Trirectangular Tetrahedron Tangent Prism

While going thru the Trèteaux Angles -- Trestle Angles -- trèteau à devers, Golden Rhombus Parallelogram Roof, Rhombicosidodecahedron Hip Rafters Study and Warlock Cut for Rotated Hip Rafters, Joe and I have looked at the trigonometric relationships of  geometric drawings of roof framing kernels folded and unfolded. Prismatic foot prints for rotated rafters. Prism Planes, folded and unfolded. Trirectangular Tetrahedrons for the hip rafter head cuts. By putting all of these drawings together it results in an geometric drawing of the tangent handrailing system. I envision W & A Mowat drawing something like these drawings if they had a 3D CAD program.

 They could start with a folded  roof framing kernel.

Develop the roof framing kernel into a prism.

 Develop the prism into a Trirectangular Tetrahedron.

Wire frame of Trirectangular Tetrahedron.

Develop the Trirectangular Tetrahedron into an roof surface with the pitch of plank for the tangent handrailing.

Then remove the one edge triangle of the Trirectangular Tetrahedron and label the direction of the ordinates for the direction of the minor and major axis of the ellipse on the roof surface.

Draw the minor axis of the ellipse on the roof surface.

Dimension the semi major axis on the roof surface. 

Semi Major axis of ellipse on roof Surface = Radius of Tangents ÷ cos(Pitch of Plank).

Then draw the ellipse on the roof surface.

Drawing showing the relationships of the roof framing kernel, prism plane, trirectangular tetrahedron  and the pitch of plank for the tangent handrailing.