Hyperbolic Geometry Chapter 9

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 Rules 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 Given two distinct points, A and B,  a unique line passing through them Any line segment can be extended indefinitely A segment has end points (closed) Given two distinct points, A and B, a circle with radius AB can be drawn 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?

What seems “wrong” with these results? Hyperbolic Circles A circle is the locus of points equidistant from a fixed point, the center Recall Activity 9.5 What seems “wrong” with these results?

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 Metric Axioms d(A, B) = 0  A = B d(A, B) = d(B, A) Given A, B, C points, d(A, B) + d(B, C)  d(A, C)

Distance on Poincarè Disk Model Formula for distance Where AM, AN, BN, BM are Euclidian distances M N

Distance on Poincarè Disk Model Now work through axioms d(A, B) = 0  A = B d(A, B) = d(B, A) Given A, B, C points, d(A, B) + d(B, C)  d(A, C)

Circumcircles, Incircles of Hyperbolic Triangles Consider Activity 9.3a Concurrency of perpendicular bisectors

Circumcircles, Incircles of Hyperbolic Triangles Consider Activity 9.3b 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 9.4 Concurrence of angle bisectors

Circumcircles, Incircles of Hyperbolic Triangles Recall Activity 9.4 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 Given any line l and any point P not on l,  exactly one line on P that is parallel to l Definition 9.4 Two lines, l and m are parallel if the do not intersect l P

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? No lines through P, parallel 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 There is a line l and a point P not on l such that … more than one line through P is parallel to l

Hyperbolic Parallel Postulate Result of hyperbolic parallel postulate Theorem 9.4 There is at least one triangle whose angle sum is less than the sum of two right angles

Hyperbolic Parallel Postulate Proof: We know  at least two lines parallel to l Note  to l, PQ Also  to PQ, m and thus || to l Note line n also || to l

Hyperbolic Parallel Postulate XPY > 0 Not R on l such that we have PQR QPR < QPY Move R towards fundamental circle, QRP  0 Thus QRP < XPY And PQR has one rt. angle and the other two sum < 90 Thus sum of angles < 180

Parallel Lines, Hyperbolic Plane Theorem 9.5 Hyperbolic Parallel Theorem Given any line l, any point P, not on l,  at leas two lines through P, parallel to l Remember parallel means they don’t intersect

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 9.7 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 9.6 The two angles of parallelism are congruent

Parallel Lines, Hyperbolic Plane Note results of Activity 9.8 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 9.9 90 angles at B and A

Quadrilaterals, Hyperbolic Plane Recall results of Activity 9.10 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 9