This has been done by others, and I don't even own a Tormek. However it was a good exercise to derive the formulas myself and I'd like to share the results that I find pleasing: the formula for the gap (green line), which does not use x or y, and the formula for y given x.
gap = (https://i.imgur.com/u36Hgwn.png)
y = (https://i.imgur.com/jImpsu8.png)
(https://i.imgur.com/hYPl81y.png)
Quote from: Mr.Wizard on February 16, 2020, 10:53:19 PM
This has been done by others, and I don't even own a Tormek. However it was a good exercise to derive the formulas myself and I'd like to share one of the results as I find it pleasing: the formula for the gap (green line). It does not use x or y. I can also calculate y given x but the formula is less clean.
gap = (https://i.imgur.com/u36Hgwn.png)
(https://i.imgur.com/hYPl81y.png)
So... not being a math whiz... break it down for me? (In particular the iR and the e^{i... } part. (What is i ?) ???
Maybe an actual example? a=20° d=140 on a 250mm diameter wheel. (I'm guessing t and r are each 6mm). :)
I took this as an excuse to play with complex numbers, so if you don't like math you're not going to like this. Nevertheless I think the form produced has an elegance that the equivalent sine/cosine expressions lack. i is the imaginary number sqrt(-1). e is Euler's number.
Plugging the numbers you gave:
theta = 20 ° (in radians)
d = 140mm
R = 125mm
t = 6mm
r = 6mm
I get 74.82mm for the gap.
Quote from: Mr.Wizard on February 17, 2020, 02:07:24 AM
I took this as an excuse to play with complex numbers, so if you don't like math you're not going to like this. Nevertheless I think the form produced has an elegance that the equivalent sine/cosine expressions lack. i is the imaginary number sqrt(-1). e is Euler's constant.
Plugging the numbers you gave:
theta = 20 ° (in radians)
d = 140mm
R = 125mm
t = 6mm
r = 6mm
I get 74.82mm for the gap.
You're right, I don't like it haha.
But hey, it works! :)
Mr. Wizard, thanks for posting your compact and elegant formula in complex numbers.
I checked the numerical results based on your formula and can confirm that they are correct. :)
Please, post your formulas for x and y also. It may be useful for some comparisons.
Jan
For cbwx34 and others who are not yet fans of the complex plane there is another equally simple if more verbose way to express this, and that is using vectors. We'll need to define a two argument function P that takes (magnitude, angle) and returns Cartesian components (x, y). (I avoid the standard r/theta to avoid confusion with their earlier use.)
P(m, a) = ( m cos(a), m sin(a) )
Then we can write:
(https://i.imgur.com/qDIdRrl.png)
|| (x, y) || is the norm (https://en.wikipedia.org/wiki/Norm_%28mathematics%29), i.e. distance, i.e. sqrt( |x|^2 + |y|^2 ). All angles are in radians.
What this represents is defining the contact point on the wheel as (0, R) and then walking (d-r, theta) and (r+t, theta-90°) from that point for the composite tool depth and rest offset respectively, giving us the coordinates of center of the tool rest relative to the origin at the center of the wheel.
(https://i.imgur.com/Js3llLc.png)
Jan, after a little more work I think the y formula reduced to the expression attached, which I'll edit into my original post. Please check it and report any errors.
Mr. Wizard, thanks for the y formula, it is correct and it can be simply derived from your original gap formula.
Your formulas are for me rare demonstration how complex numbers may be useful. Until now I used complex numbers only to solve quadratic equations or describing the phase offset between current and voltage.
Jan
Quote from: Jan on February 17, 2020, 06:02:58 PM
Your formulas are for me rare demonstration how complex numbers may be useful. Until now I used complex numbers only to solve quadratic equations or describing the phase offset between current and voltage.
That great because I put this together after learning more about complex numbers myself. The cyclic nature of the powers of
i that works for phase also works for angles.
(https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/560px-Euler%27s_formula.svg.png)
Quote from: Mr.Wizard on February 17, 2020, 06:30:25 PM
That great because I put this together after learning more about complex numbers myself. The cyclic nature of the powers of i that works for phase also works for angles.
(https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/560px-Euler%27s_formula.svg.png)
This helps... still above my head, but I will study and learn... hopefully. ???
(Keep in mind for me "complex numbers" are numbers with more than 2 commas). ;D
Thanks for the info... may not understand, but always find it interesting!
All,
I'm a bit confused. To me, this is simply basic trigonometry for a static system.
It doesn't seem like higher level math is really needed, so I don't understand how complex numbers fit into this. What am I missing?
Kind regards,
Rich
Quote from: RichColvin on February 17, 2020, 11:18:10 PM
All,
I'm a bit confused. To me, this is simply basic trigonometry for a static system.
It doesn't seem like higher level math is really needed, so I don't understand how complex numbers fit into this. What am I missing?
Kind regards,
Rich
I don't think you're missing anything. You're right, it may not be necessary, but to me, it's nice to see something being done one way, approached from a different way, and arriving at the same conclusion. Sorta validates the process IMO.
Honestly, I'll probably never "get it" (heck, I'd never heard of "Euler's number") ??? but seeing a different approach sometimes helps me understand the first approach better (if that makes sense).
But yeah... probably more theoretical than practical, at least for me. ;)
cbwx34, give this a shot: https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF
Rich, I was learning about complex numbers and this was a good exercise to try them out. I already knew that one could convert trig functions to Z-plane, but then the realization came that the complex plane representation was both simpler than the sine/cosine form and easier to work with.
Ah, OK, that makes sense now.
I am one to never stifle innovation & especially not creativity, but I think there needs to be a big caution here for those new to using the Tormek: Use of such math is not required to get really sharp tools.
I note this as I've heard some of my woodturning peers express reservations about Ornamental Turning (my real joyful hobby), saying that too much math is involved. That is far from the truth, but that perception stops some.
What Mr. Wizard has proposed is a very interesting and innovative approach, and as CB rightly notes helps to prove the other approaches to calculating the setup as right (or maybe wrong). We just need to be cautious not to frighten any away from using this wonderful sharpening method.
Kind regards,
Rich
Rich, as you know Ornamental turning is also called Complex turning and in my thinking the complex numbers can be used also here to simplify the description of some complicated ornamental shapes.
Jan
Jan, I'd like to see that.
Rich makes a good point. I compare the recent math oriented sharpening programs to the space program. While we may not think we need the technology for lunar or interplanetary travel to drive to work, we have all benefitted in many ways from the advanced technology. I have been an active forum member since before Dutchman started our forum "space program" rolling in 2014. I was able to comprehend enough of Dutchman's math concept to create the kenjig. Since then, this forum has been on the cutting edge with sharpness technology. I hope this will continue and want to encourage it.
I have read several posts and PMs from new or prospective owners who are concerned about finding the best computer program before they ever sharpen a knife. We sharpen on many levels and evolve. We need both a beginning point and goals.
Ken
Nevermind....
::)
Quote from: Mr.Wizard on February 18, 2020, 04:32:10 PM
Jan, I'd like to see that.
Mr. Wizard, I had in mind the popular Mandelbrot set which is a fractal formed in complex plane. It is an example of mathematical beauty arising from quite simple rules.
Jan
Jan, yes that's a classic beauty. Related to that: https://www.youtube.com/watch?v=ovJcsL7vyrk