Author: M Dabbs

Complex Cardioid using a construction

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z

z²

\frac{1}{z²}

Locus 1

Locus 2

P = Point(xAxis)
c = OrthogonalLine(P, xAxis)
A = Point(c)
fx: x(P)
a = 0 (slider from 0 to 10, step 0.1)
B: (x(A), y(A) + a)
cy: y(B) + a
cx: x(B) + a
C = (1, 0)
E = (0, 0)
D = (0, -1)
b = Line(A, E)
d = OrthogonalLine(E, b)
A' = Mirror(A, d)
e = Circle(A', D, A)
F = Intersect(e, yAxis, 2.0)
f = Ray(E, A)
P' = Mirror(P, f)
g = Ray(E, P')
G = Intersect(e, g)
h = Circle(E, F)
H = Intersect(h, g)
loc1 = Locus(H, A)
i = Segment(E, H)
j = Segment(E, A)
k, I = ComplexDivision(E, D, C, H)
loc2 = Locus(I, A)
l = false
m = false

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

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2.0)
b = Segment(O, A)
f = Segment(O, B)
G = Intersect(d, xAxis, 1.0)
g = Circle(D, P, G)
E = Intersect(g, yAxis, 2.0)
e = Circle(O, E)
a, h = AngularBisector(f, b)
D' = Mirror(D, a)
i = Ray(O, D')
AB = Intersect(e, i, 1.0)
j = Segment(O, AB)

Tool: Complex Multiplication (command ComplexMultiplication1)
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, 2.0)
b = Segment(O, A)
f = Segment(O, B)
G = Intersect(d, xAxis, 1.0)
g = Circle(D, P, G)
E = Intersect(g, yAxis, 2.0)
e = Circle(O, E)
a, h = AngularBisector(f, b)
D' = Mirror(D, a)
i = Ray(O, D')
AB = Intersect(e, i, 1.0)
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, 2.0)
e = AngularBisector(A, O, D)
B' = Mirror(B, e)
f = Ray(O, B')
E = Intersect(c, yAxis, 1.0)
F = Intersect(d, xAxis, 1.0)
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, 2.0)
p = Circle(O, J)
C, K = 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' = Translate(A, u)
A' = Translate(A, u)
c = Segment(O, A')