It's interesting to me that the second diagram is draw with the dimension shown as an arc. However, the description you translated provides that the measurement is straight?
Along the circ. is accurate, but the straight line is obviously not.
An investigation Begins
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
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The text does say, in regards to the method in Fig 18, "For greater accuracy, it is preferable to geometrically draw an angle whose value is near, and to add or subtract the difference."
I have to conclude that the traditional geometer's tools of compass and straightedge, while useful for producing many angles, is ultimately limited, even for carpentry.
Interestingly, as an aside, it turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone. So, toss out your straightedges everyone! In terms of what can be accomplished, it has been proven (by Pierre Wantzel in 1837) that an angle is constructible if and only if its cosine is a constructible number (see: https://en.wikipedia.org/wiki/Constructible_number and http://www.cut-the-knot.org/arithmetic/rational.shtml).
I think for producing any given angle with good accuracy, a marked rule or framing square and a knowledge of the tangent function on your calculator forms the more useful toolset. I'm keeping my compass and straightedge of course!
Re: An investigation Begins
Is there an echo in here?Brian wrote:It's interesting to me that the second diagram is draw with the dimension shown as an arc. However, the description you translated provides that the measurement is straight?
Along the circ. is accurate, but the straight line is obviously not.
The text does say, in regards to the method in Fig 18, "For greater accuracy, it is preferable to geometrically draw an angle whose value is near, and to add or subtract the difference."
I have to conclude that the traditional geometer's tools of compass and straightedge, while useful for producing many angles, is ultimately limited, even for carpentry.
Interestingly, as an aside, it turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone. So, toss out your straightedges everyone! In terms of what can be accomplished, it has been proven (by Pierre Wantzel in 1837) that an angle is constructible if and only if its cosine is a constructible number (see: https://en.wikipedia.org/wiki/Constructible_number and http://www.cut-the-knot.org/arithmetic/rational.shtml).
I think for producing any given angle with good accuracy, a marked rule or framing square and a knowledge of the tangent function on your calculator forms the more useful toolset. I'm keeping my compass and straightedge of course!
- Brian
- Deshi
Post
Re: An investigation Begins
Hah, I suppose I am repeating your same point.
I built a car for racing from my late teens through early 20's, a friend of a friend was so impressed by the work that he hired me outright to build cars for him. It was a 'shop' but mainly pet projects.
To build the main hoop, which is the major upright support, I used geometric formulas and a steel square. Everyone in that shop thought I was nuts for not building models, cardboard cutouts and all that other wasted effort which helps one to avoid math.
I applied my calculations to the steel tubing, then I would bend it up.
I could usually get within 1/4"~, which in that case allowed me to easily fit and tack weld.
So I'm glad to see that you find a similar approach very useful in carpentry, as everyone I worked with had a combined sense of being impressed and skeptical of my method.
I built a car for racing from my late teens through early 20's, a friend of a friend was so impressed by the work that he hired me outright to build cars for him. It was a 'shop' but mainly pet projects.
To build the main hoop, which is the major upright support, I used geometric formulas and a steel square. Everyone in that shop thought I was nuts for not building models, cardboard cutouts and all that other wasted effort which helps one to avoid math.
I applied my calculations to the steel tubing, then I would bend it up.
I could usually get within 1/4"~, which in that case allowed me to easily fit and tack weld.
So I'm glad to see that you find a similar approach very useful in carpentry, as everyone I worked with had a combined sense of being impressed and skeptical of my method.
- Jonnishida
- Lurker
- Location: Seattle, WA
Post
Re: An investigation Begins
Chris,
In regards to fig 18, you can find a "unit of arc" for any circumference by : uA=r/57.3
If you set your dividers to that distance and step off the appropriate number of steps along the circumference (eg 29) I think you can fairly accurately approximate the angle. This would be especially true for large angles rather than using the chord.
For example, if r=6"
uA=6/57.3=0.10471" which corresponds closely with the chord length for 1 degree
Chord=0.10472" with rounding.
We've just reinvented the protractor !
In the the dark times before we enslaved electrons, drafting was pretty much limited to calipers, dividers and straight edges. And look what was accomplished with just those tools!
Jon Nishida (lurking in the background)
In regards to fig 18, you can find a "unit of arc" for any circumference by : uA=r/57.3
If you set your dividers to that distance and step off the appropriate number of steps along the circumference (eg 29) I think you can fairly accurately approximate the angle. This would be especially true for large angles rather than using the chord.
For example, if r=6"
uA=6/57.3=0.10471" which corresponds closely with the chord length for 1 degree
Chord=0.10472" with rounding.
We've just reinvented the protractor !
In the the dark times before we enslaved electrons, drafting was pretty much limited to calipers, dividers and straight edges. And look what was accomplished with just those tools!
Jon Nishida (lurking in the background)
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
Post
Re: An investigation Begins
That's a good point Jonni, and one that the text makes no mention of in terms of using the method. Thanks for chiming in!
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
Post
Re: An investigation Begins
I came across an interesting method of drawing an ellipse that i thought was pretty cool and seems like it would be worth sharing here.
