How do you find the angle of intersection between two given polar curves?

Question

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

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

Answer 1

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.