Geometry Epiphany Flashcards
Mobius transformation map is a bijection of the
Riemann sphere onto itself
Mobius transformation form a group with respect to compoistion isomorphic to
invertible matrices up to scalar multiplication , det is not 0
Mob transformation is generated by
z-> az, z -> z + 1, z -> 1/z
Mobius transformations act on C U infinity
triply-transitively
A mobius transformation is uniquely determined by the images of
3 points
Mobius transformation takes lines and circles to
lines and circles
Mobius transformations preserve
angles between curves
fixed points of mobius transformation is a quadratic equation with respect to z and has
exactly 2 complex roots
A mobius transformations with a unique fixed point is called
parabolic
Every parabolic mobius transformations is conjugate in the group Mob to
z -> z + 1
Every non- parabolic mobius transformation is conjugate in Mob to
z -> az, a is a complex number not 0
A non - parabolic Mob transformations is elliptic if
|a| = 1
A non - parabolic Mob transformations is hyperbolic if
|a| is not 1 and a is real
A non - parabolic Mob transformations is ioxodromic of
|a| is not 1 and a is not real
Parabolic elements when fixed point is infinite is a
translation of all points by same vecotr. When Mob trans is applied iterations of trans move points along circles through the fixed point
Elliptic elements rotate points
around 2 equally good fixed points
hyperbolic and ioxodromic have
one attractive fixpoint and one repelling
Inversion with respect to circle takes a point A to
a point A’ lying on the ray OA such that |OA| . |OA’| = r^2
Inversion^2 =
identity
inversion in circle preserves
circle pointwise
if P’ = I_gamma(P) and Q’ = I(Q’) than triangle OPQ is
similar to triangle OQ’P’
lines through origin are mapped to
lines through origin
lines not through origin are mapped to
circles through origin and vice versa
circles not through origin are mapped to
circles not through origin
Inversion preserves
angles between curves and cross-ratio of four points
Inversion can be thought of as ‘reflection with respect to a
circle’
every inversion is conjugate to a
reflection by another inversion
Every Mobius transformation is a composition of an
even number of inversions and reflections
Inversion and reflection change
orientation of the plane
Since mobius transformation are even number of inversion and reflection they preserve
orientation
mobius transformations preserve
cross- ratios
Mobius transformation is determined by images of
3 points
Points z1,z2,z3 (complex numbers) are collinear iff
z1-z2/z1-z3 is real
points z1,z2,z3,z4 lie on one line or circle iff
the cross ratio is real
given 4 distinct points the cross ratio is not
1
cross ratios of four points lying on a line or a circle are preserve by
inversions and reflections
inversion takes spheres and planes to
spheres and planes
Stereographic projection takes circles to
circles and lines
stereographic projection preserves
angles and cross-ratios
Poincare disc model is
unit disc, boundary is absolute, lines and circles orthogonal to absolute, distance is cross ratio,
any two points in Poincare disc model there exists a …….. through both points
hyperbolic line. sae if one or both points are lying on absolute
if A and B are on absolute of poincare disc modal than any hyperbolic line through them are ….. to absolute at A and B
orthogonal
d\9A,B\0 on poincare disc model satisfies …. of distance
axioms - positvity, symmetry and trainagle inequality
all isometries in poincare disc model preserve the
disc and cross-ratios
l is a orthogonal line and A is not on l but in disc or absolute. then there exsists l’ ….
through A orthognal to l
l is a hyperbolic line and A is on l then there exists l’
through A and orthognal to l
every hyperbolic segments has a
midpoint
the isometry group of poincare disc acts transitively on
triples of points of the absolute and on points in disc and on flags
for c in AB d(A,C) + d(C,B) =
d(A,B)
in the right angled traingle with angle c = pi/2 d(B,C) ,
d(B,A)
triangle inequality for c not in AB d(A,C) +d(C,B)
> = d(A,B)
hyperbolic circles are represented by ….. in the poincare disc model
euclidean circles
every eunclidean cricles in the poincare disc represents a
hyperbolic line
euclidean centre of the circle with hyperbolic centre A not in o is different from
hypberolic centre A
An isometry of H^2 is uniquely determined by the image of a
flag
every isomtery of the poincare disc model can be written as either … or
az+b/cz+d mobius transformation or anit-mobius transformation
An insometry of H^2 is uniqule determined by the images of three
points of the absolute
isometries preserve the angles means
hyperbolic angles coincides with euclidean ones
the sum of angles in a hyperbolic triangle is less than
pi
in the upper half plane hyperbolic circles are represented by
euclidean circles
every isometry of the upper half plane can be written as
mobius transfomration or anti mobius where (-z)
orientation preserving isometries of the upper half plane can be written as
mobius transformation where determinant is 1
orient reversing isomtries can be written as
anti mobius transfomration where det = -1
parallel aviom for hyperbolic geomtery says
there are infinietly many line l’ disjoint from a given line l and passing through a given point A not in l
angle of parallelism depends only on the
distance d(A.l) = min(A,B) = d(A,H)
a hyperbolic polygon with all vertices on the absolute is called
ideal polygon
Klein disc: model is inside …. lines are represented by ….. isometries are …….
unit disc, chords, projective maps preserving the disc
geomtery of the kelin disc coincides with geomtery of the
poincare disc
can you light to project hemisphere model to
klein disc, Poincare disc, and upper half-plane
Klein disc model is useful for
working with lines and right angles
isometries are projective transformations preserving the
cone
2-sheet hyperboloid determines the same hyperbolic geometry as the
klein model
if <a,a> > 0 than hyperbolic line, la, intersects the
cone producing a hyperbolic line
if <a,a> = 0 than the hyperbolic line is tangent to
cone producing the point a on the absolute
<a,a> < 0 than hyperbolic line does not intersect the …. and gives no ….
cone, line
a reflection with respect to a hyperbolic line l is an isometry preserving the line l …. and swapping the …..
pointwise, half-planes
any of isometry of two sheet hyperboloid is a composition of at most
3 reflections
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
1,1,2
a non-trivial orientation - preserving isomtery of two sheet paraboloid is elliptic if
it has 1 fixed point in H^2
a non-trivial orientation - preserving isomtery of two sheet paraboloid is parabolic if it has
1 fixed point at the absolute
a non-trivial orientation - preserving isomtery of two sheet paraboloid hyperbolic if it has
2 fixed points on the absolute
