Given:
y^2 = x^2((2 + x)/(2 - x))y2=x2(2+x2−x)
Multiply both sides by 2 - x2−x:
2y^2 - xy^2 = 2x^2 + x^32y2−xy2=2x2+x3
add xy^2 - 2x^2xy2−2x2 to both sides and flip:
x^3 + xy^2 = 2y^2 - 2x^2x3+xy2=2y2−2x2
Now substitute rcos(thetarcos(θ for x and rsin(theta)rsin(θ) for y:
r^3cos^3(theta) + r^3cos(theta)sin^2(theta) = 2r^2sin^2(theta) - 2r^2cos^2(theta)r3cos3(θ)+r3cos(θ)sin2(θ)=2r2sin2(θ)−2r2cos2(θ)
Divide both sides by r^2r2:
r(cos^3(theta) + cos(theta)sin^2(theta)) = 2(sin^2(theta) - cos^2(theta))r(cos3(θ)+cos(θ)sin2(θ))=2(sin2(θ)−cos2(θ))
Divide both sides by the coefficient of r:
r = (2(sin^2(theta) - cos^2(theta)))/(cos^3(theta) + cos(theta)sin^2(theta))r=2(sin2(θ)−cos2(θ))cos3(θ)+cos(θ)sin2(θ)
