Vector Models of Square Tail Fascia
Main Pitch = 7/12       Adjacent Pitch = 12/12
Angle in Plan between Eaves = 90°
Square Tail Fascia Vector Model: Plan Angles
Vector Model of Square Tail Fascia
Square Tail Fascia superimposed on Vector Model
Terminal Points of Vectors
Main Eave : ( any x , 0 , 0 ) will suffice; Unit Vector ( 1 , 0 , 0 ) is preferred.
The diagrams show ( cos Main Plan Angle , 0 , 0 ).
Adjacent Eave : Unit Vector
( cos Plan Angle between Eaves , sin Plan Angle between Eaves , 0 )
The Ridge Line co-ordinates are with respect to Unit Runs on the x-y plane
Hip Ridge Line :
( cos Main Plan Angle , sin Main Plan Angle , tan Hip Pitch Angle )
Fascia Ridge Line :
( cos Fascia Plan Angle , sin Fascia Plan Angle , tan Fascia Ridge Pitch Angle )
Square Tail Fascia Vector Model: Angle between Ridge Lines
Vector Product Solution of Saw Blade Bevels
Main (7/12) Side Fascia Saw Blade Bevel
Perpendicular to plane bounded by Hip Rafter Ridge and Fascia Ridge
r = Hip Rafter Ridge × Fascia Ridge : ( –1 , .87047 , –.49223 )
Perpendicular to plane bounded by Main Eave and Fascia Ridge
m = Main Eave × Fascia Ridge : ( 0 , .43523 , .25389 )
Angle between perpendiculars = Dihedral Angle between defined planes
arccos ( r · m ÷ |r| |m| ) = .25388 ÷ .71258 = 69.12792°
Saw Blade Bevel = 90° – 69.12792° = 20.87208°

Adjacent (12/12) Side Fascia Saw Blade Bevel
Perpendicular to plane bounded by Hip Rafter Ridge and Fascia Ridge
r = Fascia Ridge × Hip Rafter Ridge : ( –1 , .87047 , –.49223 )
Perpendicular to plane bounded by Adjacent Eave and Fascia Ridge
a = Adjacent Eave × Fascia Ridge : ( –.86378 , 0 , –.86378 )
Angle between perpendiculars = Dihedral Angle between defined planes
arccos ( r · a ÷ |r| |a| ) = 1.28896 ÷ 1.72756 = 41.74496°
Saw Blade Bevel = 90° – 41.74496° = 48.25504°

Trigonometric Solution of Saw Blade Bevels
Sheathing Angle and Fascia Angle on the Stick
For any Compound Angle to be cut on a stick rectangular in section:
Compound Angle Formulas
Main (7/12) Side Calculations
Main Sheathing Angle = 63.25779°
Main Fascia Angle on the Stick = 49.18024°
Main Fascia Saw Blade Bevel
= arctan ( sin 49.18024° ÷ tan 63.25779° ) = 20.87256°
The Main Side Fascia matches the Ridge Lines
Angle between Ridge Lines
= arccos ( cos 49.18024° × cos 63.25779° ) = 72.89392°

Adjacent (12/12) Side Calculations
Adjacent Sheathing Angle = 39.52120°
Adjacent Fascia Angle on the Stick = 67.58488°
Adjacent Fascia Saw Blade Bevel
= arctan ( sin 67.58488° ÷ tan 39.52120° ) = 48.25487°
The Adjacent Side Fascia matches the Ridge Lines
Angle between Ridge Lines
= arccos ( cos 67.58488° × cos 39.52120° ) = 72.89392°

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Joe Bartok

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