Miter Angle of a Pipe

Date: 02/10/99 at 23:57:33
From: Bruce Payan
Subject: Miter Angle of a Pipe

We are given a set of plans that call for the placement of a pipe that 
changes both direction and elevation.

A horizontal pipe aligned north-south takes a turn to the west and a 
drop in elevation. From an overhead view of the plan, the pipe takes 
a turn to the west of 9 degrees; from the side view of the plan, the 
pipe drops 12 degrees from the horizontal.

At what miter angle do we cut the pipe to connect these two pieces?

Date: 02/13/99 at 05:53:44
From: Doctor Mitteldorf
Subject: Re: Miter Angle of a Pipe

Here's a favorite formula of mine, which you can derive from straight 
geometric considerations if your visualization skills are good:

Take a book and open it to two facing pages. Draw an angle alpha to the 
spine on the left page, and angle beta on the right page. What is the 
angle theta between these two lines, and how does it depend on the 
angle phi to which the book is open?

If the book is open flat, then the two angles add and you have the 
sum-angle formula:
     
   cos(theta) = cos(alpha)*cos(beta) - sin(alpha)*sin(beta)

If the book is closed, then the two angles subtract and you have the 
angle difference formula:
   
   cos(theta) = cos(alpha)cos(beta) + sin(alpha)sin(beta)
   [SIC, the signs are + for difference, - for sum]

The general formula, with the book open to angle phi, is

   cos(theta) = cos(alpha)(*cos(beta)- sin(alpha)*sin(beta)*cos(phi)

which means that if the book is open to a right angle, you have simply:
         
   cos(theta) = cos(alpha)*cos(beta)

So, your miter angle is that angle whose cosine is 

   cos (12)*cos (9),

which is approximately 15 degrees or more.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/