DOCSLIB.ORG

Ideas in Geometry Distributing This Document

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ideas in Geometry Distributing This Document Ideas in Geometry Distributing this Document Copyright c 2009 Alison Ahlgren and Bart Snapp. This work is licensed under the Creative Commons Attribution-ShareAlike Li- cense. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/us/ or, send a letter to Creative Commons, 171 2nd Street, Suite 300, San Francisco, California, 94105, USA. This document was typeset on December 16, 2009. Preface The aim of these notes is to convey the spirit of mathematical thinking as demonstrated through topics mainly from geometry. The reader must be careful not to forget this emphasis on deduction and visual reasoning. To this end, many questions are asked in the text that follows. Sometimes these questions are answered, other times the questions are left for the reader to ponder. To let the reader know which questions are left for cogitation, a large question mark is displayed: ? The instructor of the course will address some of these questions. If a question is not discussed to the reader’s satisfaction, then I encourage the reader to put on a thinking-cap and think, think, think! If the question is still unresolved, go to the World Wide Web and search, search, search! This document is licensed under the Creative Commons Attribution-ShareAlike (CC BY-SA) License. Loosely speaking, this means that this document is avail- able for free. Anyone can get a free copy of this document (source or PDF) from the following site: http://www.math.uiuc.edu/Courses/math119/ Please report corrections, suggestions, gripes, complaints, and criticisms to: [email protected] or [email protected] Thanks and Acknowledgments This document is based on a set of lectures originally given by Bart Snapp at the University of Illinois at Urbana-Champaign in the Fall of 2005 and Spring of 2006. Each semester since, these notes have been revised and modified. A growing number of instructors have made contributions, including Tom Cooney, Melissa Dennison, Jesse Miller, and Bart Snapp. Thanks to Alison Ahlgren, the Quantitative Reasoning Coordinator at the University of Illinois at Urbana-Champaign, for developing this course and for working with me and all the other instructors during the continual development of this document. Also thanks to Harry Calkins for help with Mathematica graphics and acting as a sounding-board for some of the ideas expressed in this document. In 2009, Greg Williams, a Master of Arts in Teaching student at Coastal Car- olina University, worked with Bart Snapp to produce the chapter on geometric transformations. A number of students have also contributed to this document by either typing or suggesting problems. They are: Camille Brooks, Michelle Bruno, Marissa Colatosti, Katie Colby, Anthony ‘Tino’ Forneris, Amanda Genovise, Melissa Peterson, Nicole Petschenko, Jason Reczek, Christina Reincke, David Seo, Adam Shalzi, Allice Son, Katie Strle, Beth Vaughn. Contents 1 Beginnings, Axioms, and Viewpoints 1 1.1 Euclid and Beyond . 1 1.1.1 The Most Successful Textbook Ever Written . 1 1.1.2 The Parallel Postulate . 5 1.2 PointsofView ............................ 12 1.2.1 Synthetic Geometry . 13 1.2.2 Algebraic Geometry . 14 1.2.3 Analytic Geometry . 17 1.3 CityGeometry ............................ 23 1.3.1 GettingWorkDone ..................... 25 1.3.2 (Un)Common Structures . 28 2 Proof by Picture 38 2.1 BasicSetTheory ........................... 38 2.1.1 Union ............................. 39 2.1.2 Intersection .......................... 39 2.1.3 Complement ......................... 40 2.1.4 Putting Things Together . 41 2.2 Logic.................................. 45 2.3 Tessellations.............................. 51 2.3.1 Tessellations and Art . 52 2.4 ProofbyPicture ........................... 56 2.4.1 Proofs Involving Right Triangles . 56 2.4.2 Proofs Involving Boxy Things . 60 2.4.3 Proofs Involving Sums . 62 2.4.4 Proofs Involving Sequences . 66 2.4.5 Thinking Outside the Box . 67 3 Topics in Plane Geometry 81 3.1 Triangles ............................... 81 3.1.1 Centers in Triangles . 81 3.1.2 Theorems about Triangles . 85 3.2 Numbers................................ 91 3.2.1 Areas ............................. 91 3.2.2 Ratios............................. 92 3.2.3 Combining Areas and Ratios—Probability . 100 4 Compass and Straightedge Constructions 113 4.1 Constructions............................. 113 4.2 Trickier Constructions . 122 4.2.1 Challenge Constructions . 123 4.2.2 Problem Solving Strategies . 127 4.3 Constructible Numbers . 130 4.4 Impossibilities............................. 140 4.4.1 Doubling the Cube . 140 4.4.2 Squaring the Circle . 