Okay, I know the title is a bit odd so I'll try to describe it in more detail here.
I would like to plot a polar graph using following conditions (let $\phi$ be some known angle $0 \le \phi \le 2\pi$ radians which we know):
- for this particular $\phi$ distance to center is known. We know $r(\phi)$.
- $r(0)=0$
- tangent angle for polar graph at angle $\phi$ is also known. We know $dx/dy (\phi)$
I would like to get polar equation that fits above criteria.
I've managed to do this for polar graph of type $r(\theta)=a(\theta)^b$. To be more precise I managed to specify $a$ and $b$ to conform to conditions specified earlier. But the problem is this formula is only good for certain conditions (if $b<0$ then for this polar graph $r(0)$ is not $0$).
I would like for this polar graph (equation) to be some kind of spiral.
Any help is much appreciated.
Thanks.
For a logarithmic spiral the angle between the spiral and the circle is constant. For $r=ae^{b\theta}$ the angle is $\frac \pi 2-\arctan \frac 1b$. You can convert your desired slope at angle $\phi$ to an angle and use that to find $b$. The below figure has the spiral $r=e^{\theta/5}$ and shows the circle at the radius corresponding to about $\frac {5\pi}4$. It also sketches the tangent line at that point. The angle between the circle and the tangent is $\frac \pi 2 - \arctan 5 \approx 12^\circ$. This would suggest the angle between the tangent and the horizontal axis is about $-57^\circ$.