Control circle to ellipse
Author: Daniela Velichová

Control circle to an ellipse is the set of all points 
symmetric to one focus of an ellipse through its tangents.

Centre of the control circle determined by 
points symmetric to focus F2 through the ellipse tangent lines 
is the control circle with centre in the other focus F1 and radius r = 2a
equal to the doubled size of the ellipse semi-major axis. 


Control circle can be generated 
by moving the tangent point T along the ellipse c.

Proof

QT = TF2

d = QF1 = QT + TF1 = TF2 + TF1 = 2a 

F1 = (-1, -1)
F2 = (0.6351752609046653, 0.656466611651154)
c = Ellipse(F1, F2, 1.7)
S = Center(c)
T = Point(c)
t, b = Tangent(T, c)
k = OrthogonalLine(F2, t)
P = Intersect(t, k)
Q = Mirror(F2, t)
a = Segment(F2, T)
d = Segment(Q, F1)
n-uholník1, m, q, p = Polygon(Q, P, T)
n-uholník2, m_1, p_1, f2 = Polygon(P, F2, T)
GMB1 = Locus(Q, T)

Tool: Nástroj1
Input: a, b, c, d
Output: f

f(x) = a x^(3.0) + b x^(2.0) + c x + d