140 4.4.3 Trisecting the Angle . 141 5 Transformations 144 5.1 Basic Transformations . 144 5.1.1 Translations.......................... 145 5.1.2 Reflections . 147 5.1.3 Rotations ........................... 149 5.2 The Algebra of Transformations . 156 5.2.1 Matrix Multiplication . 156 5.2.2 Compositions of Transformations . 157 5.2.3 Mixing and Matching . 160 5.3 The Theory of Groups . 164 5.3.1 Groups of Reflections . 164 5.3.2 Groups of Rotations . 165 5.3.3 SymmetryGroups ...................... 166 6 Convex Sets 170 6.1 BasicDefinitions ........................... 170 6.1.1 An Application . 173 6.2 Convex Sets in Three Dimensions . 177 6.2.1 Analogies to Two Dimensions . 177 6.2.2 Platonic Solids . 177 6.3 Ideas Related to Convexity . 182 6.3.1 The Convex Hull . 182 6.3.2 Sets of Constant Width . 182 6.4 Advanced Theorems . 187 References and Further Reading 189 Index 191 Chapter 1 Beginnings, Axioms, and Viewpoints Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain? —Gustave Flaubert 1.1 Euclid and Beyond 1.1.1 The Most Successful Textbook Ever Written Question Think of all the books that were ever written. What are some of the most influential of these? ? The Elements by the Greek mathematician Euclid of Alexandria should be high on this list. Euclid lived in Alexandria, Egypt, around 300 BC. His book, The Elements is an attempt to compile and write down everything that was known about geometry. This book is perhaps the most successful textbook ever written, having been used in nearly all universities up until the 20th century. Even today its heritage can be seen in scientific thought and writing. Here are three reasons this book is so important: (1) The Elements is of practical use. (2) The Elements contains powerful ideas. 1 1.1. EUCLID AND BEYOND (3) The Elements provides a playground for the development of logical thought. We’ll address each of these in turn. The Elements is of practical use. Any time something large is built, some geometry must be used. The roads we drive on every day, the buildings we live in, the malls we shop at, the stadiums our favorite sports teams compete in; in short, if it is bigger than a shack, then geometry must have been used at some point. Moreover, geometry is crucial to modern transportation—in fact, any large scale transportation. An airplane could never make it to its destination without geometry, nor could any ship at sea. People of the past were faced with the difficulties of geometry on a continuing basis. Euclid’s The Elements was their handbook to solve everyday problems. The Elements contains powerful ideas. Around 200 BC, the head librar- ian at the Great Library of Alexandria was a man by the name of Eratosthenes. Not only was Eratosthenes a great athlete, he was a scholar of astronomy, ethics, music, philosophy, poetry, theater, and important to this discussion, mathemat- ics. His nickname was Beta, the second letter in the Greek alphabet. This is because with so many interests and accomplishments, he seemed to be second best at everything in the world. It came to Eratosthenes that every year, on the longest day of the year, at noon, sunlight would shine down to the bottom of a deep well located in present day city of Aswan, Egypt. Eratosthenes reasoned that this meant that the sun was directly overhead Aswan at this time. However, Eratosthenes knew that the sun was not directly overhead in Alexandria. He realized that the situation must be something like this: Alexandria Sun’s rays Aswan center of the Earth Using ideas found in The Elements, Eratosthenes realized that if he drew imagi- nary lines from Alexandria and Aswan down to the center of the Earth, and if he could compute the angle between these lines, then to compute the circumference 2 CHAPTER 1. BEGINNINGS, AXIOMS, AND VIEWPOINTS of the Earth he would need only to solve the equation: total degrees in a circle circumference of the Earth = angle between the cities distance between the cities Thus Eratosthenes hired a man to pace the distance from Alexandria to Aswan. It was found to be about 5000 stadia. A stadia is an ancient unit of measurement which is the the length of a stadium. To measure the angle, he measured the angle of the shadow of a perpendicular stick in Alexandria, and found: Sun’s rays stick ground Where the lower left angle is about 83◦ and the upper right angle is about 7◦. Thus we have 360 x 360 = 5000 = x. 7 5000 ⇒ 7 · So we see that x, the circumference of the Earth is about 250000 stadia. Unfortunately, as you may realize, the length of a stadium can vary. If the length of the stadium is defined to be 157 meters, the length of an ancient Egyptian stadium, then it can be calculated that the circumference of the Earth is about 39250 kilometers.
