This article relates to the book "The Geometry of Complex Numbers". I'm working through some stuff in this book and in the related book "Indras Pearls", and I'll put the stuff I work out here. Perhaps eventually this article will be cleaned up, but for now it's a work in prgress.
Much of the stuff in the book is about Möbius transformations. I'll put here things not specifically related to that.
We can multiply this by a constant A to get an equation in the form
This can be usefully put into a hermitian matrix
Which specifies the circle uniquely up to multiplication by a real. The point of this is that to transform a circle by Möbius transformation , you simply do
and this nicely handles lines.
The extended complex plane is the set of all complex numbers, plus the point at infinity. A nice way to model this is to use stereographic projection to project complex numbers onto a sphere. If we do this, then circles on our complex plane wind up as circles on our sphere. Lines wind up as circles running through the projection of the point at infinity.
A circle, of course, divides the shere into two parts. Using our equation for a circle, we see that the two parts correspond to regions in which the equation results in a positive or negative number.
Möbius transformations, interestingly, can be understood as the process of projecting the plane onto a sphere, moving the sphere (rotating it, inflating or deflating it, and moving it), and projecting it back onto the plane again.
Poles of a circle
Now, this is the bit that I am currently wortking with. For any circle on the sphere, there are two diametrically opposite points lying equidistant from all points on the circle. We will call these the positive and negative poles of the circle. When projected onto the complex plane, each will be the negative congugate reciprocal of the other.
Any circle on the sphere is uniquely specified by the location of its positive pole and the distance on the sphere from that pole of the circle. This radius is (probably) most easily stated as the dot product of the vector from the center of the sphere to the pole and to each point - the cosine of the angle between the two.
Pmurray bigpond.com 00:03, 31 Mar 2005 (UTC)
Hmmm. Another way to look at the pole is to consider it to be the tip of the cone tangent to the sphere at the given circle.
pencils of circles
Given any two crcles represented by matrixes and , the matrix is also a circle. The set of all of these circles may be called a "pencil" of circles. and are the "genrators" of this pencil. Any two circles in a pencil of circles can be used as the generators of that pencil.
If the two generators are disjoint, then the pencil will be a "hyperbolic" pencil. If they intersect, then the pencil will be "elliptical". If they are tangent, then the pencil will be "parabolic".
- TODO - insert picture here
Note every hyperbolic pencil of circles is orthogonal to an elliptical pencil. Which is to say: if two pencils of circles A and B are orthogonal, then every circle in A will be orthogonal to every circle in B.
- TODO - insert picture here
Any circle in a hyperbolic pencil of circles can be moved to any other circle by a hyperbolic möbius transformation, and any circle in an elliptical pencil of circles can be moved to any other circle by a elliptical möbius transformation.
Each pencil of circles has two thingy points (which are coincident in a parabolic pencil). These thingy points are the fixed points of the möbius transformations that move the circles in the pencil onto one another. So all the möbius transformations are related to each other in that each is some power of the other.
- where is the pole of the circle ? What is its radius on the sphere?
- What's the inverse?
- Where do two circles intersect?
- Given two points, what is the great circle between them?
- this one is easy - project the points onto the sphere. The cross product is the pole of the creat circle. But first I need to solve the above.
- given a pencil of circles, what is the pencil orthogonal to it? Where are it's two thingy points?
- the orthogonal pencil will have the same thingy points.
- given two thingy points a and b, it's easy to work out the two pencils. A hyperbolic pencil is generated by the point circles at a and b. The elliptical pencil is generated by the line through a b and the circle having a diameter at a and b. Hmm. We can treat these as being the canonical representation of the pencil.
- hmm. The two thingy points alone define the pencil. But this does not work for parabolic pencils, where the thingy points are coincident. We couls use the midpoint, the distance, and the direction ... but this does not work for pencils with a thingy point at infinity. A given moebius transformation and its powers defines a pencil of loxodromes ... which are circles for pure hyperbolic and elliptical transformations. Perhaps our notation above can be extended form just circles toany sort of loxodrome.
- what's a neat way of deriving the transformation that moves a circle onto another one? We need more information, because the question is under-specified. To specify it fully, we need the circle and two points on its circumference. Or an equivalent.
- This is the same problem as above - find the thingy points. The thingy points are simply the fixed points of the family of pure hyperbolic or pure elliptical mobius transformations that generate the pencil. The the thingy points are where the radius is at a minimum. Don't hyperbolic pencils generate an imaginary elliptical set if you push λ far enough?
I am becoming intersted in spherical geometry. The stuff above is handy. Lets take our "sphere" as being steriographically projected onto a plane (using that standard projection I used elsewhere). Any circle/point has a circle on the "opposite" side of the sphere. What's the mobius transform that reflects the sphere around its central point? This will give us the opposite circle.
- Doh - neg reciprocal
I also want the sphere to roll like a trackball. This is a subset of transformations that are pure elliptical and that have fixed points at opposite points (4 degrees of freedom - same as a quaternion). Whats the form of these? What's a natural way of handling dragging? This must be an old problem: if a user puts a mouse on a shphere and daggs it around .... two ways to do it. Privelige the pont they cick, or privelige the centre point.
on second thought
I won't be displaying the sphere as a trackball - I will be displaying it as stereographically projected. So I am looking for that transform that
- is pure elliptical
- maps the mouse down position to the current mouse drag position
- maps the straight line from the original mouse down to the centre to the straight line from the mouse drag to the centre.
Not sure how to manage this when the user drags point 0,0. May have to dither it.
Note that if one chooses point "p" to be the point at infinity, then Jordan Curves A1/B1, A2/B2 etc are simply pairs of left-handed and right-handed circles (ie, having positive and negative radii). Since we already know how to derive a transformation that carries one onto the other, generating our group becomes very easy.
Or should do.
I'd like to add a section to mobius transformations, being the form of mobius transformations that do valid transforms of the POincare and UHP models of hyperbolic geometry, and the steriographical mapping model of sherical geometry.
This last one ... if i use a mapping where the unit circle gets mapped to the girdle of the sphere, and values greater then one get mapped to the top half of the sphere,
then the set of transformations is those transformations having fixed point b = - 1/fixed point b, and k real = 0.
Hmm. I'd be better off keeping the rotations as unit quaternions. When I need to compose, I convert them into mobius transformations.
To work out that transformation (quaternion to mobius xform), take a point z, stereographically project it to the unit sphere, transformit by the quaternion, stereographically project it back, then work out from that the equivalent mobius xform (by inspection). Easy as pie (not)!