imported>Paul Murray: moved User:Pmurray bigpond.com/Complex Numbers as a 3 Vector to User:Paul Murray/Complex Numbers as a 3 Vector: name change

2006-12-24T05:46:59Z

moved User:Pmurray bigpond.com/Complex Numbers as a 3 Vector to User:Paul Murray/Complex Numbers as a 3 Vector: name change

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One useful way to express complex numbers is by [[stereographic projection|stereographically projecting]] them onto a sphere '''Γ''' , and treating it as the [[Riemann sphere]] - the extended complex plane '''C'''. Doing this produces some nice results for arithmetic and [[Möbius Transformation]]s that transparently handles the point at infinity.

== The Transformations ==

This pair of transformations is one way of performing a steriographic projection, and it will be assumes for the rest of the article:

:<math>\mathbb C = \{ z \}, \Gamma = \{ \hat z \} :</math>
:<math>\hat z = \begin{pmatrix} z_0 \\ z_1 \\ z_2 \end{pmatrix} =
\begin{pmatrix}
\frac {2 a} {1 + \bar z z} \\
\frac {2 b} {1 + \bar z z} \\
\frac {1 - \bar z z} {1 + \bar z z}
\end{pmatrix} </math>
:<math>z = a + b i = \frac {z_0 + z_1 i} {1 - z_2} </math>

Although there are many ways of projecting '''C''' onto '''Γ''', the transformation above has useful propperties. The sphere '''Γ''' is the unit sphere. The pair (z0, z1) has the same argument as z. z2 reflects the modulus of z (although not in a direct way). All |z| < 1 get mapped to z2 < 0, and all |z| > 1 get mapped to z2 > 0. This gives us a simple set of transformations:

<math>- \hat z = \begin{pmatrix} - z_0 \\ - z_1 \\ z_2 \end{pmatrix}</math>,
<math>i \hat z = \begin{pmatrix} - z_1 \\ z_0 \\ z_2 \end{pmatrix}</math>,
<math>\bar \hat z = \begin{pmatrix} z_0 \\ - z_1 \\ z_2 \end{pmatrix}</math>,
<math>{\hat z} ^{-1} = \begin{pmatrix} z_0 \\ z_1 \\ - z_2 \end{pmatrix}</math>

=== [[Circle]]s ===

A circle on '''Γ''' may be defined by the combination of a unit vector '''v''' normal to the plane of the circle, and a distance ''d'' along that vector where the center of the circle lies. The radius of the circle (in 3-space) will be √ (1-d²). Note that this is not, in general, the radus of the circle when transformed onto the complex plane, nor is the centre of the circle (where '''v''' intersects '''Γ''') the centre of the circle on the complex plane.

Using the expression for a circle on [[User:Pmurray_bigpond.com/Geometry of Complex Numbers#Circles]]:

:<math>A z \bar z + B z + \bar B \bar z + D = 0</math>, <math>\mathfrak C =
\begin{pmatrix} A & B \\ C & D \end{pmatrix} =
\bar \mathfrak C^T </math>

A circle defined by '''v''' and ''d'' on '''Γ''' will take the form:

:todo: work this out

=== [[geodesic]]s ===

A point to note is that [[geodesic]]s on the sphere are always [[great circle]]s. That is, they lie on planes passing through the center. Thus (α, β, γ) defines a set of points α z0 + β z1 + γ z2 = 0 unique up to multiplication by a constant λ which are geodesic on Γ. Projecting this back onto the complex plane:

:todo: solve the above an convert to our form for the equation of a circle

==arithmetic with these numbers==

=== multiplication by a real ===

=== addition with a real ===

=== multiplication by a complex ===

=== addition with a complex ===

==[[Möbius transformation]]s==

The whole point of this exercise is that Möbius transformations are particularly pretty on a sphere mapped in this manner.

=== geometry ===

A general möbius transformation has two fixed points γ1 and γ2 on '''Γ'''. There will be a line '''L1''' between these two points.

:<math>L1 = \lambda \hat {\gamma_1} + (1 - \lambda) \hat {\gamma_2}</math>

These two points will also each define a plane tangent to the sphere at each point. These two planes will intersect in a line '''L2'''.

:todo - derive L2

Note that if the two fixed points are diametrically opposite, then '''L2''' will be at infinity. '''L1''' and '''L2''' are always perpendicular. They will be [[skew lines]], except at the limit when γ1 and γ2 are the same point - a "parabolic" transformation.

'''L1''' and '''L2''' each define a "sheaf" of planes '''S1''' and '''S2''' which intersect the lines radially. If '''L2''' is at infinity, then this '''S2''' will be a set of parallel planes orthogonal to '''L1'''.

:todo - derive S1 and S2

Each plane in '''S1''' and a subset of the planes in '''S2''' intersect '''Γ''' in a "pencil" of circles '''P1''' and '''P2'''.

:todo - derive P1 and P2

Each circle in '''P1''' is orthogonal to each circle in '''P2'''. This, '''P1''' and '''P2''' are [[User:Pmurray_bigpond.com/Geometry of Complex Numbers|orthogonal pencils]].

''I suspect that the stuff above may be totally wrong, and that the pencils will be the intersection of a pebncil of spheres with the unit sphere.''

==== mobius ====

The [[Möbius transformation]] <math>\mathfrak H</math> can be derived from the fixed points γ1, γ2, and a characteristic constant ''k''. If <math>\mathfrak H</math> is an "elliptical" transformation, then it will transform each circle in '''P1''' onto itself and each circle in '''P2''' onto some other one. If it is "hyperbolic", the converse is true.

If <math>\mathfrak H</math> is "loxodromic", then the lokodrome traced out by a point under continuous iteration of <math>\mathfrak H</math> will join γ1 and γ2 in a spiral having some angle θ to all cicles in '''P1''' and π/2-θ to all circles in '''P2'''.

=== composition ===

User:Paul Murray/Complex Numbers as a 3 Vector - Revision history

imported>Paul Murray: moved User:Pmurray bigpond.com/Complex Numbers as a 3 Vector to User:Paul Murray/Complex Numbers as a 3 Vector: name change