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Differential Geometry:
Conformal Maps
Linear Transformations
Definition:
We say that a linear transformation M:Rn→Rn
preserves angles if M(v)≠0 for all v≠0 and:
Mv, Mw
v, w
=
Mv Mw
vw
for all v and w in Rn.
Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
If we denote by ei, the vector that has a one in
the i-th place and zero everywhere else, then:
T
e j Mei = M ij
Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
If we denote by ei, the vector that has a one in
the i-th place and zero everywhere else, then:
T
e j Mei = M ij
Since 〈ei,ej〉=0 whenever i≠j, this implies that:
Mei , Me j = 0
(M
T
M
)
ij
=0
Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
If we denote by ei, the vector that has a one in
the i-th place and zero everywhere else, then:
T
e j Mei = M ij
Since 〈ei,ej〉=0 whenever i≠j, this implies that:
Mei , Me j = 0
(M
T
M
)
ij
=0
So MTM must be a diagonal matrix.
Linear Transformations
Mv, Mw
v, w
=
Mv Mw
vw
Note:
If we denote by ei, the vector that has a one in
the i-th place and zero everywhere else, then:
T
e j Mei = M ij
Similarly, since 〈ei-ej,ei+ej〉=0, this implies that:
M (ei − e j ), M (ei + e j ) = 0
(M
T
M
) − (M
ii
T
M
)
jj
=0
Linear Transformations
Mv, Mw
v, w
=
Mv Mw
vw
Note:
If we denote by ei, the vector that has a one in
the i-th place and zero everywhere else, then:
T
e j Mei = M ij
Similarly, since 〈ei-ej,ei+ej〉=0, this implies that:
M (ei − e j ), M (ei + e j ) = 0
(M
T
M
) − (M
ii
T
M
)
jj
=0
So the diagonal entries of MTM are all equal.
Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
Thus, if M preserves angles, then MTM must be
λ  0 
of the form:


MTM =     
0  λ


Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
Thus, if M preserves angles, then MTM must be
λ  0 
of the form:


If we set
MTM =     
0  λ


N=M/λ1/2, then:
N T N = Id
Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
Thus, if M preserves angles, then MTM must be
λ  0 
of the form:


If we set
MTM =     
0  λ


N=M/λ1/2, then:
N T N = Id
So N must be a rotation/reflection.
Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
Thus, if M preserves angles, then MTM must be
λ  0 
of the form:


If we set
MTM =     
0  λ


N=M/λ1/2, then:
N T N = Id
So N must be a rotation/reflection.
And M must be a rotation/reflection composed
with a scaling transformation.
Linear Transformations
Mv, Mw
=
v, w
Mv Mw
vw
Note:
In particular, M preserves angles, if and only if
it maps circles to circles.
M
Complex Numbers
A complex number z is any number that can be
written as:
z = x + iy
where x and y are real numbers and i is the
square root of -1: 2
i = −1
Complex Numbers
Given two complex numbers, z1=x1+iy1 and
z2=x2+iy2:
• The sum of the numbers is:
z1 + z 2 = ( x1 + x2 ) + i ( y1 + y2 )
Complex Numbers
Given two complex numbers, z1=x1+iy1 and
z2=x2+iy2:
• The sum of the numbers is:
z1 + z 2 = ( x1 + x2 ) + i ( y1 + y2 )
• The product of the numbers is:
z1 z 2 = ( x1 + iy1 )( x2 + iy2 )
= x1 x2 + iy1iy2 + x1iy2 + iy1 x2
= ( x1 x2 − y1 y2 ) + i ( x1 y2 + y1 x2 )
Complex Numbers
Given a complex numbers, z=x+iy:
• The negation of the number is:
− z = − x − iy
Complex Numbers
Given a complex numbers, z=x+iy:
• The negation of the number is:
− z = − x − iy
• The conjugate of the number is:
z = x − iy
Complex Numbers
Given a complex numbers, z=x+iy:
• The negation of the number is:
− z = − x − iy
• The conjugate of the number is:
z = x − iy
• The square-norm of the number is:
2
z = z ⋅ z = x2 + y2
Complex Numbers
Given a complex numbers, z=x+iy:
• The negation of the number is:
− z = − x − iy
• The conjugate of the number is:
z = x − iy
• The square-norm of the number is:
2
z = z ⋅ z = x2 + y2
• The reciprocal of the number is:
x
1 1z
y
=
= 2 −i 2
z zz
z
z
Complex Numbers
Often, we think of the complex numbers as
living in the (real) 2D plane.
Im
Im(z)
z=x +iy
Re(z)
Re
Complex Numbers
Often, we think of the complex numbers as
living in the (real) 2D plane.
Then if c is a complex number, the function:
f ( z) = z + c
is a translation in the complex plane.
Im
Im
c
c
Re
Re
Complex Exponentials
Given θ∈R, the value of the complex
exponential eiθ is:
iθ
e = cos θ + i sin θ
Complex Exponentials
Given θ∈R, the value of the complex
exponential eiθ is:
iθ
e = cos θ + i sin θ
And any complex number, z=x+iy, can be
expressed in terms of its radius and angle:
iθ
z = re
where r=|z|, and θ=ArcTan2(y,x).
Complex Numbers
If c=eiθ0 is a complex exponential, then the
function:
re iθ  re i (θ +θ 0 )
f ( z ) = cz
is a rotation by the angle θ0.
Im
Im
c
θ0
Re
Re
Complex Numbers
More generally, if c=r0eiθ0 is any complex
number, then the function:
i (θ +θ 0 )
iθ
(
)
re

