WBahn,

Math is my downfall.

Would you share the excel file? I am pretty good with excel when it comes to financial analysis and charts, but most of the higher order math functions are over my head. I would like to see what you did.

 

Looks right. It is two halves (mirror image) of a cosine wave.
The angles at the pointed ends are both 90 degrees.

Just as you had predicted!

 

Yes, similar, but yours is better. Mine is attached to post #10.

ETA: I just looked at the scale on yours. If the scale is in inches, yours is twice as big as it should be, but the shape is correct.

Ah, you're right. I have a cell that is for the radius of the tube and I put 1.75" when I should have put half that. Of course, I could have (should have) just used 1.00" because it is trivial to scale it up and down. I was even thinking of putting multiple patterns inside each other for different ratios of the two diameters. Then you could see it go from real close to a circle to this shape. I may yet do that.

 

I had to figure all that out with pencil and paper, when I started in the 50s.

What is a computer?

 

Yes. Please do that. It would be interesting to see.

 

I had to figure all that out with pencil and paper, when I started in the 50s.

What is a computer?

I can remember giving almost a week's pay for a four-function calculator with a seven-segment LED display.

 

I had to figure all that out with pencil and paper, when I started in the 50s.

What is a computer?

Yes. Please do that. It would be interesting to see.

That's pretty much what I did just now.

First, I imagined a plane cutting down the axis of the cylinder with a circle drawn on it the same diameter as the tube. I then imagined an a radial sweeping out a quarter circle starting from going straight down the axis and ending up pointing straight out the side of the tube. I called the angle from the axis to the radial θ, which goes from 0 to ∏/2.

For a given angle of θ, the tip of the radial is x=Rsin(θ) in from the axis and y=Rcos(θ) down the tube.

Now I imagine a piece of paper wrapped around the tube with a set of axes centered above the center of by circle. If I bring a line straight up from <x,y>, it will intersect the paper at the same y value, but the value in the perpendicular direction, call it w, will not be equal to x, but to the arclength w along the top surface of the tube, which is just a circle in cross section with a vector whose angle is arcsin(x/R), which is just θ. So w=Rθ.

Thus we have a set of parametric equations for our locus of points (in one quadrant) which is <w,y> = R<θ, cos(θ)>. Putting this in terms of y(w), we simply end up with y(w) = Rcos(w/R). For R=1 (normalized distance), this becomes y=cos(w), as MrChips pointed out some time back.

To get the complete pattern, you need to cover all for quadrants, which is just

±y = ±R cos(w/R) as w/R goes from 0 to ∏/2.

or

y = ±R cos(w/R) as w/R goes from -∏/2 to +∏/2.

To plot this in Excel, set up a column that takes θ from -∏/2 to +∏/2 in whatever increments you want. In the next column, get w by multilying it by the value in the first column (i.e., θ). Then have the next column produce R*cos(θ) and have the fourth column simply be the negative of the third column. The you just plot the final three columns using the first (of the three) as the horizontal values and the final two columns as two different series. One will sweep out the top arc and the other will sweep out the bottom.

 

I got curious as to how much effort it would actually take to do it with Excel. It actually turned out to be much easier than expected. Here's what it looks like:


Of course, the pixels are not exactly square, but you get he idea.

If you plot it in a CAD program you can actually get a scaled drawing. There are other software packages that produce accurately scaled drawings - for example some of the LaTeX packages, and these have inbuilt functions like sin and sqrt as well, and are free.

 

I can remember giving almost a week's pay for a four-function calculator with a seven-segment LED display.

I started using calculators in the late 70s, sure did spoil my "Head Math".

 

I started using calculators in the late 70s, sure did spoil my "Head Math".

I had a similar experience my first year in college when I started using a calculator in virtually all courses. By the middle of the spring semester I found myself pulling out my calculator to add two simple (i.e., no carry) two-digit numbers. That was a wake-up call and I discovered that I actually had a bit of a hard time adding them in my head. From that day on I resort to a calculator only for "reasonable" problems. It's paid off nicely in keeping my mental and mechanical math skills in pretty good shape. My students are often amazed at how quickly I can do the arithmetic on the board, but unfortunately that says a lot more about their math skill than mine.

 

Never did have a need for a calculator back then and didn't own one.
I went straight from slide-rules to computers.

 

I taught school for one year straight out of college. Without a calculator, averaging grades took me more time than grading the papers.

 

I taught school for one year straight out of college. Without a calculator, averaging grades took me more time than grading the papers.

Got into computers as a career in part because of writing a short program for a professor that calculated grades at the end of the semester.