Determining an Angle From Side Views
Date: 07/24/2002 at 23:41:51 From: Bob Frost-Stevenson Subject: Compound angles I have searched everywhere, and can't come up with the answer. I need to know, what is the simple angle of two adjoining angles? I can probably explain a bit better. In the end elevation of a drawing, a rod rises from point A at 32 degrees. In the side elevation, the same rod is seen rising from point A at 48 degrees. How do I work out at what angle to cut the end of the rod? Is it simply adding the angles and dividing by two? Thank you very much.
Date: 07/25/2002 at 09:15:26 From: Doctor Rick Subject: Re: Compound angles Hi, Bob. It isn't as simple as that. The easiest way to derive the formula is to use analytic geometry. Let the rod start from the origin of a cartesian coordinate system, with the end elevation being the projection of the rod onto the y-z plane and the side elevation being the projection of the rod onto the x-z plane. I'm not sure whether your angles are the angles from the vertical or from the horizontal, but it turns out that the formula is simplest if all angles are from the vertical, so that's how I'll define the angles. The point on the rod at height 1 unit above the x-y plane has some coordinates (x,y,1). In the end elevation, this point has coordinates (y,1) and the tangent of the angle from the vertical, which I'll call alpha, is y/1. In the side elevation, the point has coordinates (x,1) and the tangent of the angle from the vertical (I'll call it beta) is x/1. The compound angle theta is the angle between the rod and the z axis. The distance between the point (x,y,1) and the z axis is r = sqrt(x^2+y^2), so tan(theta) = r/1 = sqrt(x^2 + y^2) = sqrt(tan^2(alpha) + tan^2(beta)) and our formula for the compound angle is theta = arctan(sqrt(tan^2(alpha) + tan^2(beta)) If the angles from the vertical in the two elevations are 32 degrees and 48 degrees, then alpha = 32 degrees; tan(alpha) = 0.624869 beta = 48 degrees; tan(beta) = 1.110613 tan^2(alpha) + tan^2(beta) = 0.624869*0.624869 + 1.110613*1.110613 = 1.623922 sqrt(tan^2(alpha) + tan^2(beta)) = 1.274332 theta = arctan(1.274332) = 51.87 degrees I hope this does the trick for you! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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