# formula for logarithmic spiral on a linear level

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.

• In your first paragraph, are you talking about an operation like fixing a point on the circle but increasing the radius until it looks very line-like, and "continuing that process" until it has "center at infinity"? Feb 3, 2015 at 5:44
• Incomprehensible. What is a "linear graph"? What are the "beginning" and the "end" of a circle? What do you mean by making a circle straight? Feb 3, 2015 at 6:26
• Take the beginning of the circle at 0,360 and the end at 360,360, disconnect the two ends and cut a line down to the center of the circle and "stretch" it out, lay it straight (probably physically impossible). Make a square with x axis 0-360 and y as 0-360. Then take a polar coordinate from tje circle amd lay it on the square. For example, the polar coordonates of a side of a triangle that lies within the circle is 0,360, 60,180, 120,360. Plot these in the square, etc. I can do the elements, the fib spiral is difficult Feb 3, 2015 at 13:07
• Still incomprehensible. What do you mean by 0,360? Is that the point in the $xy$-plane with $x$-coordinate $0$ and $y$-coordinate $360$? If so, how do you disconnect the "two ends" $(0,360)$ and $(360,360)$ --- they're already disconnected, they are quite far apart. Or maybe those are polar coordinates, the "beginning" at $r=0$, $\theta=360$ (degrees, presumably), the "end" far away at $r=360$, $\theta=360$? Utterly incomprehensible. Please talk your problem over with someone who can help you put it into understandable mathematical terms. Feb 3, 2015 at 23:18
• @Gerry, why are these comments so dismissive? I could have said what you were saying in a friendlier tone, and (judging by your history) I'm guessing you could have as well. Feb 4, 2015 at 1:56

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.

• Yesssss!!!! Thanku so much. As i said it has been several years since my last calc class and couldnt describe what i was trying to say in sufficient mathematical terms, but u seem to have understood and gave me exactly what i needed. Thankyou i get it now. I need a limit because i am studying sacred geometry in cycles and i want my cycles limited to360, as a circle can really extend outward forever. Everything has a cycle lile a clock, but time is linear, so that is what i am plotting. Geometrical Cycles according to the motion of time. Thankyou for the help Feb 4, 2015 at 1:44
• @ksmith: I'm still not sure I understand. I think I get the analogy you are drawing between cycles and time, but the thing is that the Fibonacci spiral is not cyclic: as you progress forward in time and make a full clock-rotation, you do not end up back at the same place you started. So if you want to capture the "extending outward forever" feature that circles have, then why don't you let your time go on forever? Feb 4, 2015 at 2:07
• Besides, what about things like $r=\sin(\frac14\theta)$? When you plot it only from 0 to 360, you get a spiral-like shape, but this apparent shape is a lie: its true nature is revealed only by a plot through four full rotations. Feb 4, 2015 at 2:08

@Eric - Wouldnt let me comment. Well its like a clock represents time, its appears like a circle and the clock once it completes its rotation it ends in the same place, but the time it counts continues to move forward, it does not end in the same place yet we still use a circularly repeating clock to tell us where time is. I know fib is not cyclical with an ending just like time. I want a limit because everything moves in geometrical patterns. If i overlay the chart i am making for lets say a forex chart (which can be used to measure many things), whether it be the circular plot or the square plot containing the circular elements, everything falls right into place according to the geometries. It really can be applied to anything. Back to the limit, because it is all geometrically connected, it does not matter where i lay it, it is all the same. So i need only a small portion of "infinity" to overlay, if that makes sense.the planetary motions as well as many other things. Im still studying it out in many aspects of our existence, but so far everything is lining up