Wednesday, March 26, 2014

Golden Rhombus Parallelogram Roof #2

The hip rafter head cuts just don't seem to fit correctly on the Golden Rhombus Parallelogram Roof test model. So, I needed to re-visit what defines the hip rafter head cut miter line that develops the top back bevel angle on the unbacked hip rafter.

Using the ground plan of a pentagon and traditional layout geometry for the bevel angles on the hip rafters at the peak of the hip rafter everything works out as expected. The geometry matches the roof framing trigonometric ratios as well.

Platonic Solid ICOSAHEDRON
Deck: Pentagon
Roof Surface Faces: 5 x Equilateral Triangles
Deck Angle = 108°
Plan DD = 54.00°
Common Rafter Slope Angle SS = 37.37737°
Hip Rafter Slope Angle R1 = 31.71747°

Dihedral Angle = 138.16038°
Hip Rafter Backing Angle = 20.91981°

Platonic Solid Edges
Hip Rafter Miter Angle = 31.71747°
Hip Rafter Bevel Angle = 58.2825°
Hip Rafter Saw Blade Bevel Angle = 54.00°

Roof Eave Angle = 108.00000
SS = Main Rafter Slope Angle = 37.37737
S = Adjacent Rafter Slope Angle = 37.37737
DD = Main Plan Angle = 54.00000
D = Adjacent Plan Angle = 54.00000
R1 = Hip Rafter Slope Angle = 31.71748
C5m Main Hip Rafter Backing Angle = 20.90516
C5a Adjacent Hip Rafter Backing Angle = 20.90516
P2m = Main Jack Rafter Side Cut Angle = 30.00000
90° - P2m = Main Roof Sheathing Angle = 60.00000
P2a = Adjacent Jack Rafter Side Cut Angle = 30.00000
90° - P2a = Adjacent Roof Sheathing Angle = 60.00000
R4 = 31.71747°
R5B = 19.96463°
A5B = 25.17126°
R5P = 26.56505°
A5P = 18.00000°
R4B = 49.49929°
R4P = 31.71747°



With a square ground plan the geometry matches the roof framing trigonometric ratios.

Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 31.74747
S = Adjacent Rafter Slope Angle = 31.74747
DD = Main Plan Angle = 45.00000
D = Adjacent Plan Angle = 45.00000
R1 = Hip Rafter Slope Angle = 23.63071
C5m Main Hip Rafter Backing Angle = 21.84290
C5a Adjacent Hip Rafter Backing Angle = 21.84290
P2m = Main Jack Rafter Side Cut Angle = 40.37702
90° - P2m = Main Roof Sheathing Angle = 49.62298
P2a = Adjacent Jack Rafter Side Cut Angle = 40.37702
90° - P2a = Adjacent Roof Sheathing Angle = 49.62298
R4 = 42.49429°
R5B = 17.19096°
A5B = 16.46543°
R5P = 17.19096°
A5P = 16.46543°
R4B = 42.49429°
R4P = 42.49429°




Here's another square ground plan, but with unequal roof slope angles.







Hip Rafter A
Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 45.00000
S = Adjacent Rafter Slope Angle = 60.00000
DD = Main Plan Angle = 60.00000
D = Adjacent Plan Angle = 30.00000
R1 = Hip Rafter Slope Angle = 40.89339
C5m Main Hip Rafter Backing Angle = 20.70481
C5a Adjacent Hip Rafter Backing Angle = 48.59038
P2m = Main Jack Rafter Side Cut Angle = 22.20765
90° - P2m = Main Roof Sheathing Angle = 67.79235
P2a = Adjacent Jack Rafter Side Cut Angle = 40.89339
90° - P2a = Adjacent Roof Sheathing Angle = 49.10661
Main
P4 = 19.28633°
R5B = 23.41322°
A5B = 34.53758°
R5P = 36.86990°
A5P = 19.10661°
R4B = 52.62876° correct
R4P = 23.57818° ... wrong

Adjacent
R4 = 52.62876°
R5B = 36.86990°
A5B = 19.10661°
R5P = 23.41322°
A5P = 34.53758°
R4B = 23.57818° ... correct
R4P = 52.62876° ... wrong

From the 3D drawing and the geometry
R4Pm = 38.27204°
R4Pa = 22.75094°



Update
I need a way of changing
R1 = Hip Rafter Slope Angle B = 35.26439
to
R1 = Hip Rafter Slope Angle B = 34.35489
_______________________________________________________________
Roof Eave Angle = 105.00000
SS = Main Rafter Slope Angle = 40.89339
S = Adjacent Rafter Slope Angle = 34.35489

DD = Main Plan Angle = 43.77559
D = Adjacent Plan Angle = 61.22441

R1 = Hip Rafter Slope Angle = 30.92781
C5m Main Hip Rafter Backing Angle = 28.20944
C5a Adjacent Hip Rafter Backing Angle = 15.76254
Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 46.02802

P2m = Main Jack Rafter Side Cut Angle = 38.27161
90° - P2m = Main Roof Sheathing Angle = 51.72839
P2a = Adjacent Jack Rafter Side Cut Angle = 24.38937
90° - P2a = Adjacent Roof Sheathing Angle = 65.61063

