Geometry of Möbius Transformations
Color Illustrations

Vladimir V. Kisil

Contents

Chapter 1  Erlangen Programme: Preview


Figure 1.1: Actions of the subgroups A and N by Möbius transformations. Transverse thin lines are images of the vertical axis, grey arrows show the derived action.


Figure 1.2: Action of the subgroup K. The corresponding orbits are circles, parabolas and hyperbolas shown by thick lines. Transverse thin lines are images of the vertical axis, grey arrows show the derived action.


  (a)(b)
Figure 1.3: K-orbits as conic sections. Circles are sections by the plane EE′, parabolas are sections by PP′ and hyperbolas are sections by HH′. Points on the same generator of the cone correspond to the same value of φ.


Figure 1.4: Decomposition of an arbitrary Möbius transformation g into a product.


(a)(b)
Figure 1.5: (a) Different EPH implementations of the same cycles defined by quadruples of numbers. (b) Centres and foci of two parabolas with the same focal length.


Figure 1.6: Different σ-implementations of the same σ-zero-radius cycles. The corresponding foci belong to the real axis.



(a)(b)(c)
Figure 1.7: Orthogonality of the first kind in the elliptic point space. Each picture presents two groups (green and blue) of cycles which are orthogonal to the red cycle Cσs. Point b belongs to Cσs and the family of blue cycles passing through b is orthogonal to Cσs. They all also intersect at the point d which is the inverse of b in Cσs. Any orthogonality is reduced to the usual orthogonality with a new (“ghost”) cycle (shown by the dashed line), which may or may not coincide with Cσs. For any point a on the “ghost” cycle, the orthogonality is reduced to the local notion in terms of tangent lines at the intersection point. Consequently, such a point a is always the inverse of itself.


Figure 1.8: Focal orthogonality for circles. To highlight both similarities and distinctions with ordinary orthogonality, we use the same notations as in Fig. 1.7.


(a) (b)
Figure 1.9: (a) The square of the parabolic diameter is the square of the distance between roots if they are real (z1 and z2). Otherwise, the negative square of the distance between the adjoint roots (z3 and z4). (b) Distance is the extremum of diameters in elliptic (z1 and z2) and parabolic (z3 and z4) cases.


Figure 1.10: The perpendicular as the shortest route to a line.


     (a)(b)
Figure 1.11: K-orbits as conic sections, separate images of sections. Circles are sections by the plane EE′, parabolas are sections by PP′ and hyperbolas are sections by HH′. Points on the same generator of the cone correspond to the same value of φ.

Chapter 2  Groups and Homogeneous Spaces

Chapter 3  Homogeneous Spaces from the Group SL2(ℝ)


Figure 3.1: Actions of the subgroups K, N′ and A′, which fix point ι in three EPH cases.

Chapter 4  The Extended Fillmore–Springer–Cnops Construction

Chapter 5  Indefinite Product Space of Cycles


(e)     
(p)     
(h)     
Figure 5.1: Linear spans of cycle pairs in EPH cases.
The initial pairs of cycles are drawn in bold (green and blue). The cycles which are between the generators are drawn in the transitional green-blue colours. The red components are used for the outer cycles. The left column shows the appearance of the pencil if the generators intersect and the right if they are disjoint.


Figure 5.2: Positive and negative cycles. Evaluation of determinants with elliptic value σ=−1 shown by dotted drawing, with the hyperbolic σ=1 by dashed and with intermediate parabolic σ=0 by dash-dotted. Blue cycles are positive for respective σ and green cycles are negative. Cycles positive for one value of σ can be negative for another. Compare this figure with zero-radius cycles in Fig. 1.6.


    
Figure 5.3: Pencils of parabolas and hyperbolas which have centres on different level.


    
Figure 5.4: Pencils spanned by circles, where one or two are imaginary—not different from a case of real circles.


        
Figure 5.5: Pencils spanned by touching circles and hyperbolas.


Figure 5.6: Pencil spanned by a circle and a line—not different from a case of two circles.

Chapter 6  Joint Invariants of Cycles: Orthogonality


(a)    (b)
Figure 6.1: Relation between centres and radii of orthogonal circles


Figure 6.2: σ-orthogonal pencils of σ-cycles. One pencil is drawn in green, the others in blue and dashed styles.




Figure 6.3: Three types of orthogonality in the three types of the point space.
Each picture presents two groups (green and blue) of cycles which are orthogonal to the red cycle Cσs. Point b belongs to Cσs and the family of blue cycles passing through b also intersects at the point d, which is the inverse of b in Cσs. Any orthogonality is reduced to the usual orthogonality with a new (“ghost”) cycle (shown by the dashed line), which may or may not coincide with Cσs. For any point a on the “ghost” cycle the orthogonality is reduced to the semi-local notion in the terms of tangent lines at the intersection point. Consequently, such a point a is always the inverse of itself.


(a)    (b)
(c)    (d)
Figure 6.4: Three types of inversions of the rectangular grid. The initial rectangular grid (a) is inverted elliptically in the unit circle (shown in red) in (b), parabolically in (c) and hyperbolically in (d). The blue cycle (collapsed to a point at the origin in (b)) represent the image of the cycle at infinity under inversion.




