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


 Clemence Marshall
 1 years ago
 Views:
Transcription
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 s Sketchpad. A cartesian coordinate representation and a number of fascinating applications of inversion are also presented. Definition Let O be the center of a fixed circle of radius r in the Euclidean plane. Let P be any point in the plane other than O. An inversion in circle C(O, r), I(O, r), is a function such that if I (O,r) (P ) = P then P OP and (OP )(OP ) = r 2. Here P is called the inverse of P, O is called the center of inversion, and r is called the radius of inversion, and r 2 is called its power. It follows from the above definition that to each point P of the plane, other than O, there corresponds a unique inverse point P. To make the inversion a transformation of the plane, we add to the plane a single ideal point Ω defined to be the inverse of the center of inversion. The point Ω is considered to lie on every line in the plane. 7.1 Constructing The Inverse of a Point: Given the circle of inversion C (O,r) and a point P, how do you construct the inverse of P? If P is inside the circle of inversion: (See Figure 7.1) Draw the ray OP. Draw a perpendicular to OP at P. This intersects the circle of inversion in two points, label one of them Q. Connect the center of the circle O to Q. Draw a perpendicular to OQ from Q. The intersection of this perpendicular with OP is P, the inverse of P. 52
2 Q O P P. Figure 7.1: Inverse of a Point Inside the Circle of Inversion Note that OP Q OQP. (OQ) 2 = r 2. Hence OQ OP = OP OQ. Therefore, OP OP = If P is outside the circle of inversion: (See Figure 7.2) Let M be the midpoint of the segment OP. Construct a circle centered at M of radius MO. It intersects the circle of inversion C (O,r) in two points, Q and R and goes through O and P. Construct the segment QR. The intersection of QR and OP is the point P, the inverse of P. Q... P O.... M P. R Figure 7.2: Inverse of a point Outside the Circle of Inversion Note that OP Q OQP. (OQ) 2 = r 2. Hence OQ OP = OP OQ. Therefore, OP OP = 53
3 Using Sketchpad: Inverting a Line: To invert a straight line in the circle of inversion C (O,r) follow the following steps: 1. Construct the circle of inversion C and the line l. 2. Construct an arbitrary point P on the line l. 3. Construct the ray OP. 4. Construct the intersection of OP and the circle, call it Q. 5. Construct the segment OP. 6. Mark O as the center of dilation. 7. Mark the ratio r. OP 8. Dilate Q by the ratio r centered at O. The image of Q is P the inverse OP image of P. 9. Hide everything except the circle, its center, the point determining its radius, P and the straight line l and the two point determining it. 10. Select everything and create a tool and call it invcirc. 11. Apply the tool a few times and use arc through three points to determine the image of the line. Inverting a Circle: To invert a circle C 1 in the circle of inversion C (O,r), replace l by C 1 in the steps above. 7.2 Inversion Using Coordinates: Theorem An inversion about x 2 + y 2 = r 2 is given by (x, y) (x, y xr 2 ) = ( x 2 + y, yr 2 2 x 2 + y ) 2 Proof. Since (x, y), (x, y ) and (0, 0) are collinear, we have y = y. Now x x d((0, 0), (x, y)) d((0, 0), (x, y )) = r 2, hence x 2 + y 2 x 2 + y 2 = r 2 and (x 2 +y 2 ) = r4 2 y2 x (x 2 + y 2 ) x 2 x 2 + y 2 (x 2 +y 2 ) = r 4. Hence x 2 = r4 x 2 + y 2 y 2 and x 2 = r4 y 2 (x 2 + y 2 ) x 2 + y 2 = r4 x 2 + y y2 x 2 2 x. Hence 2 x 2 + y2 x 2 = r4 x 2 x 2 + y and x 2 (x 2 + y 2 ) = r4 2 x 2 x 2 + y. Hence 2 x 2 = r 4 x 2 (x 2 + y 2 ) 2 and x = r2 x x 2 + y 2. Similarly, we can show that y = r2 y x 2 + y 2. Exercise 1: What is the image of (x 1) 2 + y 2 x 2 + y 2 = 1? = 1 under an inversion in 54
