Thursday, March 27, 2014

Golden Rhombus Parallelogram Roof #3

Joe Bartok's latest email on the Golden Rhombus

The key to this roof is to not only bisect the roof surface but to let it drive everything else – including the plan angles.

Quite the roof this is! First the common slope-common slope-corner angle format that's always served perfectly well in a Hip-Valley roof failed. Then the natural geometry following the 45° plan angles turned out to be a stumbling block. Is there a simple geometric or mathematical test that could have excluded these two possibilities and saved time?

If there is such a test I don’t know what it is, so I can’t honestly say I understand the geometry of the Golden Rhombus roof. And can we do this roof surface angle bisect adjustment on any roof where the slopes and plan angles conspire to create really narrow cuts at the Hip rafter peaks?

Something I have mastered in the past month or so is the fine art of using an eraser. Before the prismoidal solids, trestle (Tréteaux) angles, more claw angles on rafters and purlins, intersecting Hip rafters and the Golden Rhombus Roof the two “Undo Buttons” were virtually identical twins. Although it’s possible I may have been unconsciously gnawing on one of them in either deep thought or frustration

 “Undo Buttons”, before and after

Sim – I think I’ve got it and after all that farting around it turns out to be so easy. I can't find any flaws in the geometry or trigonometry.

Last night I revisited half-splitting the 81.7323° angle on the roof surface. Since I don't trust how the formulas apply to this roof I began with vectors calculated only from the most elementary dimensions seen in plan and elevation. At this stage I did not drag the backing angles into the math and only applied the 40.86615° roof surface angle indirectly to solve the plan angles.

Splitting the roof surface angle into equal 40.86615° parts produces 2 × 42.94378° plan angles at the 18° Hip rafter peak and 2 × 47.05622° plan angles at the peak of the 27.7323° Hip rafter.

The "R5P" angles determined the lines for "R4P", and list of perpendiculars to the unbacked Hip surfaces were used to find the "A5P" blade bevels along the R4P lines. Interesting how the R4P angles have exchanged position with respect to the Hips, compared to the previous models. The remaining angles were solved with the Compound Angle Formulas.

18° Hip rafter:
Linear algebra (vectors)
R5P = arctan (tan 18° cos 42.94378°) =13.37912°
R4P = 41.513056° (Angle of Saw Travel)
A5P = 12.153252° (Blade Bevel for R4Pm)
Compound Angle Formulas
90° – R1 = 72° (Angle on Adjoining Face)
90° – DD = 47.05622° (Blade Bevel for 90° – R1)
90° – R5P = 76.62088° (Footprint or Trace Angle)

27.7323° Hip rafter:
Linear algebra (vectors)
R5P = arctan (tan 27.7323° cos 47.05622°) = 19.70602°
R4P = 43.562996° (Angle of Saw Travel)
A5P = 19.915881° (Blade Bevel for R4Pm)
Compound Angle Formulas
90° – R1 = 62.2677° (Angle on Adjoining Face)
90° – DD = 42.94378° (Blade Bevel for 90° – R1)
90° – R5P = 70.29398° (Footprint or Trace Angle)

The developments of the models began with only the 40.86615° roof surface angle and the 26.56505° (18° Hip) and 16.04506° (27.7323° Hip) backing angles.
The R4P and 90° – R5P angles followed naturally from the model dimensions.

To summarize, the fundamental attributes of the roof are unchanged:
The C5s (backing angles) are not affected by any calculations or proposed cuts
Sum of P2s (angles on roof surface) = 81.7323°
Sum of the Blade Bevels along the Hip plumb lines = 90°

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