Load more

Recommended publications
  • Models of 2-Dimensional Hyperbolic Space and Relations Among Them; Hyperbolic Length, Lines, and Distances
    Models of 2-dimensional hyperbolic space and relations among them; Hyperbolic length, lines, and distances Cheng Ka Long, Hui Kam Tong 1155109623, 1155109049 Course Teacher: Prof. Yi-Jen LEE Department of Mathematics, The Chinese University of Hong Kong MATH4900E Presentation 2, 5th October 2020 Outline Upper half-plane Model (Cheng) A Model for the Hyperbolic Plane The Riemann Sphere C Poincar´eDisc Model D (Hui) Basic properties of Poincar´eDisc Model Relation between D and other models Length and distance in the upper half-plane model (Cheng) Path integrals Distance in hyperbolic geometry Measurements in the Poincar´eDisc Model (Hui) M¨obiustransformations of D Hyperbolic length and distance in D Conclusion Boundary, Length, Orientation-preserving isometries, Geodesics and Angles Reference Upper half-plane model H Introduction to Upper half-plane model - continued Hyperbolic geometry Five Postulates of Hyperbolic geometry: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. A circle may be described with any given point as its center and any distance as its radius. 4. All right angles are congruent. 5. For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Some interesting facts about hyperbolic geometry 1. Rectangles don't exist in hyperbolic geometry. 2. In hyperbolic geometry, all triangles have angle sum < π 3. In hyperbolic geometry if two triangles are similar, they are congruent.
    [Show full text]
  • A Historical Introduction to Elementary Geometry
    i MATH 119 – Fall 2012: A HISTORICAL INTRODUCTION TO ELEMENTARY GEOMETRY Geometry is an word derived from ancient Greek meaning “earth measure” ( ge = earth or land ) + ( metria = measure ) . Euclid wrote the Elements of geometry between 330 and 320 B.C. It was a compilation of the major theorems on plane and solid geometry presented in an axiomatic style. Near the beginning of the first of the thirteen books of the Elements, Euclid enumerated five fundamental assumptions called postulates or axioms which he used to prove many related propositions or theorems on the geometry of two and three dimensions. POSTULATE 1. Any two points can be joined by a straight line. POSTULATE 2. Any straight line segment can be extended indefinitely in a straight line. POSTULATE 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. POSTULATE 4. All right angles are congruent. POSTULATE 5. (Parallel postulate) If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate 3 defines a sphere. Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom, which also holds only in the plane: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
    [Show full text]
  • Geometry Course Outline
    GEOMETRY COURSE OUTLINE Content Area Formative Assessment # of Lessons Days G0 INTRO AND CONSTRUCTION 12 G-CO Congruence 12, 13 G1 BASIC DEFINITIONS AND RIGID MOTION Representing and 20 G-CO Congruence 1, 2, 3, 4, 5, 6, 7, 8 Combining Transformations Analyzing Congruency Proofs G2 GEOMETRIC RELATIONSHIPS AND PROPERTIES Evaluating Statements 15 G-CO Congruence 9, 10, 11 About Length and Area G-C Circles 3 Inscribing and Circumscribing Right Triangles G3 SIMILARITY Geometry Problems: 20 G-SRT Similarity, Right Triangles, and Trigonometry 1, 2, 3, Circles and Triangles 4, 5 Proofs of the Pythagorean Theorem M1 GEOMETRIC MODELING 1 Solving Geometry 7 G-MG Modeling with Geometry 1, 2, 3 Problems: Floodlights G4 COORDINATE GEOMETRY Finding Equations of 15 G-GPE Expressing Geometric Properties with Equations 4, 5, Parallel and 6, 7 Perpendicular Lines G5 CIRCLES AND CONICS Equations of Circles 1 15 G-C Circles 1, 2, 5 Equations of Circles 2 G-GPE Expressing Geometric Properties with Equations 1, 2 Sectors of Circles G6 GEOMETRIC MEASUREMENTS AND DIMENSIONS Evaluating Statements 15 G-GMD 1, 3, 4 About Enlargements (2D & 3D) 2D Representations of 3D Objects G7 TRIONOMETRIC RATIOS Calculating Volumes of 15 G-SRT Similarity, Right Triangles, and Trigonometry 6, 7, 8 Compound Objects M2 GEOMETRIC MODELING 2 Modeling: Rolling Cups 10 G-MG Modeling with Geometry 1, 2, 3 TOTAL: 144 HIGH SCHOOL OVERVIEW Algebra 1 Geometry Algebra 2 A0 Introduction G0 Introduction and A0 Introduction Construction A1 Modeling With Functions G1 Basic Definitions and Rigid
    [Show full text]
  • Points, Lines, and Planes a Point Is a Position in Space. a Point Has No Length Or Width Or Thickness
