Need Math Help on Accuracy - Page 2 - Woodworking Talk - Woodworkers Forum
Old 05-11-2016, 12:40 PM
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The 24" could be the hypotenuse or the base, depending on what we are talking about.

From what I understood, we were talking about deflection off the measurement line, in which case the 24" is off a deflected ruler and would be the hypotenuse.

If instead we have measured 24" out properly, but the tool or guide rail or whatever deflected a bit, but still cut out to where the 24" is measured out on the other axis, then the 24" is the base and the cut actually went out the hypotenuse and travelled more than 24".

Regardless, given the sizes we are talking about here, it doesn't matter.
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Old 05-11-2016, 02:47 PM
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yes, it does matter.

if one is making a 90 degree cut 24 inches long - and the miter/gauge/rail/guide/geometry challenged whatever is not 90 degrees but 89.5 degrees, the cut will be short/long depending on which way the miter/gauge/rail/guide/geometry challenged whatever is "off"

and, over 24 inches, it is not insignificant if one is doing something that requires precision cuts.
lopping off a 24" chunk of firewood, it does not matter. making an n x 24 inch frame, it matters.
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Old 05-11-2016, 03:47 PM
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If you were to come perfectly square from an edge (perfect 90°), the length obviously is 24 inches. With a 0.5° deflection, the now 24 inch length is just a hair shorter (0.009 in) in comparison to the original 24". Because of this, Jig_saw is right, 24" is the hypotenuse, but not of a right triangle. Drawing a line from the end the perfectly square line to the end of the 0.5° deflected line does not yield a 90° angle. Why is this brought up? The equations being used are for right triangles. This is not a right triangle. Now, for all practicalities, does that matter? No, because the angle is too small to worry about.

Just a little food for thought.

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Old 05-11-2016, 06:14 PM
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Quote:
Originally Posted by darins View Post

If you were to come perfectly square from an edge (perfect 90°), the length obviously is 24 inches. With a 0.5° deflection, the now 24 inch length is just a hair shorter (0.009 in) in comparison to the original 24". Because of this, Jig_saw is right, 24" is the hypotenuse, but not of a right triangle. Drawing a line from the end the perfectly square line to the end of the 0.5° deflected line does not yield a 90° angle. Why is this brought up? The equations being used are for right triangles. This is not a right triangle. Now, for all practicalities, does that matter? No, because the angle is too small to worry about.

Just a little food for thought.
"If I have a base line, then draw a line at a perfect 90 degree angle 24" long, I have zero deflection (for lack of a better word) at the end of the 24".

'plain me how a 24 inch line at 90 degrees to the base line can become a hypotenuse?
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Old 05-11-2016, 10:55 PM
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It seems those thinking about the hypotenuse are confused about the concept of the error. They see the 24" line pivoting away from true 90 degrees. Yes, that new, off-square line can be seen as a hypotenuse, but that is not the error.

In statistics, errors are calculated as orthogonal (perpendicular) to the estimator (the 24" line in the OP's example). The ratio between the error and the estimator is the relative error, and also is the tangent.

It may be easier to think of the error not as a rotation of the whole line, but rather the lateral (sideways) displacement of the far end of the line, perpendicular to the line. While the hypotenuse is interesting, in discussion of the error, it is irrelevant.
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Old 05-12-2016, 12:33 AM
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so then ....

We can't use a right triangle trigonometric function like tangent because we really don't have a 90 degree angle. It's a minute difference and for practical purposes, it wouldn't really matter ...JMO :frown2:

The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)
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Old 05-12-2016, 08:54 AM
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yes we can use a right triangle function. because as explained, the OP is looking for the distance the cut will be "off" (or displaced - per his term) from a true 90 degree angle - at 24" - if there is a 0.5 degree error in the miter gauge (for example)

and that is:
tan(0.5 degrees) * 24 = 0.008726867 * 24 = 0.209444827 inches
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Old 05-12-2016, 09:06 AM
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Well, I'm stlll confused ...

