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calculation for the hyperbolic-projective theorem

We have the the Poincare-Minkowski Map

The map m is given by
m(x,y) = (2x/(1-x²-y²),2y/(1-x²-y²), (1+x²+y²)/(1-x²-y²)).
The map m maps an h-line to the intersection of H with a plane through O.
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+α²+β², 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

We have the the Poincare-Minkowski Map The map m is given by m(x,y) = (2x/(1-x²-y²),2y/(1-x²-y²), (1+x²+y²)/(1-x²-y²)). The map m maps an h-line to the intersection of H with a plane through O. 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+α²+β², 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)/(az-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))/(az-(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/\|az-(1+γ)\|² = (\|az-(1+γ)\|²-\|z-a(1+γ²)/\|az-(1+γ)\|² = (\|a\|²\|z\|² + (1+γ)² -\|z\|² -\|a\|²(1+γ)²)/\|az-(1+γ)\|². = (((1+γ)²-\|z\|²)(1-\|a\|²)/\|az-(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)/\|az-(1+γ)\|². 1+\|Z\|² = (\|az-(1+γ)\|²+\|z-a(1+γ)\|²)/\|az-(1+γ)\|² = (\|a\|²\|z\|²+(1+γ)²+\|z\|²+\|a\|²(1+γ)² -2(1+γ)(az+az)/\|az-(1+γ)\|². = ((1+\|a\|²)(\|z\|²+1+γ)²)-2(1+γ)(az+az))/\|az-(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 - (az+az))/\|az-(1+γ)\|². Z = κ(z-a(1+γ))/(az-(1+γ)) = κ(z-a(1+γ))(az-(1+γ))/\|az-(1+γ)\|² = κ(a\|z\|²-z(1+γ)-a²z+a(1+γ)²)/\|az-(1+γ)\|² = κ(a(\|z\|²+(1+γ)²)-(1+γ)(z+a²z))/\|az-(1+γ)\|². As before, \|z\|²+(1+γ)² = 2γ(1+γ). So Z = κ(1+γ)(2γ-(z+a²z))/\|az-(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 - (az+az))/2(1+γ)(1-\|a\|²) = ((1+\|a\|²)γ -(az+az))/(1-\|a\|²). Here, 1-\|a\|² and 1+\|a\|² are real and (az+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