While going through the exercises in "Traité Théorique et Pratique de Charpente" by Louis Mazerolle it helps me to understand the Stereotomy geometry in 2D by extracting and unfolding tetrahedrons to make sure the 3D Stereotomy in the book is correct.
Here we have a Tetrahedron Extracted From an 8:12 pitched roof (33.69007°) Jack Rafter Side Cut Angle at the Hip Rafter.
Unfold each face.
First draw the line CE that is perpendicular to line AB, trait carré, square to a give line.
Next develop the slope of the tetrahedron by drawing a line from point C that is perpendicular to line CD, trait carré, square to a give line. The length of line FC = CE, then draw the line FD. The line CD is the common rafter run of the tetrahedron and the angle CDF is the slope of the tetrahedron.
By developing the slope triangle of the tetrahedron you can see all the roof framing angles developed from the unfolded tetrahedron.
The unfolded tetrahedron has
Common Rafter Slope Angle
Jack Rafter Side Cut Angle
Roof Sheathing Angle
Hip Rafter Slope Angle
Hip Rafter Backing Angle
Hip Rafter Square Tail Fascia Miter Angle
Purlin Miter Angle
Using the Roof Framing Geometry , Tetrahedron Angles, Google SketchUp Plugin and using
Rafter Pitch Angle = 66.9065291478°
Plan Angle = 39.7621583409°
we can go full circle and see the plan angle of 45° as the dihedral angle between the two planes.Next time you set your saw to 45° for the jack rafter side cut bevel angle you'll be developing a lot more than just the jack rafter side cut angle.