Following my discussion with Ian Anderson in a separate thread concerning the finding of backing cuts on a curved hip rafter, I was reacquainted with one of my French Layout texts, a volume dealing with advanced cabinetmaker's drawing, La Menuiserie.
I thought it would be worthwhile to start going through the book, and make the models and examples contained within, as double circular work remains a frontier I have yet to explore in depth. You've gotta study if you want to advance in this field, at least that is how it seems to me.
The preliminary section is called Éléments de géométrie plane, and contains a series of standard exercises for producing parallel lines, perpendicular lines, and then moves to drawing various angles, using compass and straightedge. I was intrigued to come across a different method than I had seen before for producing a desired angle, Fig. 19:
So, I opened SketchUp and drew it out, but the angle produced by the intersection of a line projected from point C through D to produce D' was not exactly 29˚, as advertised. Puzzled, I set Sketchup to a higher number of facets (360) per circle, thinking the coarse faceting of the arc initially might have led to the obvious inaccuracy. However, I still got the same result after a redraw.
So, then I turned to calculation - you can see some of my numbers pencilled in on the drawing - using the Law of Sines to work out angle D'OC (which was 119.49209...˚), then, subtracting 90˚ from that obtained the angle AOD', at 29.492˚. It supposed to be 29.0˚, so this method, for that angle at least, produces an inaccuracy of nearly half a degree.
One can argue about how accurate is accurate enough, but I would say that half a degree of error is too much. For rough framing, perhaps okay, but that is far from what this text is all about.
This is a modern text by Les Compagnons Du Devoir, so I was surprised to find a method shown so early in the book which is not anything more than capable of producing a rough angle. I read through the textual description for that Figure, thinking I might find some sort of comment to the effect that the method was only approximate, but no. Surprising.
An investigation Begins
- Chris Hall
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- Brian
- Deshi
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Re: An investigation Begins
Chris, is this the same text in which you found the saw horse? I remember you had some trouble duplicating the saw horse as described in the book and it required some changes. Knowing the quality of your investigation I am starting to think that book may have many small errors.
- Chris Hall
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- Location: Greenfield, Massachusetts
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Re: An investigation Begins
No, it's not the same text. The work was by Mazerolle and dates to 1860 or so. The sawhorse drawing you refer to had many errors, some of which can be ascribed to etching errors in the plate made by the printer, however other errors are so egregious that I suspect they are deliberate, placed there to mislead. It was one of the factors which made solving the layout so difficult.
This book is a totally modern text, part of a multi-volume series, and I highly doubt that there are deliberate mistakes in the book, especially with something so fundamentally old school as using straightedge and compass to produce angles.
This book is a totally modern text, part of a multi-volume series, and I highly doubt that there are deliberate mistakes in the book, especially with something so fundamentally old school as using straightedge and compass to produce angles.
- Chris Hall
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- Contact:
- Location: Greenfield, Massachusetts
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Re: An investigation Begins
So, I noticed another method for drawing an angle on the same page, in fact it is one shown immediately proceeding the above example, namely Fig. 18:
With this method, you are to make the circumference of the circle equal to 360. Then using the formula to relate radius to circumference, that is, 2πr, we solve to find the length of the radius at 57.2957795..... This is abbreviated on the above drawing as 57.3.
Then, if you wish to produce a 29˚ angle, by marking up a chord of 29 units with a compass, the angle will be produced.
So it says. Again, I fired up SketchUp and tried this out and found the result didn't correspond exactly.
So, back to mathematics to see what the answer should be. The formula for chord length, given a known angle and radius length, is:
Chord length = 2r sin(angle/2)
Solving for the values known for angle and radius, the chord length obtained would be 28.6914....
Again, the method produces a rough approximation, but is not an exact method. This is curious to me.
