Thursday, May 7, 2020

Resultant and Sightline Angle Calculator

Finally I'm getting around to some chair making.  And to do that, I needed to work out resultant and sightline angles.  I won't bore you with definitions of these angles - you can learn about them on Schwarz's blog, or Curtis Buchanan's or Pete Galbert's sites (among others).

However I might bore you with some trigonometry.  The result of all this is that I made an Excel spreadsheet that will calculate resultant and sightline angles, given the rake and splay angles.  If anybody wants an Excel file with these calculations, send me a message using the "Contact Me" gadget of this blog and I'll be happy to send you a copy.

NOTE: in this discussion, I'm assuming the underside of the seat is level with the floor.

Here's a bit of the background mathematics:
Diagram showing how rake and splay relate to resultant and sightline angles
I'm going to treat a chair leg as a line with length, but no width or thickness.  In the diagram above, point E is on the underside of the seat where the leg originates and point A is on the floor directly below E (EA is a plumb line).  The horizontal and vertical axes are easy enough to represent in 2-dimensional space, but the third axis coming out of the page (toward the viewer) is along the segment AB.  Line segment EC can be the right front leg of a chair and we're looking at the chair from the front.  Hopefully it is clear that point C is where the leg touches the floor.

If I mark a line directly back to the horizontal axis from point C, that line intersects the horizontal axis at point D.  The angle formed by the plumb line and the segment ED is the splay angle, and it measures s°.
Splay of the front right leg is the angle off vertical when viewing from front
Similarly, if I mark a line from point C directly over to the third axis (along segment AB), it intersects that axis at point B.  The angle formed by the plumb line and segment EB is the rake angle, and it measures r°.
Rake of the front right leg is the angle off vertical when viewing from the side
You can also think of the rake and splay in inches.  In the case of the diagram above, the leg is splayed x inches to the right and raked y inches to the front.

The resultant angle, Θ, is a function of both rake and splay.  It is the angle the leg (segment EC) makes with the the plumb line EA and is the angle at which we need to bore the mortise hole in the seat.

The sightline angle, α, is also a function of rake and splay.  I find it hard to define in words, but I've read a couple different versions.  One goes like this.  It is the angle at which, as you rotate the chair and look at the leg, the leg looks perfectly vertical.

You can see the trig calculations at the upper right in the diagram.  These are easy enough to figure out (for this mathematically inclined dude), but writing them into an Excel spreadsheet can be challenging.  But that's what I did and here is the result.
Resultant and sightline angles as a function of rake and splay angles
To use the table, say you have a front leg with rake of 2° and splay of 4°.  Find the value for rake in the first column, go over to the column under the desired splay, and read off the resultant of 4.5° and sightline of 26.5°.

And in case I ever need to find resultant and sightline angles for rake and splay angles not represented in the table, the box at bottom (just left of center) allows you to put in any rake and splay.

On the right side box at the bottom, you can also find the rake and splay in inches (distance from where the plumb line hits the floor) by entering the height at the underside of the seat.  In this example, my small chair's seat bottom was 11" off the floor, resulting in a rake of 0.7" and splay of 1.3".

I know what you're saying.  "But Matt, what if I don't know the rake and splay angles?  Can I do something if I can measure the rake and splay in inches?"  Well yes, yes you can.  The following table shows rake and splay in inches, given the height of the bottom of the seat, as well as rake and splay angles in degrees.
Table of rake and splay in inches, given height of chair (H) and rake and splay angles
You've got to use this table in reverse.  Say you have a chair with height to bottom of the seat 11".  The bottom of the leg is 1" forward of where the plumb line hits the floor.  That's a 1" rake.  You also have the bottom of the leg out 2" laterally from where the plumb line hits the floor.  Splay is 2".  In the table, find 1" rake (black entries in the table) and you know you have a rake angle of 5°.  In that row of the table, find the red entry for 2" and that is between 10° and 11°.  Call it 10 1/2°.

I realize that I could make a table with rake and splay in inches along the top row and left column so you can look up the rake and splay in degrees, but I was losing energy.

You can do a lot with this.  I was copying a chair and I liked the spread of the legs at floor level.  But in the original I thought the mortise in the seat was a little too close to the edge.  So I played with the design, moving the top of the leg away from the edge of the seat while keeping the bottom of the leg unmoved.  Based on how much I moved the top of the leg, I was able to find the new rake and splay angles, and ultimately the new resultant and sightline angles.

It's fun stuff.  Again, if anybody would like a copy of this spreadsheet, send me a message using the "contact me" gadget on this blog.

31 comments:

  1. I head hurts after reading this post Matt.

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    1. Haha! Come on, Ralph - we work our bodies in the shop, why not work the mind as well. Really, this stuff is not that tough. You just have to work with it a little bit and it gets better.