    Points, Lines, and Planes A Point is a position in space. A point has no length or width or thickness. A point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. A A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. It does not have a beginning or end. A B C D A line consists of infinitely many points. The four points A, B, C, D are all on the same line. Postulate: Two points determine a line. We name a line by using any two points on the line, so the above line can be named as any of the following: ! ! ! ! ! AB BC AC AD CD Any three or more points that are on the same line are called colinear points. In the above, points A; B; C; D are all colinear. A Ray is part of a line that has a beginning point, and extends indefinitely to one direction. A B C D A ray is named by using its beginning point with another point it contains. −! −! −−! −−! In the above, ray AB is the same ray as AC or AD. But ray BD is not the same −−! ray as AD. A (line) segment is a finite part of a line between two points, called its end points. A segment has a finite length. A B C D B C In the above, segment AD is not the same as segment BC Segment Addition Postulate: In a line segment, if points A; B; C are colinear and point B is between point A and point C, then AB + BC = AC You may look at the plus sign, +, as adding the length of the segments as numbers.
    [Show full text]
  • Historical Introduction to Geometry
    i A HISTORICAL INTRODUCTION TO ELEMENTARY GEOMETRY Geometry is an word derived from ancient Greek meaning “earth measure” ( ge = earth or land ) + ( metria = measure ) . Euclid wrote the Elements of geometry between 330 and 320 B.C. It was a compilation of the major theorems on plane and solid geometry presented in an axiomatic style. Near the beginning of the first of the thirteen books of the Elements, Euclid enumerated five fundamental assumptions called postulates or axioms which he used to prove many related propositions or theorems on the geometry of two and three dimensions. POSTULATE 1. Any two points can be joined by a straight line. POSTULATE 2. Any straight line segment can be extended indefinitely in a straight line. POSTULATE 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. POSTULATE 4. All right angles are congruent. POSTULATE 5. (Parallel postulate) If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The circle described in postulate 3 is tacitly unique. Postulates 3 and 5 hold only for plane geometry; in three dimensions, postulate 3 defines a sphere. Postulate 5 leads to the same geometry as the following statement, known as Playfair's axiom, which also holds only in the plane: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
    [Show full text]
  • Lesson 5: Three-Dimensional Space
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 M3 GEOMETRY Lesson 5: Three-Dimensional Space Student Outcomes . Students describe properties of points, lines, and planes in three-dimensional space. Lesson Notes A strong intuitive grasp of three-dimensional space, and the ability to visualize and draw, is crucial for upcoming work with volume as described in G-GMD.A.1, G-GMD.A.2, G-GMD.A.3, and G-GMD.B.4. By the end of the lesson, students should be familiar with some of the basic properties of points, lines, and planes in three-dimensional space. The means of accomplishing this objective: Draw, draw, and draw! The best evidence for success with this lesson is to see students persevere through the drawing process of the properties. No proof is provided for the properties; therefore, it is imperative that students have the opportunity to verify the properties experimentally with the aid of manipulatives such as cardboard boxes and uncooked spaghetti. In the case that the lesson requires two days, it is suggested that everything that precedes the Exploratory Challenge is covered on the first day, and the Exploratory Challenge itself is covered on the second day. Classwork Opening Exercise (5 minutes) The terms point, line, and plane are first introduced in Grade 4. It is worth emphasizing to students that they are undefined terms, meaning they are part of the assumptions as a basis of the subject of geometry, and they can build the subject once they use these terms as a starting place. Therefore, these terms are given intuitive descriptions, and it should be clear that the concrete representation is just that—a representation.
