Hyperbolic Geometry

Hyperbolic Geometry Chapter 9

Hyperbolic Lines and Segments • Poincaré disk model • Line = circular arc, meets fundamental circle orthogonally • Note: • Lines closer tocenter of fundamentalcircle 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 • 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 concludewhen 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 • Calculatethe 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 • 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 beon fundamentalcircle • “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 9.4Two lines, l and m are parallel if the do not intersect P l

Parallel Postulate in Poincaré Disk • Playfare’s postulate Says  exactlyone 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 atcenter 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 levery line through P intersects l • Let line l be the equator • All other lines (great circles) through any pointmust intersect the equator

Hyperbolic Parallel Postulate • Hyperbolic Parallel Postulate • There is a line land 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 postulateTheorem 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, mand 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 towardsfundamental circle, QRP  0 • Thus QRP < XPY • And PQR has onert. angle and the other two sum < 90 • Thus sum of angles < 180

Parallel Lines, Hyperbolic Plane • Theorem 9.5 Hyperbolic Parallel TheoremGiven any line l, any point P, not on l,  at leas two lines through P, parallel to l • Rememberparallel meansthey don’tintersect

Parallel Lines, Hyperbolic Plane • Lines outside the limiting rays will beparallel to line AB • Calledultraparallel orsuperparallel orhyperparallel • 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 betweenone of the limitingrays and CD • Theorem 9.6The two anglesof parallelismare congruent

Parallel Lines, Hyperbolic Plane • Note results of Activity 9.8 • CD is a commonperpendicular tolines AB, HF • Can be proved inthis context • If two lines do not intersect then eitherthey are limiting parallelsor have a commonperpendicular

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 aresummit angles • Note they are congruent • Side EF is the summit • You should have foundnot possible to constructrectangle (4 right angles)

Hyperbolic Geometry Chapter 9

Hyperbolic Geometry