Golden Rhombus Parallelogram Roof #5

Joes' drawings and notes of splitting the roof surface of the Golden Rhombus Parallelogram Roof. This method is similar to the Rhombicosidodecahedron Hip Rafters blog where the roof is cut into roof surface pieces, triangles, squares, pentagons, with the miter line of the roof  being  the sum of the jack rafter side cut angles at the peak  divided by 2. The hip rafters will be different widths, but for the parallelogram the hip rafters are centered on the hip rafter run line. This method also results in the width of the hip rafter backing length on the roof surface of the hip rafters being equal.

Here's a drawing showing this method. There's a difference in the width of the hip rafters, but it's not much of a difference.

There's 3 ways of mitering hip rafters and this method can be used on polyhedron's as well, where the plan angles are not equal, like the Rhombicosidodecahedron.

3 different ways of cutting the hip rafters head cuts on any roof surface.

1. The normal way using R4P, hip rafter side cut angle at peak of hip rafter, but the hip rafter miter cuts will be different lengths for plan angles that are not equal.
2. Using the hip rafter miter line in plan view, like my online script, or using traditional layout geometry.
3. Bisecting the roof surface, like Joe's drawings. 

Joe's Non-Rectangular Sections Calculator

18° Hip Entries:
Plan Angle at Peak = 40°
μ = arctan (cos 18° tan 40°) = 38.5909583° (R4P)
β = 72° (90° – R1)
α = 26.56505° (C5)
Returns (rounded off):
MIT =  38.30262° (Angle on Roof Surface)
BEV = 59.59404° (90°– Section Plane Angle)
ρ = 16.42889° (Section Plane Angle – R5P)
Projected Right Angle =  103.97707° (90° + R5P)
Supplementary Angle =  76.02293° (90° – R5P)
Blade Angle along MIT Line = 9.49278° (Section Plane Backing Angle)
Blade Angle along β Line = 50.00000° (90° – DD ... Blade Bevel along 90° – R1)
Blade Angle along μ Line = 11.45699° (A5P ... Blade Bevel along R4P)

27.7323° Hip Entries:
Plan Angle at Peak = 50°
μ = arctan (cos 27.7323° tan 50°) = 46.52926239° (R4P)
β = 62.2677° (90° – R1)
α = 16.04506° (C5)

Returns (rounded off):
MIT =  43.42969° (Angle on Roof Surface)
BEV = 59.59404° (90°– Section Plane Angle)
ρ = 11.73413° (Section Plane Angle – R5P)
Projected Right Angle =  108.67183° (90° + R5P)
Supplementary Angle = 71.32817° (90° – R5P)
Blade Angle along MIT Line =  9.49278° (Section Plane Backing Angle)
Blade Angle along β Line = 40.00000° (90° – DD ... Blade Bevel along 90° – R1)
Blade Angle along μ Line = 20.88368° (A5P ... Blade Bevel along R4P)

1.