    [Show full text]
  • Perpendicular and Parallel Line Segments
    Name: Date: r te p a h Perpendicular and C Parallel Line Segments Practice 1 Drawing Perpendicular Line Segments Use a protractor to draw perpendicular line segments. Example Draw a line segment perpendicular to RS through point T. S 0 180 10 1 70 160 20 150 30 140 40 130 50 0 120 180 6 0 10 11 0 170 70 20 100 30 160 80 40 90 80 150 50 70 60 140 100 130 110 120 R T © Marshall Cavendish International (Singapore) Private Limited. Private (Singapore) International © Marshall Cavendish 1. Draw a line segment perpendicular to PQ. P Q 2. Draw a line segment perpendicular to TU through point X. X U T 67 Lesson 10.1 Drawing Perpendicular Line Segments 08(M)MIF2015CC_WBG4B_Ch10.indd 67 4/30/13 11:18 AM Use a drawing triangle to draw perpendicular line segments. Example M L N K 3. Draw a line segment 4. Draw a line segment perpendicular to EF . perpendicular to CD. C E F D © Marshall Cavendish International (Singapore) Private Limited. Private (Singapore) International © Marshall Cavendish 5. Draw a line segment perpendicular to VW at point P. Then, draw another line segment perpendicular to VW through point Q. Q V P W Line Segments 68 and Parallel Chapter 10 Perpendicular 08(M)MIF2015CC_WBG4B_Ch10.indd 68 4/30/13 11:18 AM Name: Date: Practice 2 Drawing Parallel Line Segments Use a drawing triangle and a straightedge to draw parallel line segments. Example Draw a line segment parallel to AB . A B 1. Draw a pair of parallel line segments.
    [Show full text]
  • Properties of Straight Line Segments
    Properties of straight line segments mc-TY-lineseg-2009-1 In this unit we use a system of co-ordinates to find various properties of the straight line between two points. We find the distance between the two points and the mid-point of the line joining the two points. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • find the distance between two points; • find the co-ordinates of the mid-point of the line joining two points; Contents 1. Introduction 2 2. The distance between two points 3 3. Themidpointofthelinejoiningtwopoints 5 www.mathcentre.ac.uk 1 c mathcentre 2009 1. Introduction In this unit we use a system of co-ordinates to find various properties of the straight line between two points. We find the distance between the two points and the mid-point of the line joining the two points. Let’s start by revising some facts about the coordinates of points. Suppose that a point O is marked on a plane, together with a pair of perpendicular lines which pass through it, each with a uniform scale. We shall label the lines with the letters x and y, and we call them the x and y axes. The fixed point O is called the origin, and it is the intersection point of the two axes. This is shown in Figure 1. y A(x, y) y x O x Figure 1.
    [Show full text]
  • Solutions to Exercises
    Solutions to Exercises Solutions to Chapter 1 exercises: 1 i 1.1: Write x =Re(z)= 2 (z + z)andy =Im(z)=− 2 (z − z), so that 1 i 1 1 ax + by + c = a (z + z) − b (z − z)+c = (a − ib)z + (a + bi)z + c =0. 2 2 2 2 − a Note that the slope of this line is b , which is the quotient of the imaginary and real parts of the coefficient of z. 2 2 2 Given the circle (x − h) +(y − k) = r , set z0 = h + ik and rewrite the equation of the circle as 2 2 2 |z − z0| = zz − z0z − z0z + |z0| = r . 1.2: A and S1 are perpendicular if and only if their tangent lines at the point of intersection are perpendicular. Let x be a point of A ∩ S1, and consider the Euclidean triangle T with vertices 0, reiθ,andx. The sides of T joining x to the other two vertices are radii of A and S1,andsoA and S1 are perpendicular 1 if and only if the interior angle of T at x is 2 π, which occurs if and only if the Pythagorean theorem holds, which occurs if and only if s2 +12 = r2. 1.3: Let Lpq be the Euclidean line segment joining p and q. The midpoint of 1 Im(q)−Im(p) Lpq is 2 (p + q), and the slope of Lpq is m = Re(q)−Re(p) . The perpendicular 1 − 1 Re(p)−Re(q) bisector K of Lpq passes through 2 (p + q) and has slope m = Im(q)−Im(p) , and so K has the equation 1 Re(p) − Re(q) 1 y − (Im(p) + Im(q)) = x − (Re(p) + Re(q)) .
