Thursday, April 17, 2014

Pentahedron and Non-Rectangular Sections

Pentahedron and Non-Rectangular Sections Study by Joe Bartok

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warlock_rhombic_nr_solution.pdf

warlock_rhombic_javascript_solution.pdf
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When I compare the Golden Rhombus and the current Offset Rotated Rafters this Compound Angle Formula works all around:
μ = arctan (sin Angle at Rafter Peak ÷ tan Blade Bevel Angle)

Golden Rhombus (my study half-split the roof angle but the formula works just as well for your equal width Hips solution):
μ = arctan (sin (90° – 27.73230°) ÷ tan (90° – 47.05622°) = 43.56300°
β = 90° – R1 = 90° – 27.73230° = 62.26770°
α = C5 = 16.04506°

70° Offset 12/12 Side Rafter ... Claw Angle Version:
μ = arctan (sin 28.71825° ÷ tan 5.36467°) = 78.94194° (Angle on Upper Shoulder of Offset Rafter)
β = 28.71825° (Upper Claw Angle)
α = 13.99545° (Rotated Rafter Backing Angle)

70° Offset 12/12 Side Rafter ... Plumb Line Version:
μ = arctan (sin (90° – 43.21918°) ÷ tan 16.86990°) = 67.40640° (Angle on Upper Shoulder of Offset Rafter)
β = 90° – R1 = 90° – 43.21918° = 46.78082°
α = 13.99545° (Rotated Rafter Backing Angle)

Looking good, but a long way to go yet.
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Development of a Pentahedron describing a Non-Rectangular Section, given μ, β and α. After the first few steps the rest of the drawing falls into place.

Setting a different reference length = 1 has produced a few new relationships:
ρ = (90° – BEV) – arctan (cos YDIH ÷ tan β)
Blade Bevel for YDIH = arcsin (tan β ÷ tan (BEV + ρ))
MIT = arctan (sin μ tan β ÷ (cos α cos μ tan β + sin μ sin α))

Test firing the formulas on real Hip roof studies:

Golden Rhombus ... Half-split Roof Angle
μ = arctan (sin (90° – 27.73230°) ÷ tan (90° – 47.05622°) = 43.56300°
β = 90° – R1 = 90° – 27.73230° = 62.26770°
α = C5 = 16.04506°
ρ = Offset Rafter Slope Angle – arctan (tan Main Hip Slope Angle sin Blade Bevel
ρ = 30.79574° – arctan (tan 27.73230° sin (90° – 47.05622°) = 11.08972°
MIT = arctan (sin 43.56300° tan 62.26770° ÷ (cos 16.04506° cos 43.56300° tan 62.26770° + sin 43.56300° sin 16.04506°)) = 40.86615°

70° Offset 12/12 Side Rafter ... Claw Angle Version:
μ = arctan (sin 28.71825° ÷ tan 5.36467°) = 78.94194° (Angle on Upper Shoulder of Offset Rafter)
β = 28.71825° (Upper Claw Angle)
α = 13.99545° (Offset Rafter Backing Angle)
ρ = (90° – Trace Angle on Main Hip) – arctan (tan (90° – Upper Claw Angle) sin Blade Bevel)
ρ = (90° – 55.50766°) – arctan (tan (90° – 28.71825°) sin 5.36467°) = 24.80855°
MIT = arctan (sin 78.94194 tan 28.71825 ÷ (cos 13.99545 cos 78.94194 tan 28.71825 + sin 78.94194 sin 13.99545)) = 57.74662°

70° Offset 12/12 Side Rafter ... Plumb Line Version:
μ = arctan (sin (90° – 43.21918°) ÷ tan 16.86990°) = 67.40640° (Angle on Upper Shoulder of Offset Rafter)
β = 90° – R1 = 90° – 43.21918° = 46.78082°
α = 13.99545° (Offset Rafter Backing Angle)
ρ = Main Hip Slope Angle – arctan (tan Offset Rafter Slope Angle sin Blade Bevel)
ρ = 30.96374° – arctan (tan 43.21918 sin 16.86990°) = 15.71018°
MIT = arctan (sin 67.40640° tan 46.78082 ÷ (cos 13.99545 cos 67.40640° tan 46.78082 + sin 67.40640° sin 13.99545)) = 57.74661°

There doesn’t appear to be a pattern for the formulas for ρ in terms of the roof angles, but:
... 30.79574° and 30.96374° are really 90° – Trace Angles for their respective rafters
... Main Hip Slope Angle and Offset Rafter Slope Angle are 90° – Upper Claw Angle are the complements of the angles at their respective rafter peaks.

ρ = 90° – Trace Angle – arctan( sin Blade Bevel ÷ tan Angle at Rafter Peak)???

Penathedron and Non-Rectangular Section Test
Intersecting Hip Rafters – Warlock Cut, Upper Shoulder Hip B
Angle on Hip B Upper Left Shoulder = 37.65287°
Blade Bevel @ 37.65287° = 55.39851°

μ = arctan (sin 37.65287° ÷ tan 55.39851°) = 22.85244° = Angle on Hip B Upper Right Shoulder ... expected that, the rafter section is rectangular
β = 37.65287° = Angle on Hip B Upper Left Shoulder
α = 0° ... at first glance it might seem like this should be 45°, but there is no backing or rotation angle here

MIT =  37.65287°
BEV = 43.14862° (Trace Angle on Hip A)
ρ = 0°
Projected Right Angle = 136.85138°
Supplementary Angle = 43.14862° (= Trace Angle on Hip A)
Blade Bevel for XDIH = 55.39850°
Blade Bevel for YDIH = 26.71762° (Blade Bevel @ 22.85244°)
Blade Bevel for ZDIH = 55.39850° (= Blade Bevel for XDIH)

The code returned angles for a tetrahedron (which we already know works here). This isn't telling me anything new or lending insight as to an easier means of drawing the intersection.




Although the math takes different routes the following methods agree with one another, this is what I like to see!
– find angles and dimensions to DP lines, complete calculations with Law of Cosines
– solve as for rectangular section with Compound Angle Formulas, apply bevel with Non-Rectangular Section calculator

– solve angles with only the Roof Framing and Joinery Calculator










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