Author: M Dabbs

Complex Limacon from complex multiplication with a condition

Double-click to open in a new window

AB

Locus

Select Point A, then press SHIFT and the + sign to animate

A_1 = Intersect(xAxis, yAxis)
B_1 = Point(yAxis)
C = (1.5454999259777202, -0.7202217512879947)
A_2 = (3.6138277973455457, -1.0949711759596388)
c = Circle(C, A_2)
A = Point(c)
α = 1.6929693744344996 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
B = Rotate(A, α, C)
β = Angle(A, C, B)
a = Segment(A_1, A)
b = Segment(A_1, B)
d, D = ComplexMultiplication(A_1, B_1, A, B)
loc1 = Locus(D, A)
e = false
γ = Angle(A, C, B)
f = Segment(C, B)
g = Segment(C, A)
E, F = Intersect(c, a)

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')