Quote:
Originally Posted by darins View Post

If you were to come perfectly square from an edge (perfect 90°), the length obviously is 24 inches. With a 0.5° deflection, the now 24 inch length is just a hair shorter (0.009 in) in comparison to the original 24". Because of this, Jig_saw is right, 24" is the hypotenuse, but not of a right triangle. Drawing a line from the end the perfectly square line to the end of the 0.5° deflected line does not yield a 90° angle. Why is this brought up? The equations being used are for right triangles. This is not a right triangle. Now, for all practicalities, does that matter? No, because the angle is too small to worry about.

Just a little food for thought.
Quote:
Originally Posted by TomCT2 View Post
yes we can use a right triangle function. because as explained, the OP is looking for the distance the cut will be "off" (or displaced - per his term) from a true 90 degree angle - at 24" - if there is a 0.5 degree error in the miter gauge (for example)

and that is:
tan(0.5 degrees) * 24 = 0.008726867 * 24 = 0.209444827 inches
Quote:
Originally Posted by woodnthings View Post
We can't use a right triangle trigonometric function like tangent because we really don't have a 90 degree angle. It's a minute difference and for practical purposes, it wouldn't really matter ...JMO :frown2:

So then ....
Let's increase the angle to 30 degrees. Rotating the 24" leg out to 30 degrees and connecting it to the previous end point would not result in a 90 degree angle at either intersection...right?

The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)
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Old 05-12-2016, 12:17 PM
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and why we built the Tower of Babel out of wood is not the question either.

the question is:
if my 90 degree angle is "off" by 0.5 degrees, how much difference is that going to make at 24" from the "base line"?

in the OP's question, the 24" length is the length of the perfect 90 degree cut.
any cut line other than 90 degree becomes the hypotenuse - it is longer than 24 inches.

if you make the angle 30 degrees, at 24" from the "base line" the distance from the theoretical 90' cut to the point of the 30' cut would be:

tan(30) * 24 = 0.577... * 24 = 13.856 inches

and the question is not "how long is the hypotenuse/length along the 30' cut?"

it's the ole'
sin=O/H
cos=A/H
tan=sin/cos or = O/A
we don't know the opposite length, we don't know the hypotenuse length
we do know the angle and the length of the adjacent side.

Last edited by TomCT2; 05-12-2016 at 12:19 PM.
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Old 05-12-2016, 02:23 PM
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Quote:
So then ....
Let's increase the angle to 30 degrees. Rotating the 24" leg out to 30 degrees and connecting it to the previous end point would not result in a 90 degree angle at either intersection...right?
Nope. Maybe this diagram might help. Error is perpendicular to what you're measuring. That is a statistical definition. To get the error length, you create a new line, off from the original by the error of the angle. What you have to do is extend that new line (dashes), to a point where it intersect the perpendicular drop to the end of your original line. That distance which is square to the original line is the error. Creating an isosceles triangle by rotating a line 30 degrees and connecting the endpoints is nice looking, but it has nothing to do with the definition of error.

The error is proportional to the length of your original line- a 48" line will have twice the lateral error as a 24" line. But the relative error (error divided by length) will be the same for both lines, and that's the tangent of the angle!!!
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Old 05-12-2016, 04:58 PM
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yes - I tried to upload a sketch - failed miserably. need to re-figure out how this software handles pix...
the "error" is what the OP was looking for.

thanks for the effort!
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Old 05-12-2016, 09:11 PM
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Is the Perpendicular the same as the Opposite?

I learned

Canter Away Happily
To Other Areas

I've never had to look it up since I learned that.

P.S. You can always introduce lines to construct a right triangle.

