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Hyperbolic Geometry Chapter 11 Hyperbolic Lines and Segments • Poincaré disk model Line = circular arc, meets fundamental circle orthogonally • Note: Lines closer to center of fundamental circle are closer to Euclidian lines Why? Poincaré Disk Model • Model of geometric world Different set of rules apply • Rules Points are interior to fundamental circle Lines are circular arcs orthogonal to fundamental circle Points where line meets fundamental circle are ideal points -- this set called • Can be thought of as “infinity” in this context Poincaré Disk Model Euclid’s first four postulates hold 1.Given two distinct points, A and B, a unique line passing through them 2.Any line segment can be extended indefinitely A segment has end points (closed) 3.Given two distinct points, A and B, a circle with radius AB can be drawn 4.Any two right angles are congruent Hyperbolic Triangles • Recall Activity 2 – so … how do you find measure? • We find sum of angles might not be 180 Hyperbolic Triangles • Lines that do not intersect are parallel lines • What if a triangle could have 3 vertices on the fundamental circle? Hyperbolic Triangles • Note the angle measurements • What can you conclude when an angle is 0 ? Hyperbolic Triangles • Generally the sum of the angles of a hyperbolic triangle is less than 180 • The difference between the calculated sum and 180 is called the defect of the triangle • Calculate the defect Hyperbolic Polygons • What does the hyperbolic plane do to the sum of the measures of angles of polygons? Hyperbolic Circles • A circle is the locus of points equidistant from a fixed point, the center What seems “wrong” with these results? • Recall Activity 11.2 Hyperbolic Circles • What happens when the center or a point on the circle approaches “infinity”? • If center could be on fundamental circle “Infinite” radius Called a horocycle Distance on Poincarè Disk Model • Rule for measuring distance metric • Euclidian distance d A, B a1 b1 a2 b2 Metric Axioms 1.d(A, B) = 0 A = B 2.d(A, B) = d(B, A) 3.Given A, B, C points, d(A, B) + d(B, C) d(A, C) 2 2 Distance on Poincarè Disk Model AM / AN AM BN • Formula for d ( A, B) ln ln BM / BN AN BM distance M N Where AM, AN, BN, BM are Euclidian distances Distance on Poincarè Disk Model AM / AN AM BN d ( A, B) ln ln BM / BN AN BM Now work through axioms 1.d(A, B) = 0 A = B 2.d(A, B) = d(B, A) 3.Given A, B, C points, d(A, B) + d(B, C) d(A, C) Circumcircles, Incircles of Hyperbolic Triangles • Consider Activity 11.6a Concurrency of perpendicular bisectors Circumcircles, Incircles of Hyperbolic Triangles • Consider Activity 11.6b Circumcircle Circumcircles, Incircles of Hyperbolic Triangles • Conjecture Three perpendicular bisectors of sides of Poincarè disk are concurrent at O Circle with center O, radius OA also contains points B and C Circumcircles, Incircles of Hyperbolic Triangles • Note issue of bisectors sometimes not intersecting More on this later … Circumcircles, Incircles of Hyperbolic Triangles • Recall Activity 11.7 Concurrence of angle bisectors Circumcircles, Incircles of Hyperbolic Triangles • Recall Activity 11.7 Resulting incenter Circumcircles, Incircles of Hyperbolic Triangles • Conjecture Three angle bisectors of sides of Poincarè disk are concurrent at O Circle with center O, radius tangent to one side is tangent to all three sides Congruence of Triangles in Hyperbolic Plane • Visual inspection unreliable • Must use axioms, theorems of hyperbolic plane First four axioms are available • We will find that AAA is now a valid criterion for congruent triangles!! Parallel Postulate in Poincaré Disk • Playfair’s Postulate l l Given any line and any point P not on , exactly one line on P that is parallel to l • Definition 11.4 Two lines, l and m are parallel if the do not intersect P l Parallel Postulate in Poincaré Disk • Playfare’s postulate Says exactly one line through point P, parallel to line • What are two possible negations to the postulate? 1. No lines through P, parallel 2. Many lines through P, parallel Restate the first – Elliptic Parallel Postulate There is a line l and a point P not on l such that every line through P intersects l Elliptic Parallel Postulate • Examples of elliptic space Spherical geometry • Great circle “Straight” line on the sphere Part of a circle with center at center of sphere Elliptic Parallel Postulate • Flat map with great circle will often be a distorted “straight” line Elliptic Parallel Postulate • Elliptic Parallel Theorem Given any line l and a point P not on l every line through P intersects l • Let line l be the equator All other lines (great circles) through any point must intersect the equator Hyperbolic Parallel Postulate • Hyperbolic Parallel Postulate There is a line l and a point P not on l such that … more than one line through P is parallel to l Parallel Lines, Hyperbolic Plane • Lines outside the limiting rays will be parallel to line AB Called ultraparallel or superparallel or hyperparallel Note line ED is limiting parallel with D at Parallel Lines, Hyperbolic Plane • Consider Activity 11.8 Note the congruent angles, DCE FCD Parallel Lines, Hyperbolic Plane • Angles DCE & FCD are called the angles of parallelism The angle between one of the limiting rays and CD • Theorem 11.4 The two angles of parallelism are congruent Hyperbolic Parallel Postulate • Result of hyperbolic parallel postulate Theorem 11.4 For a given line l and a point P not on l, the two angles of parallelism are congruent • Theorem 11.5 For a given line l and a point P not on l, the two angles of parallelism are acute The Exterior Angle Theorem • Theorem 11.6 If ABC is a triangle in the hyperbolic plane and BCD is exterior for this triangle, then BCD is larger than either CAB or ABC. Parallel Lines, Hyperbolic Plane • Note results of Activity 11.8 CD is a common perpendicular to lines AB, HF • Can be proved in this context If two lines do not intersect then either they are limiting parallels or have a common perpendicular Quadrilaterals, Hyperbolic Plane • Recall results of Activity 11.9 • 90 angles at B and A` Quadrilaterals, Hyperbolic Plane • Recall results of Activity 11.10 • 90 angles at B, A, and D only • Called a Lambert quadrilateral Quadrilaterals, Hyperbolic Plane • Saccheri quadrilateral A pair of congruent sides Both perpendicular to a third side Quadrilaterals, Hyperbolic Plane • Angles at A and B are base angles • Angles at E and F are summit angles Note they are congruent • Side EF is the summit • You should have found not possible to construct rectangle (4 right angles) Hyperbolic Geometry Chapter 11