r
⋅
r
e
f ( z ) = cz
0
is a rotation by the angle θ0 followed/preceded
by a scaling by r0.
Im
Im
c
θ
Re
Re
Complex Numbers
Finally, if we consider the reciprocal function,
then the function:
1 − iθ
1
iθ
re  e
f ( z) =
r
z
is an (orientation-preserving) inversion.
Im
Im
Re
Re
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
That is, if F(x,y)=(f1(x,y),f2(x,y)) then the derivative
of F preserves oriented angles:
 ∂f1

 ∂x
 ∂f 2
 ∂x

where R is a rotation.
∂f1 

∂y 
= λR
∂f 2 
∂y 
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
That is, if F(x,y)=(f1(x,y),f2(x,y)) then the derivative
of F preserves oriented angles:
 ∂f1

 ∂x
 ∂f 2
 ∂x

∂f1 

∂y 
= λR
∂f 2 
∂y 
where R is a rotation.
Thus, a map is conformal if it sends
infinitesimally small circles to circles.
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
Im
Im
f(z)=z2.2
Re
Im
Ω
f(z)=z1/2
Re
Re
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
It does not map circles to circles.
Im
Im
f(z)=z2.2
Re
Im
Ω
f(z)=z1/2
Re
Re
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
But it does map “tiny” circles to circles.
Im
Im
f(z)=z2.2
Re
Im
Ω
f(z)=z1/2
Re
Re
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
Note, if F:Ω→R2 and G:F(Ω)→R2 are conformal
maps, then so is G°F.
Im
f(z)=z2.2
Re
Im
Ω
f(z)=z1/2
Re
Re
Conformal Maps
Definition:
Given a domain Ω⊂R2, the map F:Ω→R2 is
conformal if it preserves oriented angles.
And, if F:Ω→R2 is conformal, then so is F-1.
Im
f(z)=z2.2
Re
Im
Ω
f(z)=z1/2
Re
Re
Conformal Maps
Definition:
In a similar manner, we say that a map between
two surfaces is conformal if it preserves angles.
[Global Conformal Surface Parameterization, Gu and Yau]
The Complex Plane and the Sphere
Stereographic Projection:
A bijective map from the unit sphere onto R2+∞:
 x
z 

π ( x, y, z ) = 
,
 y−2 y−2
2
The Complex Plane and the Sphere
Stereographic Projection:
A bijective map from the unit sphere onto R2+∞:
 x
z 