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  2. Nice exercise.
    Although, if I were doing a chair with an inclined seat, I would make a drawing using descriptive-geometry to obtain the sightline direction and angle.
    The precision will be adequate (depending on drawing carefulness).
    I don't know if descriptive geometry is still taught in schools know that people have CAD. 50 years ago, I liked it.
    Sylvain

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    1. Have to admit that I had to search the internet for "descriptive geometry". I had Euclidean geometry in high school (in the '70s), but not descriptive geometry. I think I would have liked it. Perhaps if I had gotten into engineering in college (like I should have) I would have had it there. I might have to do some research on this ...

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    3. I doubt most colleges teaching engineering would have taught descriptive geometry in the '70's, I know mine didn't. From what I remember of the book my dad showed me, descriptive geometry would probably have been categorized as advanced drafting and most engineering schools didn't "lower themselves" to teaching drafting.

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  3. Reading Chris Schwarz again, he makes his chair first with an horizontal seat. Then he shortens the back legs (1" dining - 2" lounging).
    This increases the front leg rake and reduces the back leg rake. It also reduces the lateral distance between the back feet.
    It also changes the splay angles front and back.
    Note that it is the rake and splay with reference to the ground which change,not with reference to the seat.
    This changes the look and stability (the back legs are more vertical).
    One has the to calculate the "horizontal-seat" rake and splay to obtain the desired final rake and splay either numerically or graphically (quite easy).

    One can then use the numerical method to calculate the sightline.

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    1. Schwarz' 1" and 2" shortenings of the back legs seem excessive. I've typically done 3/8" to 1/2". I should have mentioned in my post that I do the calculations based on a seat that is level with the ground, but later I cut the back legs shorter to make it lean back. It's much easier (for me) to think about rake, splay, resultant and sightline with a level seat. Since the underside of the seat is my reference surface for boring the angled mortise holes, it's best for me to calculate those angles from a level seat.

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    2. I believe he referenced John Brown's method in which he used a spirit level and pitched the seat back measuring with his fingers: 1 for dining, 2 for sitting/lounging, then Schwarz added "3 for drinking"

      2 finger tips looks to be just a bit over 1" on my paws

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  4. It seems I have been a little carried away with the theory.
    After making a drawing to scale, it seems the only angle which is visibly changed (about 2.5° in my case) after cutting one inch off the back legs is the rake of the back leg with reference to the ground.
    It is the only one for which it is worthwhile to make a little sketch to evaluate the margin of tilting backwards.
    Body center of gravity:
    https://www.faa.gov/data_research/research/med_humanfacs/oamtechreports/1960s/media/AM62-14.pdf

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    1. All my life, knowing a bit of math has made my life easier. Love that stuff!

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  6. This message is for Paolo B., from Italy/Switzerland. I received your request for the spreadsheet, but when I sent it via e-mail, the message failed. Please resend your request and make sure the e-mail address you provide is correct. Thank you.

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  7. I would like the Excel file - please

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    1. The only way to get the spreadsheet to you is for you to use the "contact me" form on this blog page. Please try that.

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  8. Thank you for this post. I've built a couple of chairs and this still taught me some things I needed to know. Could you please send me the spreadsheet? Thanks.

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    1. Tony, see my response just above your comment. Please use the "contact me" form on the blog page.

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  9. I just seen a video on chair making from Woodpecker Deep Dive and the reference your spread sheet. Is it possible to send the spread sheet to me. Thank you.

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    1. Your comment does not give me your e-mail address to send it to. Please use the "contact me" form on this blog page.

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  10. Hello. Thanks for this information! You've done a really great job. I would appreciate having a copy of your EXCEL spreadsheet that calculates these angles. Regards, Leon LaPointe

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    1. Sent to the e-mail address that was your "message header" name. If that didn't work, use the "contact me" gadget in this blog page.

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  11. My brain will never stop hurting now.

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    1. Come on, it's not that bad. Maybe a beer will help! haha

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  12. Another request for a copy of the spreadsheet please.

    TIA,

    Bob

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    1. Bob, you have to use the "contact me" gadget on this page. Otherwise I have no e-mail address to send it to.

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  13. Good Morning, I would like a copy of the spreadsheet. Thank You.

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    1. You have to use the "contact me" gadget on this page. Otherwise I have no e-mail address to send it to.

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  14. Thanks. Would you please send me a copy of your spreadsheet. Are you aware of anyone who has done something similar for rocking chairs?

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    1. I can't send it without an e-mail address. The way to do get me your address is using the "contact me" gadget on this page. Don't know of anything for rocking chairs, though I don't know why they would be any different.

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  15. I would love a copy of the Excel spreadsheet for the calculations. Thanks for doing all the work :-).

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