 B.1.2 Infinitesimal Area Element...

More information

Chapter 4.1 Parallel Lines and Planes

Chapter 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 information

vector calculus 2 Learning outcomes

vector 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 information

Trainer/Instructor Notes: Transformations Terms and Definitions Geometry Module 1-2

Trainer/Instructor Notes: Transformations Terms and Definitions Geometry Module 1-2 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 information

SHINPEI BABA AND SUBHOJOY GUPTA

SHINPEI 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 non-elementary representation ρ: π 1

More information

Math 497C Sep 9, Curves and Surfaces Fall 2004, PSU

Math 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 information

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

Vector 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 information

61. Pascal s Hexagon Theorem.

61. 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 information

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 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 information

Tilings of the sphere with right triangles III: the asymptotically obtuse families

Tilings 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 information

Department 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 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 information

Contents. 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. 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 information

Triangle Congruence and Similarity A Common-Core-Compatible Approach

Triangle Congruence and Similarity A Common-Core-Compatible Approach Triangle Congruence and Similarity A Common-Core-Compatible Approach The Common Core State Standards for Mathematics (CCSSM) include a fundamental change in the geometry program in grades 8 to 10: geometric

More information

Foundations of Geometry 1: Points, Lines, Segments, Angles

Foundations 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 information

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

We call this set an n-dimensional 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 information

Double Tangent Circles and Focal Properties of Sphero-Conics

Double Tangent Circles and Focal Properties of Sphero-Conics Double Tangent Circles and Focal Properties of Sphero-Conics Hans-Peter Schröcker University of Innsbruck Unit Geometry and CAD April 10, 008 Abstract We give two proofs for the characterization of a sphero-conic

More information

Various Ways of Representing Surfaces and Basic Examples

Various 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 information

Creating Repeating Patterns with Color Symmetry

Creating Repeating Patterns with Color Symmetry Creating Repeating Patterns with Color Symmetry Douglas Dunham Department of Computer Science University of Minnesota, Duluth Duluth, MN 55812-3036, USA E-mail: ddunham@d.umn.edu Web Site: http://www.d.umn.edu/

More information

Projective Geometry: A Short Introduction. Lecture Notes Edmond Boyer

Projective 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 information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus 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. * 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 information

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

55 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 information

Wintersemester 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 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 information

Chapter 4 Circles, Tangent-Chord Theorem, Intersecting Chord Theorem and Tangent-secant Theorem

Chapter 4 Circles, Tangent-Chord Theorem, Intersecting Chord Theorem and Tangent-secant Theorem Tampines Junior ollege H3 Mathematics (9810) Plane Geometry hapter 4 ircles, Tangent-hord Theorem, Intersecting hord Theorem and Tangent-secant Theorem utline asic definitions and facts on circles The

More information

Conjectures. Chapter 2. Chapter 3

Conjectures. Chapter 2. Chapter 3 Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

More information

Geometry Enduring Understandings Students will understand 1. that all circles are similar.

Geometry 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 information

Honors Geometry Final Exam Study Guide

Honors Geometry Final Exam Study Guide 2011-2012 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 information

Exploring 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 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