π ( x, y, z ) = 
,
 y−2 y−2
Circles/lines on the plane
get mapped to circles on
the sphere, so the map
is conformal.
Möbius Transformations
The group of invertible conformal maps from
the plane into itself is called the Möbius group.
Möbius Transformations
The group of invertible conformal maps from
the planeIminto itself is called
the Möbius group.
Im
Im
f(z)=z2.2
Re
Ω
f(z)=z1/2
Re
Q: Are these maps part of the Möbius group?
Re
Möbius Transformations
The group of invertible conformal maps from
the planeIminto itself is called
the Möbius group.
Im
Im
f(z)=z2.2
Re
Ω
f(z)=z1/2
Re
Re
Q: Are these maps part of the Möbius group?
A: No, they aren’t invertible on the whole plane.
Möbius Transformations
The group of invertible conformal maps from
theImplane into itself
is
called
the
Möbius
group.
Im
Im
Im
Re
Re
Translation
Re
Scale + Rotation
Re
Inversion
Q: Are these maps part of the Möbius group?
Möbius Transformations
The group of invertible conformal maps from
theImplane into itself
is
called
the
Möbius
group.
Im
Im
Im
Re
Re
Translation
Re
Scale + Rotation
Re
Inversion
Q: Are these maps part of the Möbius group?
A: Yes. In fact, the composition of these maps
comprises the entire group.
Möbius Transformations
If we think of the plane as the set of complex
numbers, any Möbius transformation can be
expressed as a fractional linear transformation:
az + b
f ( z) =
cz + d
with ad-bc≠0.
Möbius Transformations
az + b
f ( z) =
cz + d
If c=0, then:
az + b  a  b
f ( z) =
=  z +
d
d  d
Scale+Rotation
Translation
Möbius Transformations
az + b
f ( z) =
cz + d
If c=0, then:
az + b  a  b
f ( z) =
=  z +
d
d  d
If a=0, then:
Scale+Rotation


b
1

f ( z) =
= b
cz + d
 ((cz ) + d ) 
Scale+Rotation
Inversion
Translation
Möbius Transformations
az + b
f ( z) =
cz + d
Otherwise, if a,c≠0: c
(az + b )
az + b
= a
cz + d c (cz + d )
a
cb 

cb
 cz + 
(cz + d ) + − d
a
a a
a
=
=
(cz + d )
c
c (cz + d )
cb
−d

a   cb
a a
1


= +
= +   − d 
c cz + d c   a
 ((cz ) + d ) 
f ( z) =
Möbius Transformations
az + b
f ( z) =
cz + d
Otherwise, if a,c≠0: c
(az + b )
az + b
= a
cz + d c (cz + d )
a
cb 

cb
 cz + 
(cz + d ) + − d
a
a a
a
=
=
(cz + d )
c
c (cz + d )
cb
−d

a   cb
a a
1


= +
= +   − d 
c cz + d c   a
 ((cz ) + d ) 
f ( z) =
Möbius Transformations
az + b
f ( z) =
cz + d
Otherwise, if a,c≠0: c
(az + b )
az + b
= a
cz + d c (cz + d )
a
cb 

cb
 cz + 
(cz + d ) + − d
a
a a
a
=
=
(cz + d )
c
c (cz + d )
cb
−d

a   cb
a a
1


= +
= +   − d 
c cz + d c   a
 ((cz ) + d ) 
f ( z) =
Möbius Transformations
az + b
f ( z) =
cz + d
Otherwise, if a,c≠0: c
(az + b )
az + b
= a
cz + d c (cz + d )
a
cb 

cb
 cz + 
(cz + d ) + − d
a
a a
a
=
=
(cz + d )
c
c (cz + d )
cb
−d

a   cb
a a
1


= +
= +   − d 
c cz + d c   a
 ((cz ) + d ) 
f ( z) =
Möbius Transformations
az + b
f ( z) =
cz + d
Otherwise, if a,c≠0:
Translation
Scale+Rotation

az + b a   cb
1


= +   − d 
f ( z) =
cz + d c   a
 ((cz ) + d ) 
Inversion
Scale+Rotation Translation
Möbius Transformations
az + b
f ( z) =
cz + d
Note:
1. Scaling the four coefficients does not change
the transformation:
az + b f ⋅ az + f ⋅ b
=
cz + d f ⋅ cz + f ⋅ d
Möbius Transformations
az + b
f ( z) =
cz + d
Note:
1. Scaling the four coefficients does not change
the transformation:
az + b f ⋅ az + f ⋅ b
=
cz + d f ⋅ cz + f ⋅ d
2. If we represent the map by the 2x2 matrix:
a b 


c d 
then composing maps is the same as
multiplying matrices.
Möbius Transformations
az + b
f ( z) =
cz + d
Note:
As a result, it is not uncommon to talk about the
Möbius group as the group of 2x2 matrices, with
complex entries and determinant 1.
Möbius Transformations
If we have conformal maps F,G:S→S2 from a
surface S onto the sphere, then the composition
F°G-1:S2→S2 must be conformal as well.
F
G
Möbius Transformations
If we have conformal maps F,G:S→S2 from a
surface S onto the sphere, then the composition
F°G-1:S2→S2 must be conformal as well.
Since the only conformal maps from the
sphere/plane into itself are the Möbius
transformations, this means that, up to a
Möbius transformation, the map F:S→S2 is
unique.