Figure 6.5: Focal orthogonality of cycles. In order to highlight both the similarities and distinctions with the ordinary orthogonality, we use the same notations as in Fig. 6.3.

Chapter 7  Metric Invariants in Upper Half-Planes


  (a)(b)  
Figure 7.1: (a) Zero-radius cycles in elliptic (black point) and hyperbolic (the red light cone) cases. The infinitesimal-radius parabolic cycle is the blue vertical ray starting at the focus.
(b) Elliptic-parabolic-hyperbolic phase transition between fixed points of a subgroup. Two fixed points of an elliptic subgroup collide to a parabolic double point on the boundary and then decouple into two hyperbolic fixed points on the unit disk.

Chapter 8  Global Geometry of Upper Half-Planes


(a) (b)
(c)
Figure 8.1: Compactification of ℝσ and stereographic projections in (a) elliptic (b) parabolic and (c) hyperbolic point spaces. The stereographic projection from the point S defines the one-to-one map PQ between points of the plane (point space) and the model—surfaces of constant curvature. The red point and lines correspond to the light cone at infinity—the ideal elements of the model.


[t].3@percent

t=0
-2cm→ [t].3@percent

t=0.25
-2cm→ [t].3@percent

t=0.5

[t].3@percent

t=1
-2cm→ [t].3@percent

t=2
-2cm→ [t].3@percent

t=4
Figure 8.2: Six frames from a continuous transformation from the future to the past parts of the light cone. Animations as GIF and PDF (requires Acroreader) are provided on DVD.


(a) (b)
Figure 8.3: Hyperbolic objects in the double cover of ℝh. If we cross the light cone at infinity from one sheet, then we will appear on the other. The shaded region is the two-fold cover of the upper half-plane on (a) and the unit disk on (b). These regions are Möbius-invariant.


Figure 8.4: Double cover of the hyperbolic space, cf. Fig. 8.1(c). The second hyperboloid is shown as a blue skeleton. It is attached to the first one along the light cone at infinity, which is represented by two red lines. A crossing of the light cone implies a transition from one hyperboloid to another.


System Transfer matrices
-7emPropagation in a homogeneous and isotropic medium with refractive index n:

(

      y2
V2

) = (

      1 t/n
01

) (

      y1
V1

)

-5.5em A circular boundary between two regions with refractive indices n1 and n2:

(

      y2
V2

) = (

      10
n1n2
r
1

) (

      y1
V1

)

-6.5em A ray emitted from the focal plane. The output direction v2 depends only on y1:

(

      y2
V2

) = (

      ab
c0

) (

      y1
V1

)

Figure 8.5: Some elementary optical systems and the transfer matrices

Chapter 9  Conformal Unit Disk


Figure 9.1: Action of the isotropy subgroups of ι under the Cayley transform—subgroup K in the elliptic case, N′ in the parabolic and A′ in the hyperbolic. Orbits of K and A′ are concentric while orbits of N′ are confocal. We also provide orbits of N which are concentric in the parabolic case. The action of K, N′ and A′ on the upper half-plane are presented in Fig. 3.1.


Figure 9.2: Cayley transforms in elliptic (σ=−1), parabolic (σ=0) and hyperbolic (σ=1) spaces. In picture, the reflection of the real line in the green cycles (drawn continuously or dotted) is the blue “unit cycle”. Reflections in the solidly-drawn cycles send the upper half-plane to the unit disk and reflections in the dashed cycles sends it to its complement. Three Cayley transforms in the parabolic space (σ=0) are themselves elliptic (σ=−1), parabolic (σ=0) and hyperbolic (σ=1), giving a gradual transition between proper elliptic and hyperbolic cases.


  



Figure 9.3: The unit disks and orbits of subgroups A, N and K.
(E): The elliptic unit disk.
(Pe), (Pp), (Ph): The elliptic, parabolic and hyperbolic flavours of the parabolic unit disk.
    (H): The hyperbolic unit disk.

Chapter 10  Invariant Metric and Geodesics


(a)         (b)
Figure 10.1: (a) The geodesic between w1 and w2 in the Lobachevsky half-plane is the circle orthogonal to the real line. The invariant metric is expressed through the cross-ratios [z1,w1,w2,z2]=[w1,w1,w2,w2];
(b) Extrema of curves’ lengths in the parabolic point space. The length of the blue (going up) curve can be arbitrarily close to 0 and can be arbitrarily large for the red (going down) one.


Figure 10.2: Showing the region where the triangular inequality fails (filled in red).




Figure 10.3: Showing geodesics (blue) and equidistant orbits (green) in EPH geometries. Above are written (k,[l,n],m) in ku2−2lu−2nv+m=0, giving the equation of geodesics.

Chapter 11  Unitary Rotations


        
Figure 11.1: Rotations of algebraic wheels, i.e. the multiplication by eι t: elliptic (E), trivial parabolic (P0) and hyperbolic (H). All blue orbits are defined by the identity x2−ι2y2=r2. Green “spokes” (straight lines from the origin to a point on the orbit) are “rotated” from the real axis.


Figure 11.2: Three type of “wheels”. In each case “rims” are points equidistant from the origin, “spokes” are geodesics between the origin and an points on the rims.


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