Hawkindale Angles and the Prism Plane

While studying Tangent Hand Railing system the first thing that you need to study is prisms. After drawing out the following prism for the Hawkindale angles, S, D, R1, R2, R3, R4, R5, P1, P2, P3, P5,P6,A9 and A7, I would now say that the first thing that you need to do while studying compound joinery for timber framing is learn how to draw out the prism for the roof slope and deck angle that you're working with.

Prism Plane = Unbacked shoulder of Hip Rafter & Valley Rafter
Prism Plane = Hip Rafter & Valley Rafter diamond post

Vertical Rotation Plane  = arctan(tan Vertical Plane x cos Horizontal Plane)
Horizontal Rotation Plane  = arctan(cos Vertical Plane ÷ tan Horizontal Plane )

Hinge Rotation Plane  = arctan( tan Horizontal Plane  ÷   cos Vertical Plane)


R1 = Vertical Plane of adjoining roof planes.
Hawkindale R1 = Arctan [Tan S x Sin D]
R1 = arctan(tan SS x cos D)
R1 = arctan(tan S x cos DD)
R1 Plumb Plane = S Plumb Plane rotated by the adjacent plan angle

R2 = Purlin Housing Angle.
Hawkindale R2 = Arctan [Sin S x Cos S x Cos D / Tan D]
R2 =  90° - arctan(sin(plan angle) x tan(purlin rotation angle - roof slope angle) + hip slope angle
Purlin Rotation Angle = 90°
R2 =  90° - (arctan(sin(DD) x tan(90° - SS) + R1)
R2 =  90° - (arctan(cos(D) x tan(90° - SS) + R1)
R2 =Intersection of R1 Plane and  a Perpendicular Plane to P2 Plane

R3 = Line traced on top or bottom of unbacked hip rafter following the perpendicular plane of the purlin.
Hawkindale R3 = Arctan [Sin S x Cos S x Cos D / Sin R1]
R3 = arctan( tan( Jack Rafter Side Cut Angle ) * cos ( Hip Rafter Backing Angle ))
R3 = arctan( tan( P2 ) * cos (C5))
Intersection of Prism Plane & R2 Plumb Plane

R4 = Angle of intersection of  prism plane and vertical plane.
R4 = arctan (tan P2 ÷ cos C5)
R4 = arctan (tan Jack Rafter Side Cut Angle ÷ cos Hip Rafter Backing Angle)
R4 = arctan ( cos Hip Rafter Slope Angle  ÷  tan Plan Angle)
Intersection of Hip Rafter Vertical Plumb Plane & Horizontal Plan Angle Plane


P1 = Purlin Miter Angle on Edge Perpendicular to Roof Surface
P1   = arctan(sin Vertical Plane ÷ tan Horizontal Plane )
P1 = arctan (sin S ÷ tan D)
P1 = arctan (sin Common Rafter Slope Angle ÷ tan Plan Angle)
Intersection of   Common Rafter Plumb Plane & Horizontal Plane


P2 = Jack Rafter Side Cut Bevel Angle
P2   = arctan(cos Vertical Plane ÷ tan Horizontal Plane )
P2 = arctan (cos S ÷ tan D)
P2 = arctan (cos Common Rafter Slope Angle ÷ tan Plan Angle)
Intersection of  Horizontal Plan Angle Plane & Common Rafter Plumb Plane

P2 - 90 = Roof Sheathing Angle
Hinge Rotation Plane  = arctan( tan Horizontal Plane ÷   cos Vertical Plane)
P2 - 90  = arctan( tan Horizontal Plane ÷   cos Vertical Plane)
P2 - 90  = arctan( tan Plan Angle  ÷   cos Common Rafter Slope Angle)
Intersection of  Horizontal Plan Angle Plane & Common Rafter Plumb Plane



Steps To Draw the Roof Framing Prism Plane