Construction of Jack Purlin Miter Angles

Development of Irregular Pentahedron

 Values and Descriptions of Angles SS = 26.56505° ... Common Slope Angle measured from level. DD = 59.03624° ... Plan Angle measured between the Hip run and the eave line, or the Valley run and the ridge line. R1 = 23.20706° ... Hip-Valley Slope Angle measured from level. P2 = 28.22051° ... Jack Rafter Side Cut Angle, measured on the shoulder set in the surface of the roof. 90° – P2 = 61.77949° ... Sheathing Angle, measured on the jack purlin shoulder set in the surface of the roof. R2 = 7.03919° ... Measured from a line square to the Hip-Valley rafter length, the Housing Layout Angle traced on the side (plumb) face of Hip-Valley rafter to accommodate an intersecting jack purlin. Also the miter angle at the Hip rafter foot to accommodate square fascia. 90° + R2 is measured from the line representing the Hip-Valley rafter length. Q3 = 30.24626° ... equal to R1 + R2. Measured from a plumb line on the side face of the Hip-Valley rafter, the Housing Layout Angle to accommodate a jack purlin (or square fascia). P1 = 15.02026° ... Jack purlin miter angle, where the purlin intersects the side (plumb) face of the Hip-Valley rafter. Measured on the face of the jack purlin perpendicular to the surface of the roof.

3D Model of Irregular Pentahedron

Right triangle of angle P1 ... section plane perpendicular to the surface of the roof
Note the three mutually perpendicular lines formed by the intersecting near faces

Intercept of the section plane with the plumb plane of the Hip-Valley rafter