angle generation theory

[Leigh]

...one could create his own flat and square references.

Flat and square are two vastly different things, the latter being much more difficult than the former.

[brunneng]

This would not be fun to do for say every degree but it could be done and you only need basic tools. It's the same way as a technique for laying out an index plate.



Tightly wrap a piece of paper around the disk that is to be your reference. Mark where it overlaps.
Continue those marks up the page at right angles to your base line.
Lay a straightedge at an angle on those extensions so that the number of divisions desired can be spaced off. (ex. if you wanted 7 divisions you would place the zero on one extension and rotate around that point until the 7 was lined up on the other extension.)
Now drop a line from each division down to the base line at a right angle to the base line.
Rewrap paper around disk and go from the center to each mark.

All depends on how accurate you want to be.

Kevin

[Wirecutter]

Keep in mind that we are talking about mathematically perfect methods, not approximate mechanical methods.

Evan -
It's a little disappointing to hear you talk this way. Does this mean we won't see you hand-filing a protractor to "good enough" precision, say a couple of arc-seconds? I mean really, if it can be done with patience, rigor, and a bit of time with hand tools, I figure you'd be on it already. What gives?

-Mark

[kap pullen]

Suppose you could drill and ream four holes in an accurate square pattern
around the center.

Like a bolt circle.

You could use toolmakers buttons, the lathe back gear,dividers or etc to locate the holes.

Than you can use pins in the center hole, and outer holes, with space blocks, shims, adjustable parrells, or a height gauge, to accurately layout the lines on any angle increment you desire.

Use the pins and space blocks like a sin bar.

You could go up 45 degrees with one outer hole and than start back down with the adjacient hole.

The Imported spin index tools use a series of holes on 10 degree increments on one plate, and the other plate 11 (or 9?) degree increments to produce a mechanism that will index on one degree increments like a venier caliper.

I am guessing the hole spacings of the above but the procedure is the same.

In high school geometry we tried to prove a tri-sected angle but were told it was impossible.

That may be what Evan is saying to be impossible.

Kap pullen

[JCHannum]

To produce a degree wheel of reasonable accuracy, such as a manual rotary table or spindex is easily done with pencil and paper as brunneng describes.

This will suffice for most shop work. If more accuracy is needed, a rotary table or dividing head with a set of dividing plates will be required. These plates can be made in the shop by using a similar procedure.

Kap's suggestion of a square sine bar is another good idea for producing an accurate angle.

The Guy Lautard books have a lot of similar ideas for set up, layout and inspection using basic shop tools. He includes the math to prove their accuracy. Many of these are the methods prior generations of machinist and artisans used.

[dberndt]

if we bisect a circle with a reference straight edge we could rotate and measure off the reference edge to determine the angle moved. Using a method like this how accurate would our measuring instrument have to be to mark a 8" circle with 1degree markings that were within +/- .1 degree. By my potentially faulty calculations it seems quite easily doable given a long enough reference surface.

So I guess using a dial indicator across a few inches of travel you could mark out about 15 1 degree markings on either side of the reference before you ran out of travel. Assuming you were measuring the deviation across the entire 8" reference and using an indicator with 1" travel. A +/- .1degree deviation at this distance would be .014"

This might work well enough for an index plate or some such relatively simple thing.

So how do they do it at the factory. If you're making an ultra precise dividing plate for instance how would you generate the angles?


Nice contribution Marv. Not sure I'm sold on the practicality of actually doing it but it does seem like an idea that would work.

Kap, that seems like an answer that makes sense. Good thinking.

Any of you telescope and/or optics monkies know of a way to do it optically?

[JCHannum]

So how do they do it at the factory. If you're making an ultra precise dividing plate for instance how would you generate the angles?

They don't need to be concerned with angles, just the number and spacing of holes.

This can be accomplished by the drawing table method to produce a dividing plate. Since most indexing heads and rotary tables use a ratio of 40, 60 or 90 to 1, any error is reduced by that much. If a plate that is somewhat inaccurate is used to produce another plate the second plate will be more accurate.

[Rich Carlstedt]

I am not sure of what angles you guys want, but trisecting a part(60 deg) is simple, all it takes is a straight edge.

Take your round stock and slice 6 additional pieces.
Arrange them around the part in a circle, touching one another.
It will give you a honeycomb appearance.
Now with the straight edge, hold it tangent to two rounds with one round between, and scribe your line. by doing every other one, you get a perfect equilateral triangle, or 60 degrees.

Do it again, using the other rounds, and you get a image that looks like the star of David. Now drawing a line to the apex of the star, produces a 30 degree angle.
All without dividers, or compasses.
Rich

[Magic9r]

Evan,
how about constructing an equilateral triangle with one point at the centre of a circle & then bisecting the angle created?
That gives you 60 and 30 degrees
Do I win a prize?

There's some good basic stuff here;
http://www.neiu.edu/~ebhunt/Trig/GTM2.HTM

Also Check out
http://mathworld.wolfram.com/AngleTrisection.html

Regards,
Nick

[kf1002002]

It was proven long ago that an angle cannot be EXACTLY trisected. The construction for bisecting an angle has been well known from ancient times: I expect the surveyors who laid out the first pyramid in the Egyptian old kingdom used it and I have no doubt that much later the disciples of Pythagoras tried mightily to find a trisection algorithm and finally succeeded only in proving it impossible using the tools available to them. Various people posting to this thread have described ingenious ways to lay out ANY angle but in all cases they depend on more or less modern tools and they get only approximations; good approximations, very good approximations and good enough for all practical purposes approximations but nevertheless approximations in the mathematical sense.
Since we are on a mathematical binge here, I have a challenge for all of you who like such things: What is the sum of all the integers from 1 to 100? That is, 1+2+3.....98+99+100? You can do this in your head in a few seconds once you find out how. Gauss did it at about 6 years old when his school teacher assigned the problem to his class and my grandson at 12 years caught on soon as I showed him how. Probably some of you already know this so go ahead and post the method if you wish.

Ken