You decide first upon the sizes of the major and minor axes of the ellipse. First you draw a circle with a compass so as to produce a diameter equal to the minor axis:
Then you draw a circle with diameter equal to the major axis:
Then you construct a series of radial lines, at an equal spacing. The text suggests 4 divisions per quadrant (16 in total for the circle), however I put the lines in every 10˚:
Now, where the radial line crosses the smaller circle, construct a horizontal line going outwards, and where that same radial line meets the larger circle, drop a vertical line down to intersect with the horizontal, like this:
Repeat for each radial line:
And all the way around the circle similarly:
Then it is a matter of connecting the dots and you will produce the required ellipse:
All in all, a simple and elegant method I thought. Give it a try and see if it works for you.
You decide first upon the sizes of the major and minor axes of the ellipse. First you draw a circle with a compass so as to produce a diameter equal to the minor axis:
Then you draw a circle with diameter equal to the major axis:
Then you construct a series of radial lines, at an equal spacing. The text suggests 4 divisions per quadrant (16 in total for the circle), however I put the lines in every 10˚:
Now, where the radial line crosses the smaller circle, construct a horizontal line going outwards, and where that same radial line meets the larger circle, drop a vertical line down to intersect with the horizontal, like this:
Repeat for each radial line:
And all the way around the circle similarly:
Then it is a matter of connecting the dots and you will produce the required ellipse:
All in all, a simple and elegant method I thought. Give it a try and see if it works for you.
- Gadge
- Sweeper of Floors, Maker of Tea
- Location: Sydney, Australia
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
Post
Re: An investigation Begins
Glad you fellows liked that way of producing an ellipse. Seems a lot better than most of the other methods i've come across.
I backtracked slightly from ellipses and looked at how to make curved miters where piece meet with a in-curved corner:
Following on from ellipses, my study has since moved through the construction of "les anses de panier", which literally means the basket handles. These are curved vaults formed from an odd number of centers (3, 5, 7, etc.) and are entirely plotted by circular arcs. That's where I learned the French have a word for the versed sine/versine, aka sagitta, which is flèche. It's funny that we don't seem to have a word for that aspect of basic geometry in English. We have chord, but nothing to describe a perpendicular line raised from the middle of the chord to touch the circle perimeter line.
Here's one of the drawings to produce such a vault, as might be found on a bridge, or interior vaulting work in wood:
After anse de panier, it was a look at converting a drawing of a circular arc on a horizontal over to the same displacement atop a sloped, or rampant line. Here's an illustration:
There were method for working out the shape in situations where you know the width but not the height, or where you had the height and width as constraints and needed to determine which curve would work.
Then it was onto 'Le Chapeau de Gendarme', a shape which evokes the head covering of a Napoleonic-era policeman. Several methods were explored, here's one of them:
After that, the study turns to a look at spirals, first with the Archimedean spiral:
That was followed by 2-, 3-, 4-, and 6-point spirals. They were easy enough and I find this sort of stuff fun to draw. Up next is volutes, but that can wait for another day as I feel I covered some good ground today. The section ends with volutes, so not much further to go, and then it will be on descriptive geometry for various solids.
If anyone is interested in details about how to do any of the geometrical forms mentioned above, I'd be happy to post up a step-by-step.
I backtracked slightly from ellipses and looked at how to make curved miters where piece meet with a in-curved corner:
Following on from ellipses, my study has since moved through the construction of "les anses de panier", which literally means the basket handles. These are curved vaults formed from an odd number of centers (3, 5, 7, etc.) and are entirely plotted by circular arcs. That's where I learned the French have a word for the versed sine/versine, aka sagitta, which is flèche. It's funny that we don't seem to have a word for that aspect of basic geometry in English. We have chord, but nothing to describe a perpendicular line raised from the middle of the chord to touch the circle perimeter line.
Here's one of the drawings to produce such a vault, as might be found on a bridge, or interior vaulting work in wood:
After anse de panier, it was a look at converting a drawing of a circular arc on a horizontal over to the same displacement atop a sloped, or rampant line. Here's an illustration:
There were method for working out the shape in situations where you know the width but not the height, or where you had the height and width as constraints and needed to determine which curve would work.
Then it was onto 'Le Chapeau de Gendarme', a shape which evokes the head covering of a Napoleonic-era policeman. Several methods were explored, here's one of them:
After that, the study turns to a look at spirals, first with the Archimedean spiral:
That was followed by 2-, 3-, 4-, and 6-point spirals. They were easy enough and I find this sort of stuff fun to draw. Up next is volutes, but that can wait for another day as I feel I covered some good ground today. The section ends with volutes, so not much further to go, and then it will be on descriptive geometry for various solids.
If anyone is interested in details about how to do any of the geometrical forms mentioned above, I'd be happy to post up a step-by-step.
- Chris Pyle
- Deshi
- Location: St. Louis, MO
Post
Re: An investigation Begins
Chris, do you have planned application for circular/elliptical forms in your work?
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