calculation for the hyperbolic-projective theorem

We have the the Poincare-Minkowski Map
  1. The map m is given by
    m(x,y) = (2x/(1-x2-y2),2y/(1-x2-y2), (1+x2+y2)/(1-x2-y2)).
  2. The map m maps an h-line to the intersection of H with a plane through O.
  3. The inverse is given by
    m-1(x,y,(1+x2+y2)½) = (x/(1+(1+x2+y2)½), y/(1+(1+x2+y2)½).

The elements of H(2) are best explained in terms of the complex variable
z = x+iy. The formulae above implicitly involve |z|, so it is natural to work
using complex numbers. We try to avoid using "z" for a coordinate.

A point P(α,β,γ) lies on H provided that γ2 = 1+α22, and γ > 0.
We write z = α+iβ, so γ = (1+|z|2)½. Then m-1(P) = z/(1+γ).

We also observe that for complex Z with |Z| < 1, m(Z) is the point Q.
Since Q is on H, it is of the form (a,b,c) with c = (1+a2+b2)½. This is
determined by a and b, and hence by the complex W = a+ib. The above
result shows that W = 2Z/(1-|Z|2), and c = (1+|Z|2)/(1-|Z|2).

The case hεH+(2).
Then h(z) = κ(z-a)/(a*z-1), where |κ| = 1 and |a| < 1.
Thus, hom-1(P) = Z = κ(z/(1+γ)-a)/(a*(z/(1+γ)-1)
Clearing fractions, Z = κ(z-a(1+γ))/(a*z-(1+γ)).

In preparation for the calculation of m(Z), we look at 1±|Z|2 and Z.

  • 1-|Z|2
    = 1-|z-a(1+γ)|2/|a*z-(1+γ)|2
    = (|a*z-(1+γ)|2-|z-a(1+γ2)/|a*z-(1+γ)|2
    = (|a|2|z|2 + (1+γ)2 -|z|2 -|a|2(1+γ)2)/|a*z-(1+γ)|2.
    = (((1+γ)2-|z|2)(1-|a|2)/|a*z-(1+γ)|2.
    Now, (1+γ)2 = 1+2γ+γ2 = 1 +2γ + (1+|z|2) = 2+2γ+|z|2.
    So 1-|Z|2 = 2(1+γ)(1-|a|2)/|a*z-(1+γ)|2.
  • 1+|Z|2
    = (|a*z-(1+γ)|2+|z-a(1+γ)|2)/|a*z-(1+γ)|2
    = (|a|2|z|2+(1+γ)2+|z|2+|a|2(1+γ)2 -2(1+γ)(a*z+az*)/|a*z-(1+γ)|2.
    = ((1+|a|2)(|z|2+1+γ)2)-2(1+γ)(a*z+az*))/|a*z-(1+γ)|2.
    Now, |z|2+(1+γ)2 = 2(1+|z|2+γ) = 2γ(1+γ).
    So 1+|Z|2 = 2(1+γ)((1+|a|2)γ - (a*z+az*))/|a*z-(1+γ)|2.
  • Z
    = κ(z-a(1+γ))/(a*z-(1+γ))
    = κ(z-a(1+γ))(az*-(1+γ))/|a*z-(1+γ)|2
    = κ(a|z|2-z(1+γ)-a2z*+a(1+γ)2)/|a*z-(1+γ)|2
    = κ(a(|z|2+(1+γ)2)-(1+γ)(z+a2z*))/|a*z-(1+γ)|2.
    As before, |z|2+(1+γ)2 = 2γ(1+γ).
    So Z = κ(1+γ)(2γ-(z+a2z*))/|a*z-(1+γ)|2.
Thus W
= 2Z/(1-|Z|2)
= 2κ(1+γ)(2γ-(z+a2z*))/2(1+γ)(1-|a|2)
= κ(2γ-(z+a2z*))/(1-|a|2).
Since 1-|a|2 is real, the real and imaginary parts of W
(i.e. the x- and y- coordinates of the final image) are
real linear combinations of α, β and γ.

Also, (1+|Z|2)/(1-|Z|)
= 2(1+γ)((1+|a|2)γ - (a*z+az*))/2(1+γ)(1-|a|2)
= ((1+|a|2)γ -(a*z+az*))/(1-|a|2).
Here, 1-|a|2 and 1+|a|2 are real and (a*z+az*) is a
real number (it is of the form w+w*), so this number
is also a real linear combination of α, β and γ.

Thus, overall mohom-1 maps x to Ax for a real matrix A.

The case h indirect.
But then h(z) = h'(z*), with h'εH+(2).
Replacing z by z* in the above analysis, we obtain the result
for this case as well.

z* denotes the complex
conjugate of z

γ2 = 1+|z|2

the hyperbolic-projective theorem