The focus is on concise, generic trigonometric methods of resolving the angles required for accurate joinery in a Hip / Valley roof. Diagrams are used extensively to supplement the abstract trigonometry. The following ideas are explored:

All kernels are equivalent (Cognate kernels)
The sine, cosine, and tangent of any kernel angles may be resolved in terms of two other angles
Compound angle relationships on the stick
The location of right angles on the roof, on the kernel, and on the stick
Definition, measurement and relationships between Dihedral, Saw blade, Miter and Bevel angles
General solutions of Deck angles
Utilizing a unit reference length to determine trig functions of angles
Extracting kernels from the stick
Extracting kernels from Interlocking kernels
Extracting sticks from kernels
Superimposing sticks on kernels
Models other than tetrahedral kernels

    ANGLE CALCULATORS   contains links to Excel worksheets and Javascript calculators which will solve the complex roof framing and joinery angles based on the theory in this section.