Author: M Dabbs

Full Mobius Transformation (z on a circle)

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z

a

b

d

c

az

az + b

cz

cz + d

z' = (az + b)/(cz + d)

Constructions

z' = \frac {az + b} {cz + d}\

z

A_1 = Point(yAxis)
B_1 = Intersect(xAxis, yAxis)
C_1 = (4.616771072584529, -1.022543872402018)
D_1 = (7.699250847748288, -1.6715181273943016)
c = Circle(C_1, D_1)
E = Point(c)
a = Segment(B_1, E)
A = (-1.3479210556224703, -0.8582512664062997)
B = (-1.0063351301485721, -4.9883356380452515)
D = (-5.974857682496183, 4.638176807128245)
C = (0.9810738907904724, 0.3838793716806031)
b, F = ComplexMultiplication(B_1, A_1, A, E)
loc1 = Locus(F, E)
G, d = ComplexAddition(B_1, F, B)
loc2 = Locus(G, E)
e, H = ComplexMultiplication(B_1, A_1, C, E)
I, f = ComplexAddition(B_1, H, D)
loc3 = Locus(H, E)
loc4 = Locus(I, E)
g, J = ComplexDivision(B_1, A_1, G, I)
loc5 = Locus(J, E)
h = false
K, L = Intersect(c, a)
xA: x(A)
yA: y(A)
xB: x(B)
yB: y(B)
xC: x(C)
yC: y(C)
xD: x(D)
yD: y(D)
compA = If(yA < 0, xA - abs(yA) * ί, xA + abs(yA) * ί)
compB = If(yB < 0, xB - abs(yB) * ί, xB + abs(yB) * ί)
compC = If(yC < 0, xC - abs(yC) * ί, xC + abs(yC) * ί)
compD = If(yD < 0, xD - abs(yD) * ί, xD + abs(yD) * ί)
text3: "z'=\frac{(" + (LaTeX[compA]) + ")z+(" + (LaTeX[compB]) + ")}{(" + (LaTeX[compC]) + ")z+(" + (LaTeX[compD]) + ")}"

Tool: Complex Multiplication (command ComplexMultiplication)
Select Points at O(0, 0), P(0, -1), A and B for A*B
Input: O, P, A, B
Output: j, AB

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
b = Segment(O, A)
f = Segment(O, B)
G = Intersect(d, xAxis, 1)
g = Circle(D, P, G)
E = Intersect(g, yAxis, 2)
e = Circle(O, E)
a, h = AngularBisector(f, b)
D' = Mirror(D, a)
i = Ray(O, D')
AB = Intersect(e, i, 1)
j = Segment(O, AB)

Tool: Complex Division (command ComplexDivision)
Select Points O(0, 0), P(0, -1), A and B for A/B
Input: O, P, A, B
Output: l, C

A/B

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
e = AngularBisector(A, O, D)
B' = Mirror(B, e)
f = Ray(O, B')
E = Intersect(c, yAxis, 1)
F = Intersect(d, xAxis, 1)
g = Segment(F, E)
G = Midpoint(g)
h = OrthogonalLine(G, g)
i = Segment(P, F)
H = Midpoint(i)
j = OrthogonalLine(H, i)
I = Intersect(h, j)
k = Circle(I, F)
J = Intersect(k, xAxis, 2)
p = Circle(O, J)
C = Intersect(p, f)
l = Segment(O, C)

Tool: Complex Addition (command ComplexAddition)
Select Points O(0, 0), A and B for A + B
Input: O, A, B
Output: C', c

A + B

u = Vector(O, B)
C' = Translate(A, u)
c = Segment(O, C')

Tool: Complex Subtraction (command ComplexSubtraction)
Select Points O(0, 0), A and B for A - B
Input: O, A, B
Output: c, A'

A - B

u = Vector(B, O)
A'_1 = Translate(A, u)
A' = Translate(A, u)
c = Segment(O, A')