tangen and sashigane

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craigak
Hopper I
Hopper I
Posts: 26
Joined: Fri Jul 15, 2011 6:27 pm

tangen and sashigane

Sun Aug 21, 2011 3:30 pm

Hi to all,
Frustrated here in Chicago. I have done fine on the math but the layout for hopper 1 eludes me.
Here is what I am doing:
chu ko:
10 on the long arm, 4.4 on the short arm

tangen
10 on the long arm, 2.15 on the short arm
sashigane place on top edge of board, outside edge

I am under the impression that this should produce a slope of 85.13˚...it doesn't.

I have a sliding crosscut tablesaw so I set the crosscut fence to the chu ko slope and then set the blade tilt to 85.13˚.

The resulting cut seems wrong.

Any help would be appreciated.

Craig
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Chris Hall
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Re: tangen and sashigane

Sun Aug 21, 2011 8:05 pm

Craig,

you didn't mention what the common slope that you are working with is, but I infer is is 10 / 4.9.

Given that, as you mentioned, we obtain a:

- chūkō of 4.40015...

- tangen of 2.15607...

Then you said,

"10 on the long arm, 2.15 on the short arm sashigane place on top edge of board, outside edge. I am under the impression that this should produce a slope of 85.13˚...it doesn't."

Your impression is a bit erroneous I'm afraid. Let's look at what a tangen slope triangle would look like:
CSG Hopper 2 tangen triangle.jpg
CSG Hopper 2 tangen triangle.jpg (39.39 KiB) Viewed 5690 times
In order to find an angular value, we can use tangent, that is, opposite / adjacent. The opposite is 2.15607... and the adjacent is 10:

ARCTAN (2.15607 / 10) = ARCTAN (0.215607) = 12.16714376˚

Now, looked at from the other end of the tangen slope triangle, in reference to a plumb line, opposite and adjacent would be reversed and we would obtain a much steeper angle. You suggested 85.13˚. All we need to do is subtract the angle value we just found from 90˚:

90 - 12.16714376 = 77.832856˚

So 85.13˚ is certainly not associated to the tangen. It does however happen to be the angle we found for the shōchūkō. That angle is for the mortise top/bottom surfaces, something you would not be able to cut with a tablesaw. So that looks to be one point of confusion.

Then you said,

"I have a sliding crosscut tablesaw so I set the crosscut fence to the chu ko slope and then set the blade tilt to 85.13˚. The resulting cut seems wrong."

That is not surprising! But here's the thing, even if you had determined the correct tangen slope value of 77.832˚, you cannot simply set up your saw as described and execute the cut to obtain the butt joint miter. Why not? because tangen is laid out on the edge of the board, while the table saw tilt is referenced perpendicular to the saw blade. Since you are running the board on the fence set to the reverse value of the chūkō, that means that the lead edge of the board, upon which you have presumably marked the tangen slope, cannot be perpendicular to the blade. Therefore the blade would need to be set at some angle other than the 77.732˚ tangen value. This is the counter-intuitive nature of compound joinery my friend.

Let's take a look with some drawings at the problem in case my words were inadequately clear. The first task was to set the saw's miter gauge to the chūkō slope:
set chuko.jpg
set chuko.jpg (369.32 KiB) Viewed 5690 times
Then I place the board on the miter gauge, and I indicate my lay out, chūkō on the face of the board, and tangen on the edge:
set up tangen cut.jpg
set up tangen cut.jpg (224.46 KiB) Viewed 5690 times
Let's check that tangen layout on the board edge to ensure that it conforms to the numbers mentioned previously which I obtained by calculation:
tangen on board edge.jpg
tangen on board edge.jpg (119.42 KiB) Viewed 5690 times
Next, I am going to set the sawblade to the same tangen slope angle value:
set sawblade.jpg
set sawblade.jpg (262.94 KiB) Viewed 5690 times
I tilt the blade to that angle, and now let's run the board into the blade to see what happens - here's a view from the backside perspective as the board approaches the blade:
start cut.jpg
start cut.jpg (318.5 KiB) Viewed 5690 times
Running the board into the blade now, and let's see how the cut line looks, here from the operator's perspective:
overcut.jpg
overcut.jpg (147.97 KiB) Viewed 5690 times
It would appear that setting the sawblade to the tangen slope produced a slight overcut, yes?

Let's look again at the board edge and the tangen slope we had laid out:
tangen on board edge.jpg
tangen on board edge.jpg (119.42 KiB) Viewed 5690 times
Now I'll place the appropriate triangles in there to indicate what sort of angle we would need on the sawblade. The sawblade is set perpendicular to the blade's face, and that perpendicular happens to be sloped at the chūkō angle in relation to the board edge:
sawblade angle.jpg
sawblade angle.jpg (164.21 KiB) Viewed 5690 times
The actual sawblade angle (the red triangle in the above picture) we would need to produce the tangen butt miter on our board has an opposite that is slightly smaller than the tangen slope angle on the board edge. This makes sense, as the sawblade when set at the tangen angle, produced an overcut.

I'll leave it to you to make the button pushes on the calculator to determine what sawblade angle you need.

Remember, the entire method I am showing here in TAJCD is not for tablesaw-jutsu, but for using a framing square and hand tools. You would probably obtain the result quickly enough by laying out your lines, then attacking the problem with a handsaw and cleaning up the miter with a plane. Of course, cut it however you please.

I might add in closing, that the issue of the difference in setting a table saw or chopsaw relative to the lines laid out on the wood itself was brought up recently by roberts, in this thread: http://www.thecarpentryway.com/Carpentr ... ?f=15&t=64

Hope the above explanation clears up any confusion on this matter.

Cheers,
~C
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Chris Hall
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Re: tangen and sashigane

Tue Aug 23, 2011 8:08 pm

No comment? I did spend about two hours writing and illustrating that reply, and wonder if it made good sense or if there is some additional clarification I might be able to provide?
roberts
Double
Double
Location: Nr Brussels, Belgium
Posts: 27
Joined: Sat Jul 02, 2011 2:57 pm

Re: tangen and sashigane

Wed Aug 24, 2011 3:29 pm

Well, I am certainly going to reply. This is a superb set of figures. The one "start cut.jpg" shows the essense of what happens when a tilted blade meets a skewed workpiece; the basis of all compound cuts.

I need more time to go over the numbers but one observation for craigak: whatever angle you think is the one you should use (e.g. to tilt the blade to), you will almost certainly have to use its complement (You can find this is in the geometry section of Murphy's standard work.) This is because angles are calculated the other way around to what you expect, and machine scales are always opposite to the way the author has assumed they are going to be. ;)

I understand your frustration, Chicago !
craigak
Hopper I
Hopper I
Posts: 26
Joined: Fri Jul 15, 2011 6:27 pm

Re: tangen and sashigane

Wed Aug 24, 2011 9:27 pm

Hi Chris,
I am just back in the office to start making sense of this. Thank you for the extra work you did here.
Craig
Matt J
Hopper I
Hopper I
Location: Maynard, Massachusetts
Posts: 91
Joined: Sat Jul 02, 2011 9:27 am

Re: tangen and sashigane

Mon Aug 29, 2011 9:44 pm

I have to say, if this doesn't demonstrate an excellent value for $20/month, I don't know what does! Awesome! I've struggled with this kind of thing a lot.

Craig, one suggestion I would make is to mark out the slopes/cuts on a piece of wood as if you're going to cut it by hand, first. I had to mess around with the layout lines for awhile before I got them to "work", but when they did I knew it. Having the lines laid out correctly would probably make setting up the table saw more intuitive. If you tried that already, then... what Chris said. :)

-Matt

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