Polar Coordinates – Find the Equation?

The locus of such points P is a cardioid r = a(1 + cos 2θ) /follow the links in Sources below to a Wiki article on the subject - the red one/

The derivation: see a picture I made for an explanation:

http://farm4.static.flickr.com/3561/3316657455_4d7...

Take an arbitrary point T(r, θ) on the circle with center Z(a, 0). The central angle is twice the inscribed one, hence

angle(Zx, ZT) = 2θ = angle(Ox, OP)

OZTP is a trapezoid with 2 right angles and

|OP| = |OP'| + |P'P| = a cos 2θ + a, so the polar coordinates of P are

P(a(1 + cos2θ), 2θ)

P.S. Yes, NorthStar, just take the polar angle θ' = angle(Ox, OP)

as parameter in P(a(1 + cos2θ), 2θ) above:

θ' = 2θ and it becomes P(a(1 + cosθ'), θ')

for polar to rectangular or vice versa the relation is x = rcos(theta) y = r sin( theta) and x^2+ y^2 = r^2 cos (theta) = x/r, sin (theta) = y/r we get sec ( theta ) = a million/cos (theta) = a million/ (x/r) = r /x r/x =2 ie sqrt (x^2+y^2) /x = 2 x^2+y^2 = 4x^2 y^2 = 3x^2 y = +/- sqrt(3) x