Author: M Dabbs

Complex Parabola using complex multiplication with a condition

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Select Point C, then press SHIFT and the + or - keys to animate

Drag Line

C - Drag Point

Locus

A_1 = Intersect(xAxis, yAxis)
B_1 = Point(yAxis)
A_2 = (-21.297806103868535, 1.1366197465109178)
B_2 = (-20.29780610386855, 1.1366197465109178)
Drag = Line(A_2, B_2)
C = Point(Drag)
b = OrthogonalLine(C, Drag)
D = Point(b)
c = Circle(C, D)
B = Point(c)
A = Point(c)
d = Segment(A_1, A)
e = Segment(A_1, B)
f, AB = ComplexMultiplication(A_1, B_1, A, B)
loc1 = Locus(AB, C)
g = false

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')