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Geometry > Geometry Epiphany > Flashcards

Geometry Epiphany Flashcards

1
Q

Mobius transformation map is a bijection of the

A

Riemann sphere onto itself

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2
Q

Mobius transformation form a group with respect to compoistion isomorphic to

A

invertible matrices up to scalar multiplication , det is not 0

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3
Q

Mob transformation is generated by

A

z-> az, z -> z + 1, z -> 1/z

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4
Q

Mobius transformations act on C U infinity

A

triply-transitively

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5
Q

A mobius transformation is uniquely determined by the images of

A

3 points

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6
Q

Mobius transformation takes lines and circles to

A

lines and circles

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7
Q

Mobius transformations preserve

A

angles between curves

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8
Q

fixed points of mobius transformation is a quadratic equation with respect to z and has

A

exactly 2 complex roots

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9
Q

A mobius transformations with a unique fixed point is called

A

parabolic

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10
Q

Every parabolic mobius transformations is conjugate in the group Mob to

A

z -> z + 1

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11
Q

Every non- parabolic mobius transformation is conjugate in Mob to

A

z -> az, a is a complex number not 0

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12
Q

A non - parabolic Mob transformations is elliptic if

A

|a| = 1

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13
Q

A non - parabolic Mob transformations is hyperbolic if

A

|a| is not 1 and a is real

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14
Q

A non - parabolic Mob transformations is ioxodromic of

A

|a| is not 1 and a is not real

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15
Q

Parabolic elements when fixed point is infinite is a

A

translation of all points by same vecotr. When Mob trans is applied iterations of trans move points along circles through the fixed point

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16
Q

Elliptic elements rotate points

A

around 2 equally good fixed points

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17
Q

hyperbolic and ioxodromic have

A

one attractive fixpoint and one repelling

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18
Q

Inversion with respect to circle takes a point A to

A

a point A’ lying on the ray OA such that |OA| . |OA’| = r^2

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19
Q

Inversion^2 =

A

identity

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20
Q

inversion in circle preserves

A

circle pointwise

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21
Q

if P’ = I_gamma(P) and Q’ = I(Q’) than triangle OPQ is

A

similar to triangle OQ’P’

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22
Q

lines through origin are mapped to

A

lines through origin

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23
Q

lines not through origin are mapped to

A

circles through origin and vice versa

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24
Q

circles not through origin are mapped to

A

circles not through origin

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25
Q

Inversion preserves

A

angles between curves and cross-ratio of four points

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26
Q

Inversion can be thought of as ‘reflection with respect to a

A

circle’

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27
Q

every inversion is conjugate to a

A

reflection by another inversion

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28
Q

Every Mobius transformation is a composition of an

A

even number of inversions and reflections

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29
Q

Inversion and reflection change

A

orientation of the plane

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30
Q

Since mobius transformation are even number of inversion and reflection they preserve

A

orientation

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31
Q

mobius transformations preserve

A

cross- ratios

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32
Q

Mobius transformation is determined by images of

A

3 points

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33
Q

Points z1,z2,z3 (complex numbers) are collinear iff

A

z1-z2/z1-z3 is real

34
Q

points z1,z2,z3,z4 lie on one line or circle iff

A

the cross ratio is real

35
Q

given 4 distinct points the cross ratio is not

A

1

36
Q

cross ratios of four points lying on a line or a circle are preserve by

A

inversions and reflections

37
Q

inversion takes spheres and planes to

A

spheres and planes

38
Q

Stereographic projection takes circles to

A

circles and lines

39
Q

stereographic projection preserves

A

angles and cross-ratios

40
Q

Poincare disc model is

A

unit disc, boundary is absolute, lines and circles orthogonal to absolute, distance is cross ratio,

41
Q

any two points in Poincare disc model there exists a …….. through both points

A

hyperbolic line. sae if one or both points are lying on absolute

42
Q

if A and B are on absolute of poincare disc modal than any hyperbolic line through them are ….. to absolute at A and B

A

orthogonal

43
Q

d\9A,B\0 on poincare disc model satisfies …. of distance

A

axioms - positvity, symmetry and trainagle inequality

44
Q

all isometries in poincare disc model preserve the

A

disc and cross-ratios

45
Q

l is a orthogonal line and A is not on l but in disc or absolute. then there exsists l’ ….

