Wednesday 1 June 2016

How I Understand Sight Lines

With starting my seating build I've done some things wrong. Right from the beginning.

I think that I'm starting to understand the parameters which are necessary for the leg layout.

Frankly spoken, even if it was pretty clear what I would like to build and I could figure out most of the dimension and construction details, I was a bit lost about the leg angles.
The legs of my little stool shall have all the same angle for the splay. A shallow rake angle at the front and a wider rake angle at the back.

I found the angles just by eyeballing it. A splay of 7° looked nice to me. And for the front a rake of 7° looked fine too. As mentioned the rake angle at the back should be a bit wider so I decided to try out 10°.

Now I've got the parameters. What I still needed was the sight lines and the resulting angles to drill the holes.
But that was exactly what I was struggling with.
So I read the related chapter in Peter Galbert's "Chairmaker's Notebook" a few times again and watched his videos about this stuff (Sight Lines).

Watching it over and over again brought me to the following knowledge:
The resulting angle is a ratio between the splay and the rake angle in degrees.
My splay should be 7°. So I measured the length of 7° from the center of my leg mortise. Peter is using a special ruler for this. You can find it in his book or on his blog. I've done that with a simple protractor, which is used in school or for technical drawings. At the end it's not more than a ruler which is showing the degree minutes. 

Protractor to measure the resulting angle

I made a mark at this point. Now I measured the rake length in degrees. 7° from the end point of the previous measurement.
If you are now drawing a line between the center point and the end of the rake length, then you have got your sight line. The measurement between these two points (in degrees) is the value of the resulting angle.
In my case it is about 10°.

Understanding sight lines

The resulting angle is the value of the angle you have to drill the holes in relation to the sight line.

With all these findings I had to realize that I had done a few things completely wrong.
My assumption was that the angle for drilling the leg mortices has to be the same if I want to have the same splay. Just the sight line has to be changed.
As mentioned - it took a while to understand the parameters.

Unfortunately I had already drilled all leg mortices. I recognized that the test-leg looked somehow strange.
That was the reason to dive into this topic ones more.

So, my seat plank is toast now. I have used it for some additional tries of drilling the holes right.
Fortunately I still had the cut off from the used board and could make a new one.  With the now right layout - hopefully.

Splay and rake angle, resulting angle and sight line

Before I could drill the new leg mortices I had to make another drilling jig. Now with the right angles.
Ten degrees for the front and 13° for the back of the stool.

New drilling jig

This type of jig is really helpful. It has to be positioned along the sight line and then you can lean the auger against it. While drilling the holes you just have to watch the auger to be perpendicular.

I haven't done the legs yet, so that I couldn't verify the theory from above. I have made some new leg mortices in the old plank and just put some sticks into it. It seems to look better. If you now have a look from the front or back then the legs look like they are aligned.

That said, maybe I should make the legs first, so that I could experiment a bit more, before I will drill the holes into the new plank.

Stay tuned!


  1. If you look at the post (3rd picture)
    you will see that the measurements are proportional to the tangent of the angles.
    The way you use your protactor gives measurements proportional to [Tan a / (1+ Tan a)].

    The greater the angle, the greater the error.
    For 7° and 7°, calculus gives 9.85°; so 10° is a good approximation
    For 7° and 10°, calculus gives 12.13° which is further away from 13°.

    It is easy to print the 2nd picture of the blog

    Note that the drilling angle given by the Galbert rule is only valid if the bottom of the seat is horizontal.


    1. Hi Sylvain,
      thanks for the hint. Now I think I got it.
      What was my error? I didn't recognize that on Peter's ruler are half units.
      So I held my protector to that ruler and thought "fine, that fits".
      But your are right, that is wrong. Just double checked it.
      How embarrassing :-(
      Anyway, I think the rest of my description is just right. But I have to use Peter's ruler.
      So I have to correct my drilling jig about a half degree.
      Thanks for your help.


  2. I'm glad you understand it Stefan. I'm still trying to visualize the flat drawing in pic one to something in 3D. I'm losing so far.

