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Hyperbolic Geometry

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Hyperbolic Geometry Chapter 11 – PowerPoint PPT presentation

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Title: Hyperbolic Geometry


1
Hyperbolic Geometry
  • Chapter 11

2
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?

3
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

4
Poincaré Disk Model
  • Euclids 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

5
Hyperbolic Triangles
  • Recall Activity 2 so how do you find ?
    measure?
  • We find sum of angles might not be 180?

6
Hyperbolic Triangles
  • Lines that do not intersect are parallel lines
  • What if a triangle could have 3 vertices on the
    fundamental circle?

7
Hyperbolic Triangles
  • Note the angle measurements
  • What can you concludewhen an angle is 0? ?

8
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

9
Hyperbolic Polygons
  • What does the hyperbolic plane do to the sum of
    the measures of angles of polygons?

10
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?
11
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

12
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)

13
Distance on Poincarè Disk Model
  • Formula for distance
  • Where AM, AN, BN, BM are Euclidian distances

14
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)

15
Circumcircles, Incircles of Hyperbolic Triangles
  • Consider Activity 11.6a
  • Concurrency of perpendicular bisectors

16
Circumcircles, Incircles of Hyperbolic Triangles
  • Consider Activity 11.6b
  • Circumcircle

17
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

18
Circumcircles, Incircles of Hyperbolic Triangles
  • Note issue of ? bisectors sometimes not
    intersecting
  • More on this later

19
Circumcircles, Incircles of Hyperbolic Triangles
  • Recall Activity 11.7
  • Concurrence of angle bisectors

20
Circumcircles, Incircles of Hyperbolic Triangles
  • Recall Activity 11.7
  • Resulting incenter

21
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

22
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!!

23
Parallel Postulate in Poincaré Disk
  • Playfairs Postulate
  • Given any line l and any point P not on l, ?
    exactly one line on P that is parallel to l
  • Definition 11.4Two lines, l and m are parallel
    if the do not intersect

24
Parallel Postulate in Poincaré Disk
  • Playfares 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

25
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

26
Elliptic Parallel Postulate
  • Flat map with great circle will often be a
    distorted straight line

27
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

28
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

29
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 ?

30
Parallel Lines, Hyperbolic Plane
  • Consider Activity 11.8
  • Note the congruent angles, ?DCE ? ?FCD

31
Parallel Lines, Hyperbolic Plane
  • Angles ?DCE ?FCD are called the angles of
    parallelism
  • The angle betweenone of the limitingrays and CD
  • Theorem 11.4The two anglesof parallelismare
    congruent

32
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

33
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.

34
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

35
Quadrilaterals, Hyperbolic Plane
  • Recall results of Activity 11.9
  • 90? angles at B and A

36
Quadrilaterals, Hyperbolic Plane
  • Recall results of Activity 11.10
  • 90? angles at B, A, and D only
  • Called a Lambert quadrilateral

37
Quadrilaterals, Hyperbolic Plane
  • Saccheri quadrilateral
  • A pair of congruent sides
  • Both perpendicular to a third side

38
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)

39
Hyperbolic Geometry
  • Chapter 11
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