A

through A orthognal to l

46
Q

l is a hyperbolic line and A is on l then there exists l’

A

through A and orthognal to l

47
Q

every hyperbolic segments has a

A

midpoint

48
Q

the isometry group of poincare disc acts transitively on

A

triples of points of the absolute and on points in disc and on flags

49
Q

for c in AB d(A,C) + d(C,B) =

A

d(A,B)

50
Q

in the right angled traingle with angle c = pi/2 d(B,C) ,

A

d(B,A)

51
Q

triangle inequality for c not in AB d(A,C) +d(C,B)

A

> = d(A,B)

52
Q

hyperbolic circles are represented by ….. in the poincare disc model

A

euclidean circles

53
Q

every eunclidean cricles in the poincare disc represents a

A

hyperbolic line

54
Q

euclidean centre of the circle with hyperbolic centre A not in o is different from

A

hypberolic centre A

55
Q

An isometry of H^2 is uniquely determined by the image of a

A

flag

56
Q

every isomtery of the poincare disc model can be written as either … or

A

az+b/cz+d mobius transformation or anit-mobius transformation

57
Q

An insometry of H^2 is uniqule determined by the images of three

A

points of the absolute

58
Q

isometries preserve the angles means

A

hyperbolic angles coincides with euclidean ones

59
Q

the sum of angles in a hyperbolic triangle is less than

A

pi

60
Q

in the upper half plane hyperbolic circles are represented by

A

euclidean circles

61
Q

every isometry of the upper half plane can be written as

A

mobius transfomration or anti mobius where (-z)

62
Q

orientation preserving isometries of the upper half plane can be written as

A

mobius transformation where determinant is 1

63
Q

orient reversing isomtries can be written as

A

anti mobius transfomration where det = -1

64
Q

parallel aviom for hyperbolic geomtery says

A

there are infinietly many line l’ disjoint from a given line l and passing through a given point A not in l

65
Q

angle of parallelism depends only on the

A

distance d(A.l) = min(A,B) = d(A,H)

66
Q

a hyperbolic polygon with all vertices on the absolute is called

A

ideal polygon

67
Q

Klein disc: model is inside …. lines are represented by ….. isometries are …….

A

unit disc, chords, projective maps preserving the disc

68
Q

geomtery of the kelin disc coincides with geomtery of the

A

poincare disc

69
Q

can you light to project hemisphere model to

A

klein disc, Poincare disc, and upper half-plane

70
Q

Klein disc model is useful for

A

working with lines and right angles

71
Q

isometries are projective transformations preserving the

A

cone

72
Q

2-sheet hyperboloid determines the same hyperbolic geometry as the

A

klein model

73
Q

if <a,a> > 0 than hyperbolic line, la, intersects the

A

cone producing a hyperbolic line

74
Q

if <a,a> = 0 than the hyperbolic line is tangent to

A

cone producing the point a on the absolute

75
Q

<a,a> < 0 than hyperbolic line does not intersect the …. and gives no ….

A

cone, line

76
Q

a reflection with respect to a hyperbolic line l is an isometry preserving the line l …. and swapping the …..

A

pointwise, half-planes

77
Q

any of isometry of two sheet hyperboloid is a composition of at most

A

3 reflections

78
Q

a non-trivial orientation - preserving isomtery of two sheet paraboloid has either .. fixed point in H^2, or … fixed point on the absolutw=e or … fixed point on the absolute

A

1,1,2

79
Q

a non-trivial orientation - preserving isomtery of two sheet paraboloid is elliptic if

A

it has 1 fixed point in H^2

80
Q

a non-trivial orientation - preserving isomtery of two sheet paraboloid is parabolic if it has

A

1 fixed point at the absolute

81
Q

a non-trivial orientation - preserving isomtery of two sheet paraboloid hyperbolic if it has

A

2 fixed points on the absolute

Geometry (2 decks)

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