I am trying to plot the contents of a circle, which include geometric elements and spirals, on a linear graph. For example, take a circle, take the beginning and the end and make it straight. What would it look like? For instance there are triangles in my circle that when plotted on a cartesian graph become something like an x^2 plot. My circle is a 360 degree circle around, but also contains radial degrees from center to edge with max 360 limit.
Here is my issue. The elements are not hard to plot, but My circle contains fibonacci spirals that i am trying to plot on a linear level as well, but cant seem to do it unless i get a ruler for every point i want to plot.
I know how to convert cartesian to polar and vice versa, i know my angles, but that is very tedious. I am looking for a formula that will give me r for every degree without using x and y. For instance, the x,y coordinate is 0,360. My spiral coordinate at that point is t,r = 0,360. Next i want the polar coordinate r for t=1, 0*=89, r=?, then t=2, 0*=88, r=? As it moves to the center of the circle around the spiral.
Again, i am not looking for x,y coordinates. I want a formula for this, is this posaible?
Most things i find are not limiting r but i am limiting r to 360.
I am studying cycles that is why r needs to be limited to 360. Like trying to fit a.circle in a square, if that makes sense
I have spent hours googling and brainstorming to no avail. Please note it has been about 7 years since my last calc class, but i do remember or catch on quickly, and learn best by an example.
Best Answer
From your description in the comments, it appears that you are taking $(r,\theta)$ coordinates and drawing them as though the $\theta$-axis is horizontal and the $r$-axis is vertical. [and for some reason you are plotting $\theta$ in degrees and you want your maximum $r$ to be 360; not sure what's up with that but it doesn't change the math.]
This is really only a change in perspective and there is no mathematical operation happening here in some sense, just exploring a biological limitation. We are creatures that are literally hard-wired to see everything in a Cartesian way. Even when you recognize a plot as polar you are seeing this from the Cartesian perspective; if you truly saw things from a polar perspective then whenever someone drew the secant curve you would react to it in the same way that I would if I saw someone draw bunch of lines, many of which overlapping.
Concretely this means that if you know $r=f(\theta)$, for instance with the Fibonacci spiral (a special case of the logarithmic spiral) you have $r=ae^{b\theta}$, then you can plot this on your square by simply forgetting your associations of "$r=$ radius" and "$\theta=$ angle", which might be easier if you replace them with $x$ and $y$. Then you will see that your square contains the Cartesian exponential curve $y=ae^{bx}$.
(Note: there is one interesting mathematical quirk which is that this change of coordinates is not a bijection: every point on the $\theta$-axis represents the same point, namely the center of the circle. If you try translating a curve that passes through the center of the circle you will see that strange things happen: in particular a continuous curve becomes apparently discontinuous. The "discontinuity" is just a biological problem: your eyes do not recognize nontrivial quotient spaces and so the points on the $\theta$-axis, although they are the same, appear to be different. But the mathematical framework you need to say that these two distinct pairs of numbers represent the same spatial point is quite nontrivial.)
FWIW: I don't see any particularly good reason to limit yourself to $0\leq\theta\leq 360$ just because you are studying circles, except that you now are quotient-ing by a more complex equivalence relation.