hyperbolic geometry chapter 11. hyperbolic lines and segments poincaré disk model line = circular...

Hyperbolic Geometry

Chapter 11

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

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 11.2

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

Distance on Poincarè Disk Model

• 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

Distance on Poincarè Disk Model

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 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

Given any line l and any point P not on l, 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

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

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 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

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 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 limitingrays and CD

• Theorem 11.4The two anglesof parallelismare congruent

Hyperbolic Parallel Postulate

• Result of hyperbolic parallel postulateTheorem 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 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 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 aresummit angles Note they are congruent

• Side EF is the summit

• You should have foundnot possible to constructrectangle (4 right angles)

Hyperbolic Geometry

Chapter 11