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December 13th, 2018, 05:59 AM   #1
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Spiral segments

I have a problem. Magnetic floppy disk media stores data in tracks, which are concentric circles, beginning at the inner radius and extending out to the edge of the disk. Each track is divided into 18 segments, and this is done on side A and side B. You can imagine these extending out like pizza slices, with the inner-most portion being smaller, and the outer-most portion being larger.

CDs and DVDs are more like record players, in that they start on the inner-most track and go out continually in a spiral without having concentric circles. This is more like a continuous writing medium that winds up being however many inches or meters long.

I would like to know how to calculate the start and end points of each logical segment, beginning on a spiral with radius R1=1.2 inches, continuing around 80 full turns as it extends out to a radius R2=3.3 inches, marking the point at each segment length corresponding to the inner-most track's segment length. In this way, the continuous spiral line will begin at a 1.2 inch radius, and every segment length, it will calculate it's radius and rotational theta relative to its starting point.

I'm not sure how to do it. I can calculate the inner-most track theta as 2*pi/18, and I know my starting radius of 1.2", and that it will need to be at 1.2 + ((3.3" - 1.2") / 80) at the end of the first revolution, and then repeating outward that much each time. But, the arc segment lengths will be increasing continually, and the value for theta will be decreasing continually, and there will be more and more segments the further out it goes... so that's where I get lost.

How would I solve this problem? Note: This is a thought experiment on how to take a 1.44MB 3.5" floppy disk, and extend its capacity to (I think) around 3.75MB. With many-100s of GB SD cards and USB thumb drives available today, I realize it's not a viable product. But, it's more of a "would it work?" thought. If I can get all of the math sorted out, I'd like to actually program an Arduino to manipulate a physical floppy drive and see if it would work.

Thank you in advance.
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December 17th, 2018, 08:34 AM   #2
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I thought of another way to explain this problem.

Suppose you have matchboxes, and you line them up in a circle such that you're constantly on a spiral so that when you complete the first revolution there's a little bit of space between the matchboxes on the inner spiral and the next one. You repeat this out for 80 revolutions, always putting matchbox to matchbox around the distance.

In the inner circle, your diameter is such that you need 18 matchboxes, but as you get further and further out you need more and more for each revolution.

How can I write a function to tell me at what value of theta (0 to 80*2*pi) each matchbox starts? And how can I write a function to take a given matchbox number and resolve back to theta and the radius of that matchbox?

Is that clearer?
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