Determining Compound-Rake by Construction
Where the rake is the combination of two angles (Compound-Rake) the actual slope of the rake is a transcendent trigonometric function of those two angles. I ran off with the Theater to avoid having to deal with Algebra; and Trig is Algy’s bigger, meaner older brother. This construction derives the correct angle (true dip perpendicular to the strike line) using only a compass and straightedge. No calculation required.
Note: If the slope angles are given as Run/Rise this is very straightforward (but you will have to measure.) Mark the two unit rises and draw the two angles. If the slope angles are given in degrees or you are dealing with wild numbers (or to fit an as-built); begin with the angles, copied with a bevel gauge or generated from a protractor, and extend them through the perpendicular to obtain AC and BC. To obtain ½ AC or BC use this construction; or, if you have easy numbers you can measure.
The method for determining compound-rake by construction
On a line of suitable length select an origin (O).
At one unit of run, erect a perpendicular.
From the origin erect the first angle (AOC).
From the origin strike an arc through AO with the radius ½ of AC.
From this intersection erect a perpendicular to AO a suitable distance.
Repeat this process for the second angle (BOC) using ½ BC for the radius struck through BO.
From this intersection erect a perpendicular to BO to where it intersects the perpendicular from AO at D.
Strike an arc with the radius DO from C to establish point E.
the angle EOC is the “true dip” The actual slope (rake)
If you are solving the “Stovepipe Problem” (previous post) to determine the ellipse needed to pass a cylinder through a compound raked plane; Use this derived angle in the previous construction for the “Angle of Rake.” That will give the correct ellipse for the compound rake.
This construction will sometimes also be useful in the following: Determining the Counter Rake for Work That Lies Diagonally Athwart the Plane of the Rake (next post)