4 Y X Figure 7.3: Inverse of (x 1) 2 + y 2 = 1 in x 2 + y 2 = 1 Answer. Well, x = x and y = y. Hence x x = = 1 and x 2 +y 2 x 2 +y 2 x 2 +1 (x 1) 2 2 y = y. Hence the image of a circle going through the center of the circle of 2x inversion is a line going through the points of intersection of the two circles. Exercise 2: What is the image of x = 1 2 under an inversion in x2 + y 2 = 1? Y X Figure 7.4: Inverse of x = 1 2 in x2 + y 2 = 1 Answer: (x 1) 2 + y 2 = 1 Exercise: What is the image of x = 1 under an inversion in x 2 + y 2 = 1? Answer: (x 1/2) 2 + y 2 = 1/4. Exercise 3: What is the image of x 2 +2x+y 2 = 0 under an inversion in x 2 +y 2 = 1? Answer: x = 1/2. 55
5 Theorem If two circles are orthogonal, (their tangents at the points of intersection are perpendicular), and if a diameter AB of one circle meets the other circle in the points C and D, then OP 2 = OC OD. P A O C B O D Figure 7.5: Orthogonal Circles are Inverses Proof. OP O is a right triangle, hence (OP ) 2 + (P O ) 2 = (OO ) 2. But OO = OC + CO = OC + P O, hence (OP ) 2 + (P O ) 2 = (OC + P O ) 2. Hence (OP ) 2 = (OC) OC P O, which implies that (OP ) 2 = OC (OC + 2P O ). Hence (OP ) 2 = OC OD. Theorem A circle orthogonal to the circle of inversion inverts into itself, and, a circle through a pair of inverse points is orthogonal to the circle of inversion. T P O P A Figure 7.6: Orthogonal Circles are Inverses Proof. Given the circle of inversion C (O,r) and an orthogonal circle centered at A. Let T be one of the points of intersection of the two circles. Now if a line through O meets this orthogonal circle at P and P then OP OP = OT 2 = R 2. Hence P and P are inverse points. Theorem If P P and Q, Q are pairs of inverse points with respect to some circle C (O,r), Then P Q = P Q r2 OP OQ. Proof. If O, P, Q are noncollinear then, OP = OQ and P OQ = Q OP, OQ OP hence OP Q OQ P. Hence P Q = OQ. Hence P Q = OQ P Q OQ = r2 P Q. P Q OP OP OQ OP OQ 56
6 Q. Q... O P P Figure 7.7: P Q P Q = r2 OP OQ 7.3 Applications of Inversion: Given three noncoaxial concurrent circles, construct a circle C tangent to all three circles. Figure 7.8: A Circle Tangent To Three NonCoaxial Circles Solution: Invert the circles about a unit circle centered at the point of concurrency of the circles creating a triangle. Now construct the inscribed circle and invert this circle in the circle of inversion to create the required circle. Ptolemy s Theorem: In a cyclic convex quadrilateral, the product of the diagonals is equal to the sum of the products of the two pairs of opposite sides. Proof: Invert the circle and the convex quadrilateral about a circle centered at one of the vertices of the quadrilateral, say A. Now B D = B C + C D. Hence, r 2 BC AB AC + CD AC AD = BD AB AD 57 r2 r2
7 B B A C C D D Figure 7.9: Ptolemy s Theorem Hence, BC AD + CD AB = BD AC. Homework Find the image of the objects below under the specified inversion. (See Figures 7.10 and 7.11 ) 2. Prove that the inverse of the circumcircle C c of a triangle ABC with respect to the incircle C i, as a circle of inversion, is the nine point circle of the triangle XY Z determined by the points of contact of C i with the sides of ABC. 58
8 Figure 7.10: Inversion  HW 59
9 Figure 7.11: Inversion  HW 60
Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationI. Vectors and Geometry in Two and Three Dimensions
I. Vectors and Geometry in Two and Three Dimensions I.1 Points and Vectors Each point in two dimensions may be labeled by two coordinates (a,b) which specify the position of the point in some units with
More informationAlgorithms for Designing PopUp Cards
Algorithms for Designing PopUp Cards Zachary Abel Erik D. Demaine Martin L. Demaine Sarah Eisenstat Anna Lubiw André Schulz Diane L. Souvaine Giovanni Viglietta Andrew Winslow Abstract We prove that every
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationFoundations of Data Science 1
Foundations of Data Science John Hopcroft Ravindran Kannan Version /4/204 These notes are a first draft of a book being written by Hopcroft and Kannan and in many places are incomplete. However, the notes
More informationWHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? 1. Introduction
WHICH SCORING RULE MAXIMIZES CONDORCET EFFICIENCY? DAVIDE P. CERVONE, WILLIAM V. GEHRLEIN, AND WILLIAM S. ZWICKER Abstract. Consider an election in which each of the n voters casts a vote consisting of
More informationMetric Measurements on a Plane from a Single Image
