On Sun, 10 Mar 2013 10:09:02 -0700, Christoph J. WALTHER wrote:
> With a miter saw I want to cut the end of a beam (fig. 1 in the sketch I posted here: http://goo.gl/6Fqtk ) so that two miter cuts (arrow heads, fig. 3) at 54.4° result at their intersection in a given angle (base angle, fig. 2 and fig. 4) of 10.3° in my example (for a more visual description one might check this video here: https://www.youtube.com/watch?v=R31PlxhhBFg where, however, it gets omitted to mention that bevel angle need to be applied). I did a test cut with setting the bevel angle at 5°, tilting right and left respectively for each of the cuts, what resulted in a base angle of approx. 8°. So the approach seems to be correct, but I need to be able to control it and hence I need help in calculating the bevel angle to set at the miter saw when cutting the two miter angles. Any help, even hints pointing into the right direction, is greatly appreciated! TIA!!!
For a board that is s units high, the heel of the cut (in Figure 2) is set in by a distance x = s sin(b), where b is your 10.3° (10.3 degrees) angle. Let a=54.4°. If your board is sitting on a horizontal plane T and you drop a vertical line segment of length s from the corner near the lower 54.4° label in Figure 3, it will impinge T at a point P a distance x from the heel. On T we can draw a right triangle with side x cos(a), angle a, hypotenuse x, and side u = x sin(a). The first side mentioned is part of an arrow line on T, and would be like in Figure 3 but with the arrow lines translated distance x to the right. Anyhow, we have a right triangle with side s next to the bevel angle and side u opposite the bevel angle, which gives bevel angle m = arctan(u/s).
The following Python 2 code computes m = 8.27° with s=3.5 and a, b as above:
from math import sin, pi, atan2 def d2r(d): return d*pi/180 s, a, b = 3.5, 54.4, 10.3 x = s * sin(d2r(b)); print x u = x * sin(d2r(a)); print u m = atan2(u, s); print m*180/pi
The above code prints out: 0.625807752907 0.508844760158 8.27194676088
which indicates that the heel inset is about 5/8" and the bevel angle 8.27°
Question: Why is your main angle 54.4° instead of 54° ? It appears that 54.4° gives you an arrow angle of 180-2*54.4 = 71.2° instead of a proper pentagon angle of 72°.