I am starting to wonder if there are geometrically accurate methods for producing any angle using straightedge and compass? Of course, one can take a 90˚ right angle and use the compasses to bisect the angle again and again, to obtain 45˚ (and 135˚ as the complement), 22.5˚, 11.25˚, and so forth. It is easy to find 60˚ by constructing an equilateral triangle with the compass, then bisecting to obtain 30˚, 15˚, 7.5˚, and so on. But let's say you want a 29˚ angle....
With this method, you are to make the circumference of the circle equal to 360. Then using the formula to relate radius to circumference, that is, 2πr, we solve to find the length of the radius at 57.2957795..... This is abbreviated on the above drawing as 57.3.
Then, if you wish to produce a 29˚ angle, by marking up a chord of 29 units with a compass, the angle will be produced.
So it says. Again, I fired up SketchUp and tried this out and found the result didn't correspond exactly.
So, back to mathematics to see what the answer should be. The formula for chord length, given a known angle and radius length, is:
Chord length = 2r sin(angle/2)
Solving for the values known for angle and radius, the chord length obtained would be 28.6914....
Again, the method produces a rough approximation, but is not an exact method. This is curious to me.
I am starting to wonder if there are geometrically accurate methods for producing any angle using straightedge and compass? Of course, one can take a 90˚ right angle and use the compasses to bisect the angle again and again, to obtain 45˚ (and 135˚ as the complement), 22.5˚, 11.25˚, and so forth. It is easy to find 60˚ by constructing an equilateral triangle with the compass, then bisecting to obtain 30˚, 15˚, 7.5˚, and so on. But let's say you want a 29˚ angle....
- Brian
- Deshi
Post
Re: An investigation Begins
Interesting, and thanks for clarifying above.
For novelty I tried it with a few random angles;
Method applied to 14 degrees produced 14.6
Applied to 29 produced 29.4
Applied to 52 produced 51.9
Applied to 71 produced 70.9
It work well when I applied it to 90![Laughing :lol:](./images/smilies/icon_lol.gif)
Seems it can be relied upon with 1/2 a degree.....so maybe it's still useful but not useful when extreme accuracy is needed.
For novelty I tried it with a few random angles;
Method applied to 14 degrees produced 14.6
Applied to 29 produced 29.4
Applied to 52 produced 51.9
Applied to 71 produced 70.9
It work well when I applied it to 90
![Laughing :lol:](./images/smilies/icon_lol.gif)
Seems it can be relied upon with 1/2 a degree.....so maybe it's still useful but not useful when extreme accuracy is needed.
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
Post
Re: An investigation Begins
It appears the method is more accurate for higher angles than for lower ones, based on your calcs.
If it is just 2 pieces of wood being joined, then perhaps +/-0.5˚ tolerance might be acceptable, depending upon circumstances, however it is often the case than when non-90˚ angles are involved, there are multiple pieces and the error accumulates. A 0.1˚ error on each piece of a hexagon or octagon miter, for example, will produce a noticeable gap when the frame is assembled - consider that if two pieces are mitered with the same 0.1˚ error, then when put together the error is doubled. So, obtaining accuracy higher than 0.5˚ is not really in the realm of 'extreme' at all it seems to me.
Also, if the joined piece is long, an error in the miter is magnified more at the other end of the stick, just like a very tiny error in aiming a rocket at a far away target will result in a miss.
If it is just 2 pieces of wood being joined, then perhaps +/-0.5˚ tolerance might be acceptable, depending upon circumstances, however it is often the case than when non-90˚ angles are involved, there are multiple pieces and the error accumulates. A 0.1˚ error on each piece of a hexagon or octagon miter, for example, will produce a noticeable gap when the frame is assembled - consider that if two pieces are mitered with the same 0.1˚ error, then when put together the error is doubled. So, obtaining accuracy higher than 0.5˚ is not really in the realm of 'extreme' at all it seems to me.
Also, if the joined piece is long, an error in the miter is magnified more at the other end of the stick, just like a very tiny error in aiming a rocket at a far away target will result in a miss.
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
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Let's look at that angle of 71˚ - I'll work through the steps so we're on the same page.
