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Hyperbolic Geometry

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Hyperbolic Geometry Hyperbolic Geometry Eric Lehman February 2014 2 The course follows the chapter called "Non-Euclidean Geometry" in the book "Geo- metry" by David A. Brannan, Matthew F. Esplen and Jeremy J. Gray and gives alternative proofs of some theorems in a more geometric and less computational presentation. In the two usual models of Hyperbolic Geometry the lines are parts of Euclidean circles orthogonal to a border wich is a circle or a line. Therefore the course begins with some complements in Euclidean and Conformal Geometry about circles, inversion and circle bundles. These tools give a geometric insight in basic concepts in Hyperbolic Geometry such as Möbius trans- formations, parallelism and ultraparallelism, orthogonality, asymptotic triangles and area defect. Hyperbolic Geometry might be useful, but most of all it is beautiful ! II. Reading Hyperbolic Geometry 32 4 Non-Euclidean Geometry 33 § 1. chapter 6. Non-Euclidean Geometry : the two usual mo- dels of a hyperbolic plane . 33 § 2. 6.1.1 Non-Euclidean Geometry : the two usual models of a hyperbolic plane . 35 § 3. 6.1.2 Existence of hyperbolic lines . 36 Table of contents § 4. 6.1.3 Inversion preserves inversion points . 38 5 The group of Hyperbolic Geometry 39 § 1. 6.2.1 Non-Euclidean Transformations and Möbius Transformations . 39 § 2. 6.2.2 The Canonical Form of a Hyperbolic Transfor- I. Circles in Euclidean Geometry 1 mation . 42 § 3. 6.2.3 The Canonical Form of a Hyperbolic Transfor- 1 Power of a point with respect to a circle 3 mation . 43 § 1. Equations of a circle . .3 § 2. Orthogonal circles . .6 6 Distance 45 § 1. 6.3.1 The distance formula . 45 2 Reflection in a circle 11 § 2. 6.3.2 Midpoints and 6.3.4 Reflection . 47 § 1. Definition of inversions . 11 § 3. 6.3.2 Circles . 47 § 2. Images of lines and circles . 14 § 4. 6.3.2 Reflections . 47 § 3. Description of inversions using complex numbers . 17 § 5. 6.3.2 Midpoints . 47 § 6. 6.4.1 Triangles . 47 3 Circle bundles 19 § 7. 6.4.2 Orthogonal lines . 47 § 1. Different kinds of circle bundles . 19 § 2. Orthogonal bundles . 22 § 3. General definitions and properties . 25 3 4 TABLE OF CONTENTS Thème I Circles in Euclidean Geometry Chapitre 1 Power of a point with respect to a circle § 1. Equations of a circle § 2. Orthogonal circles § 1. Equations of a circle 1.1 Circle of center .a; b/ and radius R We suppose given an orthonormal frame .O; {; j / of a Euclidean plane P. E Let us denote by C.; R/ or simply by C theE circle with center .a; b/ and radius R. A point M.x; y/ belongs to that circle if M R or M 2 R2 0 or by Pytagoras theorem D D .x a/2 .y b/2 R2 0 (1.1) C D That characteristic relation is called an equation of C . We define the function fC by 2 2 2 fC .x; y/ .x a/ .y b/ R D C The equations of C are ˛fC .x; y/ 0 D where ˛ is any real number different from 0. When ˛ 1, we say that the equation is the normal equation of C . We may write the normal equationD of C as x2 y2 2ax 2by c 0; where c a2 b2 R2 (1.2) C C D D C Interpretation of c. If c > 0, c is the square of the distance from the origine of coordinates O to a contactpoint T of a tangent to the circle C through O. This is also true if c 0, since then O is on the circle C . D 3 4 THÈME I. CIRCLES IN EUCLIDEAN GEOMETRY.CH.1. POWER OF A POINT WITH RESPECT TO A CIRCLE T C R pc .a; b/ O If c < 0, c is the square of the length of the segment OB where B is a point common to the circle C and to the line through O orthogonal to the line O (what happens when O ?). D C B R ......... p c ............. .................................................................................................. .a; b/ O 1.2 Power of a point with respect to a circle Definition. Let C be a circle with center .a; b/ and radius R and let M.x; y/ be any point. We denote by d the distance from M to the center of the circle. The power of the point M with respect to the circle C denoted by PC .M / is the number 2 2 PC .M / d R D Using the notations of the preceding paragraph, we have Proposition.P C .M / fC .x; y/. D Remark. c PC .O/ fC .0; 0/. D D Theorem 1. (See proof on next page) Let C be a circle, M.x; y/ a point and d a line through M . If d intersects C in two points P and Q, then MP MQ if M is outside C PC .M / D MP MQ if M is inside C Remark 1. The important consequence of this theorem is that the product MP MQ is constant when turning the line d around the point M . § 1. EQUATIONS OF A CIRCLE 5 Remark 2. Using the scalar product,we have in all cases PC .M / !MP !MQ D In particular PC .M / 0 if and only if M belongs to C . D Q C C P P M M Q Theorem 2. Let C be a circle, M.x; y/ a point outside C and d a line through M tangent to C at a contact point T . Then 2 PC .M / MT D C T R p PC .M / d M Proof of theorem 2. Let be the center of the circle C and R its radius. Denote the length M by d. The angle M T is a right angle and thus by Pythagoras’ theorem 2 2 2 2 2 MT M T d R PC .M / D D D Proof of theorem 1. Let P 0 be the point opposite to the point Q on the circle C . Since QP 0 is a diameter of C , the line PP 0 is orthogonal to the line PQ. Thus, the point P is the orthogonal projection of1P 0 on the line MQ, and !MP !MQ MP!0 !MQ D Let us write !MQ and MP! as !MQ !M !Q and MP! !M P!, we have 0 D C 0 D C 0 P! !Q and and so 0 D 2 2 2 2 !MP !MQ .!M !Q/ .!M !Q/ !M !Q d R PC .M / D C D D D 6 THÈME I. CIRCLES IN EUCLIDEAN GEOMETRY.CH.1. POWER OF A POINT WITH RESPECT TO A CIRCLE 1.3 Use of complex numbers We use the same notations as above and put z x iy and w a ib D C D C We denote the complex conjugate of a complex number z by z. Then z x iy and N N D x2 y2 zz ; .x a/2 .y b/2 .z w/.z w/ and a2 b2 ww C D C D C D The equation of C may be written .z w/.z w/ R2 0 D or zz .wz wz/ c 0 C C D 2 where c ww R PC .O/ R. D D 2 § 2. Orthogonal circles 2.1 Angle of intersecting circles For any curves of class C1, intersecting in a point T , one defines the angle of these two curves as the angle of their tangents in T . If two circles are intersecting, there are two points of intersection, and by the reflection through the line joining the centers, one sees that the angles are the same if one looks at nonoriented angles and that oriented angles are opposite. Definition. Two circles are orthogonal if they are intersecting and if their angles at the intersecting points are right angles. Remark. Since the tangent to a circle through a point T of a circle is orthogonal to the corresponding radius, we see that two circles with centers and 0 are orthogonal if the angle T 0 is a right angle. § 2. ORTHOGONAL CIRCLES 7 T 0 2.2 Orthogonality conditions Theorem. Let C and C 0 be two circles with respective centers .a; b/ and 0.a0; b0/ and respective radii R and R0, intersecting in points T and T 0. The circles C and C 0 are ortho- gonal if and only if one of the following equivalent conditions is fulfilled : 1. 2 R2 R 2 0 D C 0 2. The triangle T 0 is rectangle in T 2 3. PC ./ R 0 D 2 4. PC . / R 0 D 0 5. .a a /2 .b b /2 R2 R 2 0 C 0 D C 0 Theorem. Two circles C and C 0 with real equations 2 2 2 2 x y 2ax 2by c 0 and x y 2a0x 2b0y c0 0 C C D C C D are orthogonal if and only if 2.aa bb / c c 0 C 0 D C 0 Proof. The condition 5 above may be written 2 2 2 2 2 2 a b R a0 b0 R0 2aa0 2bb0 C C C D C We get the result from c a2 b2 R2 and c a 2 b 2 R 2. D C 0 D 0 C 0 0 Comment. Why is it convenient to use equations that are not necessarily normal ? The normal equation x2 y2 2ax 2by c 0 can only be used for genuine circles ; the equation ˛.x2 Cy2/ C2Ax C2By C D0 will describe a genuine circle if ˛ 0 AND will describeC a line if ˛ 0 and .A;C B/D .0; 0/.
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