Radford's cyclopedia of construction:carpentry, building and architecture, based on the practical experience of a large staff of experts in actual construction work, Volume 8 (Google eBook) printed in 1903
authors, William A. Radford, Alfred Sidney Johnson
It's probably on of the best book on Roof Framing in the last 100 years.It has resources for finding the plan angles for unequal pitched roofs with a framing square and finding the back bevel cuts for hip and jack rafters using a form of shiften from plan view.
Hexenschnitt -- The Witches Cut, Square Tail Fascia
Radford has several drawings in the book that deal with square tail fascia. He uses the Co-Pitch to find the hip rafter miter angle. Radford's drawing look a lot like the German method that I use. Or the German method looks a lot like Radfords method.
Preface from the book:
Open Ways to Knowledge. — Of late years a number of books have been published, in which the authors and compilers have made commendable efforts to simplify matters pertaining to the arts of carpentry and joinery, and the mechanic of today has not the difficulties of his predecessors to contend with. The workman of old could excuse his ignorance of the higher branches of his trade, by saying that he had no means of acquiring a knowledge of them. Books were beyond his reach, and trade secrets were guarded so jealously, that only a limited few were allowed to know them, and unless he was made of better stuff than...
Here's a couple of drawings I reproduced from Radfords drawings in his book on the Jack Rafter Shift Dimension in Profile for different polygon plan angles.The jack rafter shift dimension is different for each of the polygon plan angle, but the dimension stays the same for all of the different roof pitches for each type of polygon.
This drawing shows the Plumb Line Shift Dimension for:
- Square Pyramid (4 sides) = 90°
- Pentagon (5 sides) = 108°
- Hexagon (6 sides) = 120°
- Octagon (8 sides) = 135°
- Decagon (10 sides) = 144°
- Dodecagon (12 sides) = 150°
- Hexadecagon (16 sides) = 157.5°
Drawing of the Jack Rafter Plumb Line Shift Dimension for Equal Sided Octagon Roofs.