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angle generation theory

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  • angle generation theory

    Ok, admittedly a bad title. But nothing better came to mind.

    In recent weeks and months we've had discussions about how to generate reference flat surfaces from scratch and how to create right angle blocks.

    So if one had some time on their hands and a few basic tools, rouge or whatever marking agent, scraper, etc one could create his own flat and square references.

    What I'd like to discuss next is how one using only basic tools again (no rotary table, etc) could markout all the markings on a round, a line every 1 degree, 5 degrees, whatever and do it to the same kind of accuracy that could be achieved with the above methods. Meaning good enough to use as a reference in the homeshop atleast.

    I've been giving this some thought over the last week or so and don't really have it solved. I have an idea that would work within reason but I'd really like to hear other peoples ideas on the subject.

  • #2
    For your proposed problem for dividing a circle you could use basic geometrical equipment.
    Use a dividers to scribe a small accurate circle, use a wire to mark a relatively large circle outside the original.
    Use the dividers to subdivide the larger circle to degrees using trial & error, transfer back to the smaller circle with the wire. You could quickly construct 180, 90, 45, 30 & 15 degree angles then subdivide if you felt it appropriate to ease or speed the process
    Marking out on a large figure or long line & transferring back to a smaller one is a standard geometrical technique for subdivision,


    • #3
      I don't think you are going to find a solution. It is mathematically impossible to trisect an angle with basic tools like compass and ruler. Since you need to trisect an angle to produce the angles that are factors of three (30آ°, 60آ°) then it can't be done.
      Free software for calculating bolt circles and similar: Click Here


      • #4
        Attached is the text file included with my DPLATE program. It describes a very
        old technique for generating any number of circular divisions with little more
        equipment than a lathe and a micrometer. Lautard also describes this
        technique in one of his "Readers".

        ================================================== =============================

        I want an accurate division plate with 14 divisions. Not a problem - just whip
        out the dividing head and .... But wait! The dividing head doesn't have a 14
        hole plate (we'll pretend). What to do? I need a 14 hole plate to make a 14
        hole plate. The proverbial chicken-egg problem.

        Well, they didn't have a dividing head when they made the first plate. There
        must be some way to do it that doesn't require a dividing head. It's either
        that or I have to admit to myself that the dividing head sitting in front of me
        doesn't really exist.

        The technique described here requires only a lathe and a bit of mathematics.
        I can't provide an historical reference that proves this was how the first
        accurate plate was made but, nevertheless, it's a fascinating technique and may
        serve you in good stead some day.

        Imagine you've turned yourself a top-hat shaped piece of steel. Now imagine
        you've turned 14 circular disks. When you paste these disks on the brim of the
        hat, touching (what I'll call) the 'crown' of the hat, they all just fit,
        simultaneously touching the adjacent disks and the crown. Voila, a '14 hole'
        dividing plate. All that's required is to make a suitable detent to locate
        between adjacent disks and you've got a dividing plate. With such a
        contrivance it would be easy to use it as a locator to drill a more
        conventional dividing head plate.

        If the diameter of the crown of the hat is known, it's only a bit of
        elementary trigonometry to compute the required disk diameter for any number
        of divisions. The attached .jpg file illustrates the arrangement and the math
        that must be done. While straightforward, a small program makes things
        easier. After all, the program will never transpose a digit, forget a term,
        take a cosine instead of a sine or make any of those other idiotic mistakes
        we're all so prone to make. Moreover, you'll probably want to experiment a
        bit to get a combination of crown diameter and disk diameter that fits your
        available stock. That means solving the same equation several times. So much
        easier with a program than a calculator.
        Regards, Marv

        Home Shop Freeware - Tools for People Who Build Things

        Location: LA, CA, USA


        • #5

          Very good Marv, some people on this board act like that things are "Mathematically Impossible" without a computer. I for one was doing trig and manual layout work long before software. When it comes to computer accuracy; that is only good as the Machinist setting up and controlling each facet of installing the tooling. I hate CAM for that reason. Nowadays, if you can push a button,Bang! you're a machinist! Maybe Evan built the very first rotary table....but I doubt it.


          • #6
            It's trivial to bisect an angle with simple tools and then to bisect that further. Using this technique we can lay out gears. Using those gears and turning them in multiples of three turns we can mechanically trisect any angle we want.
            Free software for calculating bolt circles and similar: Click Here


            • #7
              So you say it's impossible to get accuracy without comp. help?


              • #8
                I say it is impossible to trisect an angle by using simple tools such a pecil, ruler and compass. That is mathematically proven. You can bisect an angle easily but all such further bisections simply produce numbers that are factors of two. There is no method to trisect an angle and that has been proven rigourously. Keep in mind that we are talking about mathematically perfect methods, not approximate mechanical methods.

                A simple example is to try and calculate the exact value of 1/3. It cannot be represented exactly.

                Free software for calculating bolt circles and similar: Click Here


                • #9
                  Why don't you come over for dinner sometime and I will show you how to do these impossible things you say can't be done. You do a lot of typing and writing, you are better at those than I; but I can show you in person in 15 min. how to achieve these things. Trig works for everybody.


                  • #10
                    Tell you what Millman. You find a way to trisect an angle with a ruler, compass and pencil and you will be famous. You might even get rich too on the talk show circuit. Let me know when you have it figured out.
                    Free software for calculating bolt circles and similar: Click Here


                    • #11
                      Originally posted by dberndt
             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.
                      The entire content of this post is copyright by, and is the sole property of, the author. No assignment
                      of title nor right of publication shall ensue from presentation of this material on any computer site.


                      • #12
                        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.



                        • #13
                          Originally posted by Evan
                          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?

                          The curse of having precise measuring tools is being able to actually see how imperfect everything is.


                          • #14
                            angle layout

                            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


                            • #15
                              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.
                              Jim H.