TR26579, Dartmouth College, Computer Science Metric Measurements on a Plane from a Single Image Micah K. Johnson and Hany Farid Department of Computer Science Dartmouth College Hanover NH 3755 Abstract
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationMODULES OVER A PID KEITH CONRAD
MODULES OVER A PID KEITH CONRAD Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to K n for some n 0. When we replace the scalar field K with a commutative
More informationMath 2001 Homework #10 Solutions
Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
More informationx 2 if 2 x < 0 4 x if 2 x 6
Piecewisedefined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6 Piecewisedefined Functions Example Consider the function f defined by x if x < 0 f (x) =
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More information(0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order 4; (1, 0) : order 2; (1, 1) : order 4; (1, 2) : order 2; (1, 3) : order 4.
11.01 List the elements of Z 2 Z 4. Find the order of each of the elements is this group cyclic? Solution: The elements of Z 2 Z 4 are: (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order
More informationSwitching Algebra and Logic Gates
Chapter 2 Switching Algebra and Logic Gates The word algebra in the title of this chapter should alert you that more mathematics is coming. No doubt, some of you are itching to get on with digital design
More informationMatthias Beck Gerald Marchesi Dennis Pixton Lucas Sabalka
Matthias Beck Gerald Marchesi Dennis Pixton Lucas Sabalka Version.5 Matthias Beck A First Course in Complex Analysis Version.5 Gerald Marchesi Department of Mathematics Department of Mathematical Sciences
More information1. Graphing Linear Inequalities
Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means
More informationSome Basic Techniques of Group Theory
Chapter 5 Some Basic Techniques of Group Theory 5.1 Groups Acting on Sets In this chapter we are going to analyze and classify groups, and, if possible, break down complicated groups into simpler components.
More informationhow to use dual base log log slide rules
how to use dual base log log slide rules by Professor Maurice L. Hartung The University of Chicago Pickett The World s Most Accurate Slide Rules Pickett, Inc. Pickett Square Santa Barbara, California 93102
More informationA Modern Course on Curves and Surfaces. Richard S. Palais
A Modern Course on Curves and Surfaces Richard S. Palais Contents Lecture 1. Introduction 1 Lecture 2. What is Geometry 4 Lecture 3. Geometry of InnerProduct Spaces 7 Lecture 4. Linear Maps and the Euclidean
More informationMEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationA mini course on additive combinatorics
A mini course on additive combinatorics 1 First draft. Dated Oct 24th, 2007 These are notes from a mini course on additive combinatorics given in Princeton University on August 2324, 2007. The lectures
More informationTHIS CHAPTER INTRODUCES the Cartesian coordinate
87533_01_ch1_p001066 1/30/08 9:36 AM Page 1 STRAIGHT LINES AND LINEAR FUNCTIONS 1 THIS CHAPTER INTRODUCES the Cartesian coordinate sstem, a sstem that allows us to represent points in the plane in terms
More informationHow many numbers there are?
How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationCS488/688. Sample Exam Questions
CS488/688 Sample Exam Questions University of Waterloo School of Computer Science Computer Graphics Lab January 14, 2015 This is a set of questions covering the course material. They have been asked on
More informationOrthogonal Bases and the QR Algorithm
Orthogonal Bases and the QR Algorithm Orthogonal Bases by Peter J Olver University of Minnesota Throughout, we work in the Euclidean vector space V = R n, the space of column vectors with n real entries
More informationAn Introduction to 3Dimensional Contact Topology. XiaoSong Lin
An Introduction to 3Dimensional Contact Topology XiaoSong Lin 2 Contents Preface 5 1 Basics 7 1.1 Contact structures............................... 7 1.2 Symplectic structures..............................
More information