Hyperbolic Geometry Chapter 9. Hyperbolic Lines and Segments Poincaré disk model Line = circular arc, meets fundamental circle orthogonally Note:

Hyperbolic Geometry

Chapter 9

Hyperbolic Lines and Segments

• Poincaré disk model Line = circular arc, meets fundamental circle

orthogonally

• Note: Lines closer to

center 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

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


• Lines that do not intersect are parallel lines

• What if a triangle could have 3 vertices on the fundamental circle?


• Note the angle measurements

• What can you concludewhen an angle is 0 ?


• 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

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

1 1 2 2,d A B a b a b


• Formula for distance

Where AM, AN, BN, BM are Euclidian distances

M

N

/( , ) ln ln

/

AM AN AM BNd A B

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)

/( , ) ln ln

/

AM AN AM BNd A B

BM BN AN BM

Circumcircles, Incircles of Hyperbolic Triangles

• Consider Activity 9.3a Concurrency of perpendicular bisectors


• Consider Activity 9.3b Circumcircle


• 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


• Note issue of bisectors sometimes not intersecting

More on this later …


• Recall Activity 9.4 Concurrence of angle bisectors


• Recall Activity 9.4 Resulting incenter


• 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?

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


• Flat map with great circle will often be a distorted “straight” line


• 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 pointmust 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


• 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


• 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


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 TheoremGiven any line l, any point P, not on l, at leas two lines through P, parallel to l Remember

parallel meansthey don’tintersect


• Lines outside the limiting rays will beparallel to line AB

Calledultraparallel orsuperparallel orhyperparallel

Note line ED is limiting parallel with D at


• Consider Activity 9.7 Note the congruent angles, DCE FCD


• Angles DCE & FCD are called the angles of parallelism The angle between

one 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


• Recall results of Activity 9.10

• 90 angles at B, A, and D only• Called a Lambert quadrilateral


• Saccheri quadrilateral A pair of congruent sides Both perpendicular to a third side


• 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

Documents

Hyperbolic Geometry Chapter 9. Hyperbolic Lines and Segments Poincaré disk model Line = circular arc, meets fundamental circle orthogonally Note: