Saturday, April 5, 2014

Trirectangular Tetrahedron Study Joe Bartok

The Rules of the Trirectangular Tetrahedron Game
The Hip rafter are perpendicular to one another.

The sums of the Plan Angles are right angles: DDa + DDb = 90°
dda + ddb = 90°

For a given roof, the sum of the angles on the roof surface between the Hip rafter ridge lines is
fixed:

Σ P2 = arctan (cos SS ÷ tan DDa) + arctan (cos SS ÷ tan DDb)

Backing Angles Formulas:

C5a = arctan (sin R1a ÷ tan DDa) C5b = arctan (sin R1b ÷ tan DDb)
The Backing Angles are also related by the formulas: C5a = arctan (cos Σ P2 ÷ tan C5)
C5b = arctan (cos Σ P2 ÷ tan C5a)

The angles at the Hip rafter feet – R4B, R5B and A5B – are defined by the formulas for a standard
Hip-Valley Roof.

For a given roof of this type: R5Ba = R5Bb



Definitions of Angles, Formulas
and an Example Calculation

SS = Common Rafter Slope Angle
 = 31.71748°

DD = Plan Angle at Hip Rafter Foot
 = 31.71748°

dd = Plan Angle at Hip Rafter Peak
 = 40°
α = C5 = Hip Rafter Backing Angle
= 26.56505°

β = 90° – R1
= 90° – Hip Rafter Slope Angle
= 72°

μ = arctan (sin β   tan dd) = arctan (cos R1 tan dd)
arctan (sin 72° tan 40°) = 38.59096°

r1 = Section Plane Slope Angle
= arctan (tan SS sin (DD + dd))
= arctan (tan 31.71748° sin (31.71748° + 40°))
= 30.40596°

Blade Bevel for XDIH
= Blade Bevel along MIT Line
= Section Plane Backing Angle
= arctan (sin R1 ÷ tan (DD + dd))
= arctan (sin 30.40596° ÷ tan (31.71748° + 40°))
= 9.49278°




Projected Right Angles
= 90° ± arctan (cos dd ÷ tan ß)
= 90° ± arctan (cos dd tan R1)
= 90° + arctan (cos 40° ÷ tan 72°) = 103.97707°
= 90° – arctan (cos 40° tan 18°) = 76.02293°


ρ = r1 – arctan (cos dd tan R1)
= 30.40596° – arctan (cos 40° tan 18°) = 16.42889°

Blade Bevel for YDIH
= Blade Bevel along ß Line
= 90° – dd = 50°

Blade Bevel for ZDIH
= Blade Bevel along μ Line
= arctan (cos Projected Right Angle tan dd)
= arctan (cos 76.02293° tan 40°) = 11.45699°
= arccos (sin ß ÷ sin Projected Right Angle)
= arccos (sin 72° ÷ sin 76.02293°)
= 11.45699°

hα = hρ = .286364
hα =  (sin MIT)2 + (cos MIT tan μ) 2  – 2 sin MIT cos MIT tan μ cos α
hρ =  1 + (cos MIT ÷ cos μ) 2  – 2 cos MIT cos ρ ÷ cos μ





Non-Rectangular Sections Calculator Examples

Plan Angles at Peak arbitrarily set at 40° and 50°

18° Hip Entries:
Plan Angle at Peak = 40°
μ = arctan (cos 18° tan 40°) = 38.59096°
ß = 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 – (90° – 76.02293°) Projected Right Angle =  103.97707°
Supplementary Angle = 76.02293°
Blade Angle along MIT Line = 9.49278° 
Blade Angle along ß Line = 50.00000° 
Blade Angle along μ Line = 11.45699°


27.7323° Hip Entries: Plan Angle at Peak = 50°
μ  = arctan (cos 27.7323° tan 50°) = 46.52926°
ß = 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 – (90° – 71.32817°) Projected Right Angle =  108.67183°
Supplementary Angle = 71.32817°
Blade Angle along MIT Line =  9.49278° 
Blade Angle along ß Line = 40.00000° 
Blade Angle along μ Line = 20.88368°



90° Hip Run intersection bisected (45°)
Equal Hip Rafter Widths (solution by Sim Ayers)


18° Hip Entries:
Plan Angle at Peak = 45°
μ  = arctan (cos 18° tan 45°) = 43.563° (R4P)
ß = 72° (90° – R1)
α = 26.56505° (C5)

Returns (rounded off):
MIT =  42.64537° ... Angle on Roof Surface
BEV = 58.97305° ... 90°– Section Plane Angle
ρ = 18.08763° ... Section Plane Angle – (90° – 77.06068°) Projected Right Angle =  102.93932°
Supplementary Angle = 77.06068°
Blade Angle along MIT Line =  6.93760° 
Blade Angle along ß Line = 45.00000° 
Blade Angle along μ Line = 12.62142°


27.7323° Hip Entries: Plan Angle at Peak = 45°
μ  = arctan (cos 27.7323° tan 45°) = 41.51305° (R4P)
ß = 62.2677° (90° – R1)
α = 16.04506° (C5)

Returns (rounded off):
MIT =  39.08693° ... Angle on Roof Surface
BEV = 58.97305° ... 90°– Section Plane Angle
ρ = 10.63443° ... Section Plane Angle – (90° – 69.60748°) Projected Right Angle =  110.39252°
Supplementary Angle = 69.60748°
Blade Angle along MIT Line =  6.93759° 
Blade Angle along ß Line = 45.00000° 
Blade Angle along μ Line = 19.21087°


Standard Hip-Valley Angles
18° Hip Entries:
Plan Angle at Peak = 90° – DD = 58.28252°
μ = arctan (cos 18° tan 58.28252°) = 56.98260° (R4P)
ß = 72° (90° – R1)
α = 26.56505° (C5)

Returns (rounded off):
MIT =  54° (P2 ... Angle on Roof Surface)
BEV =  58.28253° (90°– SS ... Section Plane Angle)
ρ  = 22.02375° (SS – R5P)
Projected Right Angle =  99.69372° (90° + R5P) 
Supplementary Angle = 80.30628° (90° – R5P) Blade 
Angle along MIT Line =  0.00000°
Blade Angle along ß Line = 31.71748° (90° – DD ... Blade Bevel along 90° – R1)
Blade Angle along μ Line = 15.24016° (A5P ... Blade Bevel along R4P)

27.7323° Hip Entries:
Plan Angle at Peak = Peak = 90° – DD = 31.71748°
μ = arctan (cos 27.7323° tan 31.71748°) = 28.68049° (R4P)
ß = 62.2677° (90° – R1)
α = 16.04506° (C5)

Returns (rounded off):
MIT =  27.73230° (P2 ... Angle on Roof Surface) 
BEV = 58.28253° (90°– SS ... Section Plane Angle)
ρ  =  7.62264° (SS – R5P)
Projected Right Angle = 114.09484° (90° + R5P) 
Supplementary Angle = 65.90516° (90° – R5P) Blade 
Angle along MIT Line =  0.00000°
Blade Angle along ß Line = 58.28252° (90° – DD ... Blade Bevel along 90° – R1)
Blade Angle along μ Line = 14.16082° (A5P ... Blade Bevel along R4P)











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