Geometry Epiphany

Question 1

Mobius transformation map is a bijection of the

Answer 1

Riemann sphere onto itself

Question 2

Mobius transformation form a group with respect to compoistion isomorphic to

Answer 2

invertible matrices up to scalar multiplication , det is not 0

Question 3

Mob transformation is generated by

Answer 3

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

Question 4

Mobius transformations act on C U infinity

Answer 4

triply-transitively

Question 5

A mobius transformation is uniquely determined by the images of

Answer 5

3 points

Question 6

Mobius transformation takes lines and circles to

Answer 6

lines and circles

Question 7

Mobius transformations preserve

Answer 7

angles between curves

Question 8

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

Answer 8

exactly 2 complex roots

Question 9

A mobius transformations with a unique fixed point is called

Answer 9

parabolic

Question 10

Every parabolic mobius transformations is conjugate in the group Mob to

Answer 10

z -> z + 1

Question 11

Every non- parabolic mobius transformation is conjugate in Mob to

Answer 11

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

Question 12

A non - parabolic Mob transformations is elliptic if

Answer 12

|a| = 1

Question 13

A non - parabolic Mob transformations is hyperbolic if

Answer 13

|a| is not 1 and a is real

Question 14

A non - parabolic Mob transformations is ioxodromic of

Answer 14

|a| is not 1 and a is not real

Question 15

Parabolic elements when fixed point is infinite is a

Answer 15

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

Question 16

Elliptic elements rotate points

Answer 16

around 2 equally good fixed points

Question 17

hyperbolic and ioxodromic have

Answer 17

one attractive fixpoint and one repelling

Question 18

Inversion with respect to circle takes a point A to

Answer 18

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

Question 19

Inversion^2 =

Answer 19

identity

Question 20

inversion in circle preserves

Answer 20

circle pointwise

Question 21

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

Answer 21

similar to triangle OQ’P’

Question 22

lines through origin are mapped to

Answer 22

lines through origin

Question 23

lines not through origin are mapped to

Answer 23

circles through origin and vice versa

Question 24

circles not through origin are mapped to

Answer 24

circles not through origin

Question 25

Inversion preserves

Answer 25

angles between curves and cross-ratio of four points

Question 26

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

Answer 26

circle'

Question 27

every inversion is conjugate to a

Answer 27

reflection by another inversion

Question 28

Every Mobius transformation is a composition of an

Answer 28

even number of inversions and reflections

Question 29

Inversion and reflection change

Answer 29

orientation of the plane

Question 30

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

Answer 30

orientation

Question 31

mobius transformations preserve

Answer 31

cross- ratios

Question 32

Mobius transformation is determined by images of

Answer 32

3 points

Question 33

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

Answer 33

z1-z2/z1-z3 is real

Question 34

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

Answer 34

the cross ratio is real

Question 35

given 4 distinct points the cross ratio is not

Answer 35

1

Question 36

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

Answer 36

inversions and reflections

Question 37

inversion takes spheres and planes to

Answer 37

spheres and planes

Question 38

Stereographic projection takes circles to

Answer 38

circles and lines

Question 39

stereographic projection preserves

Answer 39

angles and cross-ratios

Question 40

Poincare disc model is

Answer 40

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

Question 41

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

Answer 41

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

Question 42

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

Answer 42

orthogonal

Question 43

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

Answer 43

axioms - positvity, symmetry and trainagle inequality

Question 44

all isometries in poincare disc model preserve the

Answer 44

disc and cross-ratios

Question 45

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

Answer 45

through A orthognal to l

Question 46

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

Answer 46

through A and orthognal to l

Question 47

every hyperbolic segments has a

Answer 47

midpoint

Question 48

the isometry group of poincare disc acts transitively on

Answer 48

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

Question 49

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

Answer 49

d(A,B)

Question 50

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

Answer 50

d(B,A)

Question 51

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

Answer 51

>= d(A,B)

Question 52

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

Answer 52

euclidean circles

Question 53

every eunclidean cricles in the poincare disc represents a

Answer 53

hyperbolic line

Question 54

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

Answer 54

hypberolic centre A

Question 55

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

Answer 55

flag

Question 56

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

Answer 56

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

Question 57

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

Answer 57

points of the absolute

Question 58

isometries preserve the angles means

Answer 58

hyperbolic angles coincides with euclidean ones

Question 59

the sum of angles in a hyperbolic triangle is less than

Answer 59

pi

Question 60

in the upper half plane hyperbolic circles are represented by

Answer 60

euclidean circles

Question 61

every isometry of the upper half plane can be written as

Answer 61

mobius transfomration or anti mobius where (-z)

Question 62

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

Answer 62

mobius transformation where determinant is 1

Question 63

orient reversing isomtries can be written as

Answer 63

anti mobius transfomration where det = -1

Question 64

parallel aviom for hyperbolic geomtery says

Answer 64

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

Question 65

angle of parallelism depends only on the

Answer 65

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

Question 66

a hyperbolic polygon with all vertices on the absolute is called

Answer 66

ideal polygon

Question 67

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

Answer 67

unit disc, chords, projective maps preserving the disc

Question 68

geomtery of the kelin disc coincides with geomtery of the

Answer 68

poincare disc

Question 69

can you light to project hemisphere model to

Answer 69

klein disc, Poincare disc, and upper half-plane

Question 70

Klein disc model is useful for

Answer 70

working with lines and right angles

Question 71

isometries are projective transformations preserving the

Answer 71

cone

Question 72

2-sheet hyperboloid determines the same hyperbolic geometry as the

Answer 72

klein model

Question 73

if > 0 than hyperbolic line, la, intersects the

Answer 73

cone producing a hyperbolic line

Question 74

if = 0 than the hyperbolic line is tangent to

Answer 74

cone producing the point a on the absolute

Question 75

< 0 than hyperbolic line does not intersect the .... and gives no ....

Answer 75

cone, line

Question 76

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

Answer 76

pointwise, half-planes

Question 77

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

Answer 77

3 reflections

Question 78

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

Answer 78

1,1,2

Question 79

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

Answer 79

it has 1 fixed point in H^2

Question 80

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

Answer 80

1 fixed point at the absolute

Question 81

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

Answer 81

2 fixed points on the absolute