Scope and Purpose
A friend who works in the log building industry once asked me if there was a formula for the angle between the intercept of two planes. Investigation of this question led to what was the ultimate 3-D puzzle. It wasn't always clear where to begin, and the road from start to finish often meandered, but eighteen years later, here's the answer ... so far ...
Others have done similar work. The timber framers of the middle ages, using only geometry, ratio and proportion, erected structures that have withstood the test of time to this day. Trigonometric formulas and diagrams detailing the applications of the various angles found in a complex Hip or Valley roof may be found at the Timber Framers Guild (Hawkindale Angles) and Steel-Link (Martindale and McKibben-Gray books) websites. Another excellent online resource is SBE Builders Online Tools ... the web based calculators are accompanied by diagrams and text descriptions of the developmental geometry underlying the script. Links have been inserted to these websites in order to facilitate research and give the reader alternate viewpoints and approaches to the mathematics of material joinery.
While every effort has been made to provide accurate information in this website, typographical and other errors can and do happen. This site is the result of independent research; the contents are excerpts from my notes and worksheets. If there are any mistakes, they are my mistakes, and in no way reflect on the efforts of any other individuals or organizations.
The math presented will hopefully give the reader insight into the theory required to calculate angles and dimensions in a Hip/Valley roof. However, considering the custom nature of log and timber joinery, it is the responsibility of the user to bridge the gap between theory and application. Geometry, vector analysis and trigonometry offer different means to the same end, and should be used to complement one another to test solutions.
Furthermore, creating matching lines and planes in a roof may satisfy aesthetic considerations, but does not guarantee the integrity of the joints. Common sense dictates that any proposed joinery be engineered by a qualified individual to ensure a safe, sound structure.