"When I have your wounded." -- Major Charles L. Kelley, callsign "Dustoff", refusing to recognize that an LZ was too hot, moments before before being killed by a single shot, July 1, 1964.
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Old 05-13-2016, 10:06 AM
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"perpendicular" unfortunately conveys a sense of orientation - so it can be confusing.

the "opposite" is the triangle side opposite the angle in question
while
the "adjacent" is the triangle side that "adjoins" the angle in question, the other triangle side that "adjoins" the angle in question is the hypotenuse.

in jdonhowe's sketch, the angle at the pointy bottom is the angle in question - in the case of the OP, 0.5 degrees.
"opposite" that angle is labeled "error" - and the vertical line is the "adjacent" - the line on a slant which is continued with a dotted segment is the hypotenuse.
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Old 05-13-2016, 11:53 AM Thread Starter
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Very fascinating thread. And in a sense, an eye opener in different ways, like how the question could be viewed. I wouldn’t call myself a ‘machinist’ but I have done machining and a lot of high tolerance work ‘back in the day.’ For example I’m factory trained on overhauling a Jaguar V-12 engines as well other brands of engines, transmissions and the like.

So one of my concerns is being anal in the rough woodworking I do: does it really need to be within plus/minus 0.001” for example. Not ever working in degrees, though, when I randomly picked 0.5 degree, I had no idea it was such a large error at 24”. One of my eye openers.

So even if I were to cut the end off of a 6” board off using a combo square with an inaccuracy of 0.05 degree, that’s still about a 0.052” error if I did the math right (0.209” error on a 24” board divided by the four 6” boards in it). So if I cut both ends off four 6” boards to make a box (with the ruler part always facing the end of the board), I’ll wind up with some sort of trapezoid box! If that sounds anal, let me know, I can take it.

I just make crude things (and most of my things are out of cedar fence board for example) but I’d still like things square when they’re suppose to be.

I have a table saw I’ve use once every couple of years since it’s in a storage shed, but it (the error) makes me want to build a cross cut sled using the 5-cut method and use the right angles from that for my 90 degree layout lines… (or hit the lottery so I could afford to buy Starrett combo squares; I should be able to trust them)

So how do you folks insure that your 90 degree cuts are 90 degrees?
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Old 05-13-2016, 03:34 PM
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There are several ways...

I make a lot of crosscuts using a Radial Arm Saw which I have set to 90 degrees... permanently. I used a framing square and a blank sanding/reference blade squaring the fence by adjustment to the blade. I did not attemtpo to adjust the arm to square the fence, since the fence is much morwe easily adjustable.

On the table saw, there are a few ways. I use an precision Incra miter gauge accurate to .1 degrees. I also use a older Craftsman miter gauge with an attached fence, set square to the blade using a known reference engineering square. You can make a crosscut at 90 degrees, flip one piece over and mate them together. If the edges do not form a straight line the cut is off. Readjust the miter gauge by 1/2 the error.

You can make a crosscut then check it afterward using a known square. I also have a digital angle gauge for checking and making any desired angle from 90 degrees to 5 degrees.

Woodworking is NOT machining, so allow for more error, "allowance" ...is the technical term I think? When metal parts don't fit, it's real obvious. When wood parts don't fit, there are ways...sandpaper, hammers, planes, chisels, etc..

The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)

Last edited by woodnthings; 05-13-2016 at 05:44 PM.
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Old 05-13-2016, 04:18 PM
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Actually, a lot of woodworking is to pretty tight tolerances. One thousandth is light, two is heavy light, and woodworkers use light to tell when joinery is tight. We use tight tolerances, we just have different ways of measuring and achieving it.

A bucket or a barrel that isn't tight wouldn't be much use.

I cut two square joints, flip them to double any error, and see if there's light. The only way they'll get within a thousandth of an inch (light) is if the angles are both exactly ninety degrees. We used to call it "proving the saw".

"When I have your wounded." -- Major Charles L. Kelley, callsign "Dustoff", refusing to recognize that an LZ was too hot, moments before before being killed by a single shot, July 1, 1964.
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