Hip Rafter A side Cut Angle R4P = P2m
Hip Rafter B side Cut Angle R4P = P2a



_____________________________________________________________________



Hip Rafter B
Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 45.00000
S = Adjacent Rafter Slope Angle = 45.00000
DD = Main Plan Angle = 45.00000
D = Adjacent Plan Angle = 45.00000
R1 = Hip Rafter Slope Angle = 35.26439
C5m Main Hip Rafter Backing Angle = 30.00000
C5a Adjacent Hip Rafter Backing Angle = 30.00000
P2m = Main Jack Rafter Side Cut Angle = 35.26439
90° - P2m = Main Roof Sheathing Angle = 54.73561
P2a = Adjacent Jack Rafter Side Cut Angle = 35.26439
90° - P2a = Adjacent Roof Sheathing Angle = 54.73561
R4 = 39.23152°
R5B = 26.56505°
A5B = 24.09484°
R5P = 26.56505°
A5P = 24.09484°
R4B = 39.23152°... correct
R4P = 39.23152°... wrong

R4P = 24.15203°... from drawing




Hip Rafter C
Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 45.00000
S = Adjacent Rafter Slope Angle = 60.00000
DD = Main Plan Angle = 60.00000
D = Adjacent Plan Angle = 30.00000
R1 = Hip Rafter Slope Angle = 40.89339
C5m Main Hip Rafter Backing Angle = 20.70481
C5a Adjacent Hip Rafter Backing Angle = 48.59038
P2m = Main Jack Rafter Side Cut Angle = 22.20765
90° - P2m = Main Roof Sheathing Angle = 67.79235
P2a = Adjacent Jack Rafter Side Cut Angle = 40.89339
90° - P2a = Adjacent Roof Sheathing Angle = 49.10661
Main
P4 = 19.28633°
R5B = 23.41322°
A5B = 34.53758°
R5P = 36.86990°
A5P = 19.10661°
R4B = 52.62876° correct
R4P = 23.57818° ... wrong

Adjacent
R4 = 52.62876°
R5B = 36.86990°
A5B = 19.10661°
R5P = 23.41322°
A5P = 34.53758°
R4B = 23.57818° ... correct
R4P = 52.62876° ... wrong

From the 3D drawing and the geometry
R4Pm = 38.27204°
R4Pa = 22.75094°




Hip Rafter D
Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 60.00000
S = Adjacent Rafter Slope Angle = 60.00000
DD = Main Plan Angle = 45.00000
D = Adjacent Plan Angle = 45.00000
R1 = Hip Rafter Slope Angle = 50.76848
C5m Main Hip Rafter Backing Angle = 37.76124
C5a Adjacent Hip Rafter Backing Angle = 37.76124
P2m = Main Jack Rafter Side Cut Angle = 26.56505
90° - P2m = Main Roof Sheathing Angle = 63.43495
P2a = Adjacent Jack Rafter Side Cut Angle = 26.56505
90° - P2a = Adjacent Roof Sheathing Angle = 63.43495
R4 = 32.31153°
R5B = 40.89339° ... slope angle of adjoining hip rafter
A5B = 33.21091°
R5P = 40.89339° ... slope angle of adjoining hip rafter
A5P = 33.21091°
R4B = 32.31153° ... correct
R4P = 32.31153° ... wrong

From the 3D drawing and the geometry
R4Pm = 68.45626°
R4Pa = 68.45626°







Golden Rhombus Parallelogram



Hip Rafter A
Roof Eave Angle = 63.43490
SS = Main Rafter Slope Angle = 31.71750
S = Adjacent Rafter Slope Angle = 31.71750
DD = Main Plan Angle = 31.71745
D = Adjacent Plan Angle = 31.71745
R1 = Hip Rafter Slope Angle = 18.0000
C5m Main Hip Rafter Backing Angle = 26.56508
C5a Adjacent Hip Rafter Backing Angle = 26.56508
P2m = Main Jack Rafter Side Cut Angle = 54.0000
90° - P2m = Main Roof Sheathing Angle = 36.00
P2a = Adjacent Jack Rafter Side Cut Angle = 54.0000
90° - P2a = Adjacent Roof Sheathing Angle = 36.00
R5B = 15.45045°
A5B = 9.34970°
R5P = 9.69372°
A5P = 15.24017°
R4B = 30.44636°... correct
R4P = 56.98263°... wrong

From the 3D drawing and traditional layout geometry
R4P = 43.56300°



Hip Rafter B
Roof Eave Angle = 116.56510
SS = Main Rafter Slope Angle = 31.71750
S = Adjacent Rafter Slope Angle = 31.71750
DD = Main Plan Angle = 58.28255
D = Adjacent Plan Angle = 58.28255
R1 = Hip Rafter Slope Angle = 27.73233
C5m Main Hip Rafter Backing Angle = 16.04506
C5a Adjacent Hip Rafter Backing Angle = 16.04506
P2m = Main Jack Rafter Side Cut Angle = 27.73227
90° - P2m = Main Roof Sheathing Angle = 62.26773
P2a = Adjacent Jack Rafter Side Cut Angle = 27.73227
90° - P2a = Adjacent Roof Sheathing Angle = 62.26773
R5B = 15.45045°
A5B = 23.31858°
R5P = 24.09488°
A5P = 14.16082°
R4B = 55.07574°... correct
R4P = 28.68045° ... wrong

From the 3D drawing and traditional layout geometry
R4P = 41.51304°





Joe Bartok's pentahedron models .
Development of the Pentahedron Extracted from the Peak of the Hip Rafter











The gap in between the models of the pentahedron is the reason the hip rafter head cuts don't work. The plumb hip rafter head cut angles or the bevel angles on top of the hip rafters need to be changed.



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