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|
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|
|Rake of the front right leg is the angle off vertical when viewing from the side|
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|
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|
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.