Re: high school trigonometry?


"ram" <ram@xxxxxxxxxxxxxxxx> wrote in message
news:PJdhh.198$4Q6.32@xxxxxxxxxxx
hi!

I have been working on some formulas for compound miter cuts in carpentry.

The formulas and a diagram are presented at

http://www.woodcentral.com/bparticles/miter_formula.shtml

The assertion that the formulas could be derivied with high school math
was a challenge to me. I was able to derive a formula for the miter or
crosscut angle, but the bevel or blade tilt angle is eluding me.

I used unit vectors and 3D rotation matrices and the dot product operation
to calculate the angle between two planes starting perpendicular, then
both tilted back by the slope and finally one rotated to the angular bend
(as an excess to 90 degrees). This turns out to be twice the blade tilt as
one might expect, and the formula I got is

planesAngle = ArcCos[ Cos[Theta]^2 Sin[Phi] + Sin[Theta]^2 ]

Here my Theta is the supplement of the website's B or slope, and my Phi is
the supplement to the website's A or flat miter.

But the website formula is much nicer leading me to believe that something
could be done to my formulation to simplify it.

bladeTilt = ArcSin[Sin[B] Sin[A/2]]

I have looked at all the standard trig identities and fooled around a bit,
I can only make things worse. I would appreciate any help you can give on
this; I suppose I am admitting to giving up at this juncture.

--
rob

If you have access to a structural steel or miscellaneous iron fabricator,
ask their chief draftsman if he
has a copy of Martindale's Bevel Angles ( it's a privately printed reference
book used by structural steel
detailers ). If so, he'll probably let you copy the pages in the back of
the book for the formula
derivations. Lacking that, Google is your friend, type in "hip and valley
framing geometry" ( without the
quotes ) and follow the websites given. Also, this kind of trigonometry is
sometimes called "bin and
hopper geometry" and is used for plate work in fabricating bins and hoppers
of plate stock. It isn't
trig identities that you need -- it is an understanding of the solid
geometry of a fabricated compound angle.
Dray the joints and do the math -- nothing more that simple analytic
trigonometry is required.

I hope this helps,
Michael


.

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