    [Show full text]
  • Notes on Hyperbolic Geometry
    Notes on Hyperbolic Geometry Henry Y. Chan July 2, 2013 1 Introduction For people who have taken real calculus, you know that the arc length of a curve in R2 γ :[a; b] ! R2, where γ (t) = (x (t) ; y (t)), is defined as s Z b dx2 dy 2 s = + dt: a dt dt The reason behind this formula is that locally we have (∆s)2 ∼ (∆x)2 + (∆y)2 by the Pythagorean Theorem. More precisely, when “∆t ! 000, we get (ds)2 = (dx)2 + (dy)2 : The differential ds is called the arc length element, or the Riemannian metric, of R2. The Riemannian metric introduced above is called the standard Euclidean metric. One might wonder whether there exists other kinds of metric on R2. In fact, given a positive definite form (ds)2 = a (x; y)(dx)2 + b (x; y) dxdy + c (x; y)(dy)2 (positive definite means that b2 − 4ac > 0 for all x; y), ds allows us to measure arc lengths along curves, and thus allows us to measure the \distance" between two points, i.e. a metric, and to determine the distance minimizing curves, called geodesics. Remark 1.1. In the standard Euclidean metric, the geodesics are straight lines. (You can prove this using the triangle inequality.) Definition 1.2. If (M; d) is a metric space, an isometry is a function f : M ! M such that d (f (x) ; f (y)) = d (x; y) for all x; y 2 M. 1 Proposition 1.3. Isometries send geodesics to geodesics. Proof. This is immediate from the definition of isometries.
    [Show full text]
  • Intermediate Math Circles March 25, 2009 Analytic Geometry I
    1 University of Waterloo Centre for Education in Faculty of Mathematics Mathematics and Computing Intermediate Math Circles March 25, 2009 Analytic Geometry I Analytic Geometry is the use of a coordinate system to translate a geometry problem into an alge- braic problem. Analytic Geometry is NOT using angle theorems, similar triangles, and trigonometry. Instead we will be using points and equations for lines. To solve problems using analytic y l geometry we have to understand the basics: A The coordinate system we will be using is the Cartesian coordinate system, the x xy-plane. Our problems will consist of us- B ing the cartesian plane along with points, lines and line segments. From the diagram, A and B are the points corresponding to the coordi- nates (3; 2) and (−1; −3) respectively. l is the line passing through points A and B. AB is the line segment with endpoints A and B. The difference between a line and a line segment is that a line segment has a fixed length. These are the tools we will be using to solve problems in analytic geometry. Distance Between Points For many problems we need to know the distance between two points, the length of a line segment. If d is the distance between two points A(x1; y1) and B(x2; y2) then: p 2 2 d = (x2 − x1) + (y2 − y1) Example: Find the distance between points A and B above. Solution: We have the points A(3; 2) and B(−1; −3). Substituting into our formula gives us, 2 p p p 2 2 d = (−1 − 3) + (−3 − 2) = 16 + 25p = 41 Therefore line segment AB has length 41.
    [Show full text]
  • Hyperbolic Geometry—Introduction the Basics of the Half-Plane Model
    Hyperbolic Geometry|Introduction The basics of the half-plane model of hyperbolic geometry are explored. Now that we know a lot about general geometry, what can we do that is not Euclidean| where we don't need the parallel postulate? In fact, what can we do to show that the parallel postulate is necessary in order to distinguish Euclidean geometry from something else? Imagine living in a 2-dimensional universe that is in the half-plane y > 0, but where distances are warped. The stretching factor is inversely proportional to the distance from the x-axis. So, if it takes one minute to talk from (0; 1) to (1; 1) along the horizontal line segment then it would take only 30 seconds to walk from (0; 2) to (1; 2) while going from (0; 1=3) to (1; 1=3) takes 3 minutes. This may look strange from the outside, but remember that we are inside this crazy plane, so our rulers and our legs shrink and stretch so that these distances look perfectly normal to us. This is called the hyperbolic plane. One of the more interesting things to note is that you really shouldn't travel along horizontal line segments to get from one place to another. This is because if you bow your trip a little north, you may stretch your legs to be long enough to more than make up for the extra (Euclidean) length you are covering. In fact, it turns out the geodesics in this wacky plane are pieces of circles! Specifically, if we are given two points (x1; y1) and (x2; y2): • If x1 = x2 so that the points are separated only vertically, then the geodesic between them is the vertical line segment, and the distance between them is j ln (y2=y1)j.
    [Show full text]