Sorry about the pictures being links, apparently "You need at least 10 reputation to post images". Neat, that is super helpful.
TL;DR: I have these 2 formulas that define a pair of logarithmic spirals in polar coordinates. How can I find the intersection of these 2 curves without using a graph? I've added a graph to demonstrate that they do intersect.
This is the basic form of the formula I'm using. $$r(\varphi) = ae^{k\varphi}$$
The same curve, rotated $180^\circ$ and mirrored looks like this: $$r(\varphi) = ae^{-k*(\pi + \varphi)}e^{\pi - \varphi}$$
To convert these to cartesian coordinates: $$x(\varphi) = r(\varphi)cos(\varphi), y = r(\varphi)sin(\varphi)$$ and $$x(\varphi) = r(\varphi)cos(\pi - \varphi), y = r(\varphi)sin(\pi - \varphi)$$
Demonstrates the 2 curves intersect
Background: I'm working on a spiral template for a custom desk I'm making as a personal project. I've decided [evidently] to make this as hard as possible. I'm writing a script in python to help me generate these curves - ultimately, I'll be placing circles of varying sizes at the intersections of these curves. I'm having a hard time figuring out how to find these intersections.
I've found the formula for this logarithmic curve on Wikipedia 3: $$r = ae^{k\varphi}$$
To generate the graph I screenshotted with multiple curves (should be 34 curves in total?), I had to add a way to rotate the curve around the center of the graph (assume (0,0)) as well as to flip it. After more time than I'd like to admit, I found that these formulas appear to do the trick:
Curve traveling in the counter-clockwise direction (blue lines on the graph): $$r(\varphi, \varepsilon) = ae^{k*(\varphi + \varepsilon)}e^{-k\varepsilon}$$
Curve traveling in the clockwise direction (green lines on the graph): $$r(\varphi, \varepsilon) = ae^{-k*(\pi + \varphi + \varepsilon)}e^{-k*(\pi + \varepsilon)}e^{2k*(\pi + \varphi + \varepsilon)}$$
$\varphi$ = angle from x-axis
$\varepsilon$ = angle of rotation in counter-clockwise direction around the center
a is a constant
k is a constant
I've tried a few things to see if I can figure this out myself. At this stage, I don't even think I can coherently say what I've tried.
My initial guess was that I need to work in cartesian coordinates, since I'm trying to figure out where these 2 curves intersect, but that all falls apart when I try to make that work:
$$x = r * cos(\varphi + \varepsilon)$$ $$y = r * sin(\varphi + \varepsilon)$$
$$r = \frac{x}{cos(\varphi + \varepsilon)}$$ $$r = \frac{y}{sin(\varphi + \varepsilon)}$$
$$\frac{y}{sin(\varphi + \varepsilon)} = \frac{x}{cos(\varphi + \varepsilon)}$$ $$y = x*\frac{sin(\varphi + \varepsilon)}{cos(\varphi + \varepsilon)}$$ $$y=x*tan(\varphi + \varepsilon)$$
The issue is that I feel like both x and $\varphi + \varepsilon$ are variables, so that doesn't look right.
I've started and stopped trying to find 0's by taking the derivative of the function in it's polar form, but that was ugly as well.
I'd love any help or direction I can get here. I've spent a bunch of time trying to research how to do this, but I'm just not connecting the dots somewhere. (Some might say I've been going in circles, but now I'm just spiraling...)
