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Metric Applications of Certain Projective Theorems a Thesis Submitted to the Faculty of Atlanta University in Partial Fulfillmen

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Metric Applications of Certain Projective Theorems a Thesis Submitted to the Faculty of Atlanta University in Partial Fulfillmen METRIC APPLICATIONS OF CERTAIN PROJECTIVE THEOREMS A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS BY MADELYN CORNELIA GRAY DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA JUNE 1938 T: 9^* n\ 11 TABLE OF CONTENTS CHAPTER PAGE I INTRODUCTION 1 II EXPLANATION OF SYMBOLISM AND TERMINOLOGY 3 1. Homogeneous Coordinate System 3 2. Linear Dependence 4 3. Linear Combination 6 4. Harmonic Division 7 5. Cross Ratio 8 6. Vanishing Points 8 7. Transformations and Involutions. 9 8. Isotropic Lines and Circular Points at Infinity 10 III SOME PROJECTIVE PROPERTIES OF PLANE FIGURES WITH METRIC APPLICATIONS 11 1. Linear Dependence. 11 a. Projective Theorems 11 b. Metric Applications 14 2. Harmonic Division 18 a. Projective Theorems 18 b. Metric Applications . 20 3. Vanishing Points 24 a. Projective Theorems 24 b. Metric Applications 24 4. Involutions 26 a. Projective Theorems 26 b. Metric Applications 26 BIBLIOGRAPHY 41 Ill LIST OP PIGURUS FIGURE PAGE 1 Vanishing Points of Projective Ranges 29 2 Collinear Points, One on Each Side of a Triangle. 30 3 Bisectors of the Angles Between Two Lines 31 4 Bisectors of the Interior Angles of a Triangle . 32 5 Bisector of One Interior and Two Exterior Angles of a Triangle ..... ...... 33 6 Two Sets of Concurrent Lines Through the Vertices of a Triangle 34 7 Related Collinear Points and Concurrent Lines ... 35 8 A Harmonic Set in a Complete Quadrilateral 36 9 Medians of a Triangle 37 10 An Ordinary Quadrilateral Whose Sides are Isotropic 38 11 Harmonic Conjugate of the Bisector of an Angle Between Two Lines 39 18 Pencil of Circles 40 CHAPTER I INTRODUCTION In this thesis the writer proposes to develop certain metric applications of chosen projective theorems. Metric geometry, the geometry of Euclid, concerns itself with lengths, angles and areas; whereas projective geometry concerns itself with the properties of figures which are preserved under all projections. Metric properties are preserved by the projection of rigid motion but not by all projections; thus metric geometry is a subgeometry of projective geometry. Many familiar metric theorems can be obtained by taking special cases of the more general projective theorems. The writer is interested in pointing out certain ones of these and establishing the results as special cases of more general theorems. In this treatise we shall use the analytic axjproach, that of algebraic symbol notations and computations for geometric interpretation, rather than the synthetic method which is based on formal logic. For the analytic approach the writer must introduce certain ideas and terms which would not be required in the synthetic approach to the subject. These are presented in Chapter II in as brief a manner as the writer deemed consistent with a thorough understanding of the theorems to follow. Chapter III deals with the projective theorems considered and their metric applications. These deal for the most part -1- with linear dependence, harmonic division, vanishing points, and involutions. In each section the projective theorems with proof are given and the metric applications follow. Some of the outstanding contributors to the field are: Poncelet, (1789-1867), who was the real founder of the projective geometry; Desargues, (1593-1662), whose important "Triangle Theorem" underlies many projective theories; and Klein, (1849-1925), whose development was from the point of view of the classification of figures on the basis of their behavior when subjected to certain operations. I wish to acknowledge my deep obligation to Miss Georgia A. Caldwell of Spelman College, for valuable assistance in the preparation of this manuscript, and extend to her sincere appreciation. CHAPTER II DEFINITIONS AND EXPLANATIONS OF NOTATIONS 1. Homogeneous Cartesian Coordinates. Any point P in a plane can be located uniquely by means of its distances from two perpendicular lines. These distances, say (x,yj, are called non-homogeneous Cartesian coordinates of the point. This system is called the rectangular coordinate system or Cartesian coordinate system. Homogeneous coordinates of the finite point P are any three numbers xp,X2,x3 for which (l) x^ s x, xg s y where x3 ^ 0 . "*3“ *3 We have thus assigned homogeneous coordinates for all finite points of the plane, thus exhausting all number triples