Chord length = 2r(sin(angle/2)
Filling in some numbers,
2(57.295779...) x sin (71/2)
Gives:
114.591559... x (sin 35.5)
Further:
114.591559... x 0.58070295...)
Which equals a chord of 66.543657 or so, which is quite far away from 71.
If you look at the example of a 90˚ angle with radius of 57.3, then the chord length works out to 81.028.
An alternate method in the case of a 90˚ would be to note that for a right angle having rise and run equal, the hypotenuse (i.e., the chord length) would be √2 times longer. Sure enough, 57.295779√2 = 81.028....
It seems like the inaccuracy of this method goes up with a higher angle.
Let's consider a smaller angle than 29˚ then, say 11˚:
Chord = 2r sin (11/2)
= 114.59155559... x sin 5.5
= 10.9831142... which is very close to 11.0
As suspected, this method shown in Fig. 18 is most accurate for shallower angles, the error increasing as the angle steepens.
Re: An investigation Begins
Uh, I think you went astray somewhere there in calculation.Brian wrote:Interesting, and thanks for clarifying above.
For novelty I tried it with a few random angles;
Method applied to 14 degrees produced 14.6
Applied to 29 produced 29.4
Applied to 52 produced 51.9
Applied to 71 produced 70.9
It work well when I applied it to 90
Seems it can be relied upon with 1/2 a degree.....so maybe it's still useful but not useful when extreme accuracy is needed.
Let's look at that angle of 71˚ - I'll work through the steps so we're on the same page.
Chord length = 2r(sin(angle/2)
Filling in some numbers,
2(57.295779...) x sin (71/2)
Gives:
114.591559... x (sin 35.5)
Further:
114.591559... x 0.58070295...)
Which equals a chord of 66.543657 or so, which is quite far away from 71.
If you look at the example of a 90˚ angle with radius of 57.3, then the chord length works out to 81.028.
An alternate method in the case of a 90˚ would be to note that for a right angle having rise and run equal, the hypotenuse (i.e., the chord length) would be √2 times longer. Sure enough, 57.295779√2 = 81.028....
It seems like the inaccuracy of this method goes up with a higher angle.
Let's consider a smaller angle than 29˚ then, say 11˚:
Chord = 2r sin (11/2)
= 114.59155559... x sin 5.5
= 10.9831142... which is very close to 11.0
As suspected, this method shown in Fig. 18 is most accurate for shallower angles, the error increasing as the angle steepens.
- Brian
- Deshi
Post
Re: An investigation Begins
I should have been more specific, those numbers were applied and received from your first example, not the second.
Fair point on extreme measures of accuracy. When I make a miter it is probably within a very small margin of error, anyone making gap free miters would have a similar margin, so it is a fair point to suggest 1/2 a degree is not accurate for a great deal of applications.
I'm following along with your second example but did not plug numbers.
Fair point on extreme measures of accuracy. When I make a miter it is probably within a very small margin of error, anyone making gap free miters would have a similar margin, so it is a fair point to suggest 1/2 a degree is not accurate for a great deal of applications.
I'm following along with your second example but did not plug numbers.
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
Post
Gotcha. I had misunderstood, thinking you were looking at the second method. In any case, neither appears to be reliable.
Re: An investigation Begins
Brian wrote:I should have been more specific, those numbers were applied and received from your first example, not the second.
Fair point on extreme measures of accuracy. When I make a miter it is probably within a very small margin of error, anyone making gap free miters would have a similar margin, so it is a fair point to suggest 1/2 a degree is not accurate for a great deal of applications.
I'm following along with your second example but did not plug numbers.
Gotcha. I had misunderstood, thinking you were looking at the second method. In any case, neither appears to be reliable.
- Chris Hall
- Site Admin
- Contact:
- Location: Greenfield, Massachusetts
Post
Re: An investigation Begins
I can see that the method shown in Fig. 18 would work if one could accurately measure up along the arc of the circle instead of taking a chord distance with compass. Trouble is, there is no good way to do that. It does explain though why the method is most accurate for small angles.
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