Using that az + b/cz + d = show that every fractional linear transformation arises as a
combination of horizontal translations z z+b, dilations z az and \"reflected inversions\" z 1/z.
Conclude that fractional linear transformations preserve angles and the cross-ratio.
Solution
Consider the following operations on z
\\(z\ ightarrow cz, z\ ightarrow z+d, z\ ightarrow 1/z,z\ ightarrow (b-ad/c)z,z\ ightarrow (a/c)+z\\)
All of these operations are horizontal translations, dilations, or reflected inversions, for some
constants a,b,c and d.
If c is not 0, then apply all the above operations one after another on z:
\\(z\ ightarrow cz \ ightarrow cz+d\ ightarrow \\frac{1}{cz+d} \ ightarrow \\frac{b-ad/c}{cz+d} \
ightarrow \\frac{a}{c}+\\frac{b-ad/c}{cz+d}=\\frac{az+b}{cz+d}\\)
Suppose c is zero.
Then apply the following transforms one after another on z:
\\(z \ ightarrow \\frac{az}{d} \ ightarrow \\frac{az}{d}+\\frac{b}{d}=\\frac{az+b}{d}\\)
Once again, all of these operations are horizontal translations, dilations, or reflected inversions,
for some constants a,b and d.
Therefore the transform \\(z \ ightarrow \\frac{az+b}{cz+d}\\) can be obtained as a
combination of horizontal translations, dilations, or reflected inversions.
Since each of horizontal translations, dilations, or reflected inversions preserve angles and cross-
ratios therefore a composition of many such operations also preserve angles and cross-ratios.
Therefore, the fractional linear transform preserves angles and cross-ratios..
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using that the fact that zl - z2 is the distance between two points.pdf
1. using that the fact that |zl - z2| is the distance between two points Zl and Z2, give a geometric
argument that : |z - 4t| + |z + 4i| = 10
Solution
Using the definition of Ellipse: The locus of points for whichthe sum of the distances from each
point to two fixed points isequal.
foci (0,-4) and (0,4)
the major axis 10