except those with x3 = 0. We may define an ideal point as the point of intersection of two parallel lines designated as a point at infinity. Since two non-parallel lines meet in only one point, it is advan¬ tageous to create only one ideal point on the line. There are no coordinates for such points in the non- homogeneous coordinate system described above because all number pairs have been assigned to finite points. But in the homogeneous coordinate system, we have still the number triples in which x3 s 0. Since there is but one point at infinity on all lines with a given direction, the point at infinity in the given direction, the number tripleslx-^for which -3- -4- = À *r can be assigned to the point at infinity in the direction of slope i\ as a set of homogeneous coordinates. The equation of a straight line L, in non-homogeneous coordinates,is : (2) axx +- a2y + * 0 If in the equation we make substitutions for x and y from equation (l) and clear the equation of fractions, we obtain the equation ( 3 ) a-Lx1 + a2x2 + a5x3 =. 0 as the equation in the homogeneous coordinates of the line. "■Homogeneous Cartesian coordinates were introduced in 1835 by the German geometer and physicist, Pluecker", (1801-68 2. Linear Dependence. The number pairs a^,a2 and b^, b^ are said to be proportional if there exist two numbers, k, and JL, not both zero, such that ( 1 ) ka-^ + lb = 0, ka^ ■+ j£b2s 0. Sucli numbers can be found if the determinant of their coef¬ a-^ b ficients is equal to zero: ai b 3-°- Extending the definition to two sets of three numbers we have: The number triples ( 2, ) a, i a9 , a, « bl» b2» b3 are linearly dependent if and only if two numbers, k and X, not both zero, can be found such that (3) ka^ 4 Übi = 0, ka^ 4 £bg z 0» ka^ 4 ^bj - 0; 1 W.C. Graustein, Introduction to Higher Geometry, Hew York, 1930, p.30 -5- that is, the three number triples are linearly dependent when the system (3) of the three homogeneous linear equations has solutions for k and jd, other than tne trivial solution, 0,0. Applying the idea of linear dependence to three number triples we have: The number triples (4) a^, ag, 35» b l» b^» b3, cl> c2* 03 are linearly dependent if and only if three numbers, k ,jt>, m, not all zero, exist such that ( 5 ) ka^ / bp + mcp - 0, ka0 •+■ Jt b0 + mcp r 0, '2. ^ ' 2 ^ “~2 ka„ + j[ b^ + mc^ = 0 . This will be true if the determinant of the coefficients of m, is equal to zero: a b l* l* C1 ao » b'j * C',-> - 0 '2 ’ ~2’ "2 a3* b3> b3 In general m number triples are linearly dependent if the rank of their matrix is less than m. It follows,tnerefore, that more than three number triples are always linearly de¬ pendent . The two lines (6)^5 a1x1 + a^Xj a3x3 = 0 3 = blxl + b2x2 + b3x3 = 0 arc linearly dependent if and only if two constants k, not both zero, exist such that -6- (7) 0; that is, such that kap 4 (i bp s 0, ka<j r 0, ka3 + £, b5 = 0. The definition of linear dependence for any number of lines is the same as that for two lines. Thus, the three lines, J\ s apXp + aax? + a2x3 s 0, b P* l*! , baxa , bjX3 ; 0, X Yî ^ t <=2 2 * 03x3 : 0, are linearly dependent, if and only if there exist three con¬ stants, k,^, m, not all zero, such that k<?i^ l 0, that is, such that ka^ 4/bp 4 mcp = 0, ka^ -f/bg 4 mc^ = 0, ka3 -f J!b3 4- mc^ - 0. The determinant of the coefficients of k , Jl, m, mmust then be equal to zero: al* bl* C1 - 0 a2' b2’ C2 a3* b3’ c3 3. Linear Combination. The number triple Cp, c^, c^, is called a linear combination of the number triples ap, a^, a^ and bp, b^, b3, if two numbers, k, * exist such that ( 1 ) C1 r kax 4 ( bp,. - ka^ 4 ^£b^, Cj - ka^ 4 (!. b2 » Generalizing, we have the number set hp, h, .h^ is a linear combination of the number sets a-^, a^... ,an; bf, b ..........,bQ; . ; gp, g^, gQ if numbers A, 3 .G can be found such that the n equations (2) ht r Aap-t-Bbi f Ggp ( i = 1,2,3, n) are satisfied. -7- A linear combination of two straight lines, d\ - 0, 0 is that is (kap +-Xb1)x 4- (ka2 + Xb^Jx^ +■ (ka3 + j^b3Jx3 - Q where k and JL are constants, not both zero* A linear combination of two distinct straight lines is a straight line through their point of intersection. A linear combination of the coordinates of two points is a point on the line which joins them. 4. Harmonic Division. The algebraic ratio A> in which P divides the finite line-segment PpP^ is defined as the quotient of the directed line-segments PP^ ,PPr>: Two distinct points ^^ on a finite line L separate P^P^ harmonically when the ratio in which P^ divides <^p, is the negative of the ratio in which P^ divides i^p, ^2 : h) --4 or vi The constant ratio M .
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