How does one find the angle of intersection between two given polar curves?

For example, between $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$

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    $\begingroup$ I don't understand the formula of the polar curves you gave above !! usually when having two polars curves $r_1=r_1(\theta)$ and $r_2=r_2(\theta) $, then to find the nalgle of intersection you may take $r_1=r_2$ and try to find the value of $\theta$. $\endgroup$ – Nizar Jan 15 '16 at 9:55

For a curve given with $y(x)$ in Cartesian coordinates, $\frac{dy}{dx}$ is a slope of the curve with respect to the $y=\mathrm{const.}$ line (a tangent of the angle between the curve and the 'horizontal' line).

In polar coordinates, $\frac 1r\frac{dr}{d\theta}$ is a slope of the curve given with $r(\theta)$ with respect to the $r=\mathrm{const.}$ circle.

So you need to

  • convert equations to the $r=f(\theta)$ form,
  • solve a system of equations to find an intersection point (or points),
  • then calculate $\frac 1r\frac{dr}{d\theta}$ for both functions at $\theta$ corresponding to the intersection point,
  • apply $\arctan$ to them to obtain angles
  • and finally calculate the difference between angles.
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