    1. As the discussion shows I hadn't understand it completely.
      But now the details are complete.
      Anyway, it took a while and it was helpful for me to invest some scrap to make some haptic experiences.
      And Greg's explanations and hints have been very helpful.

  3. Stefan...your making this way harder than it needs to be. Basic geometry will get you very, very close. I'm not sure why Galbert is using that Bevel Boss thing.
    These are just triangles. In you above you have two legs and your looking for the hypotenuse.
    7(squared)+7(squared)=c(squared) thus
    49+49=98(square root)
    so first resultant angle is 9.899degs

    49+100=149(square root)
    so second resultant angle is 12.207degs

    The sight line can be found by simply using you ruler...
    over 7 up 7 and over 7 up 10, respectfully.

    Hope that helps in some way.

    1. Greg - there must be some magic here. I think of the A^2 + B^2 = C^2 as applying only to the lengths of the two legs (A and B) and the length of the hypotenuse (C). Does that somehow apply to angles as well?

    2. I've been studying this problem for a few weeks now trying to come up with a short hand that works. It seems the old-timers surly had some rule of thumb to quickly lay this out. I laid out several examples in CAD using three views to find the exact resultant angles. Then I looked for commonalities. I found that if I treated the rake and splay values as units, the resulting hypotenuse was very, very close to the actual resultant angle.
      The solution that I detailed above has a mathematical error of ~.05deg. That seems close enough in these relatively short distances.

    3. Hi Greg,
      you made my day (and it is early in the morning). The Pythagorean theorem was one of my first thoughts. But I was unsure if I could reflect that to the resultant angle. If I would have double checked the calculation result with the ruler results then I would have been sure earlier.
      Man, it can be so easy.
      I'm convinced that there have been some rules of thumb in the past. I've invested some time in gathering information about plank chairs and some of the old chair makers were just eyeballing the leg layout. Nevertheless there must have been the first time for them too, building a new chair and having the same question.

      I will update my sight lines and my drilling jig now.


      Talk soon,

    4. Glad to hear that it can be of some help. As Sylvain has pointed out, there is an increasing mathematical error as the angles become larger. For example:
      @20deg resultant the error is dining table height this translates to about 1/2" where the leg meets the floor.
      @30deg resultant the error is dining table height this translates to about 1-1/2" where the leg meets the floor.

      Is there error with this simple method of layout?...yes.
      It is entirely up to you if you can live with it or not.

    5. I already understood from the video that there is an error with increasing leg angles.
      Your explanation helped to make sure that I'm on the right path of understanding.
      Can I live with that error? I think yes. Keep in mind I'm hand tool only. I doubt that I'm able to drill with a brace so precisely.

  4. The TAN function is nearly linear for small angles, that is why Gregg's method gives "good enough" results.
    Although, if 0.5° is the acceptable error, it should not be used for splay and rake angles greater than about 15°.

    1. Hi Sylvain,
      thanks for pointing out the error.
      I think it is acceptable for this stool, even if it will be a prototype to figure out all this stuff.
      But I will keep it in mind for future projects.
      And meanwhile I could print out the "magic" ruler.

  5. Stephan,
    it looks to me like you have it rather well in hand. It is a simple calculation of triangles, but rather than turning to heavy math, as you noted, using the sightline ruler, you can simply lay out the rake and splay and derive the sightline and then measure the resultant. The sightline ruler is helpful, because unlike using a ruler and pythagoras, the sightline rule accounts for the difference in the spacing as the angles increase, no need to account for any math error that occurs using the uniform spacing of a ruler. As someone mentioned, the angles are in relation to the seat, so I usually do a quick drawing of the side view with the seat tilted and add in the leg angles in whatever looks good and then measure them in relation to the seat plane.

    1. Hello Peter,
      to be very honest, that is the most unexpected comment. :-)
      But I'm very glad to read your remarks.
      It was the combination of all parameters which drove me nuts for a while. So your video and blog post was pretty helpful. But there was this last bit missing.
      Looking forward to find some time this weekend to make some progress, revise my layout and shape the legs.
      Maybe we will talk again when the project will become more complete.