Tetrahedra and Triangles defining a Hip or Valley Roof

Interlocking Tetrahedra modeling Hip-Valley Roof Angles

Overview of triangles describing intersections between components of a Hip Roof

Tetrahedron extracted from the Hip Roof ... model of the Fundamental Hip-Valley Roof Angles

Exploded view of component triangles modeling the Fundamental Hip-Valley Roof Angles
SS ... Common Slope Angle measured from level.

DD ... Plan Angle measured between the Hip run and the eave line, or the Valley run and the ridge line.

R1 ... Hip Slope Angle measured from level.

P2 ... Jack Rafter Side Cut Angle measured on the shoulder set in the surface of the roof.

90 P2 ... Sheathing Angle, measured on the jack purlin shoulder set in the surface of the roof.

Tetrahedra modeling the intersection of the Hip-Valley Rafter with plumb planes following Eaves, Walls or Corner Post

Exploded view of component triangles modeling the intersection of the Hip-Valley Rafter with plumb planes following Eaves, Walls or Corner Post
R4B ... Hip-Valley Rafter Side Cut Angle at the Hip rafter foot or Valley rafter peak. Created on the unbacked upper shoulder of the Hip-Valley as the rafter intercepts a plumb plane following the Hip eave line or Valley ridge line.

R4P ... Hip-Valley Rafter Side Cut Angle at the Hip rafter peak or Valley rafter foot. Created on the unbacked upper shoulder of the Hip-Valley as the rafter intercepts a plumb plane following a wall line at right angles the Hip or Valley eave line.

R5B ... Housing Layout Angle measured from level, at the Hip Rafter foot or Valley rafter peak. Traced on a plumb plane following an eave line by the intersecting unbacked shoulder of the Hip rafter.

R5B ... Housing Layout Angle measured from level, at the Hip rafter peak or Valley rafter foot. Traced on a plumb plane following a wall line at a right angle the Hip eave line by the intersecting unbacked shoulder of the Hip rafter.

P6 ... Housing Layout Angle as measured from the upper shoulder of a supporting or principal common rafter, to accommodate the intersecting unbacked shoulder of a Valley rafter. Tenon Layout Angle on the plumb face of a jack rafter to form a tenon plane following the unbacked Hip-Valley shoulder. Also the angle created on the plumb face of a jack rafter by the shelf cut to match the housing on the Hip-Valley rafter (chamfer or square cut parallel to the unbacked Hip-Valley shoulder).

Tetrahedra modeling the intersection of a Jack Rafter and Jack Purlin with the side (plumb) face of the Hip-Valley Rafter

Exploded view of component triangles modeling the intersection of a Jack Rafter and Jack Purlin with the side (plumb) face of the Hip-Valley Rafter
C5 ... Backing Angle. Complement of the dihedral angle measured between the side (plumb) face of the Hip-Valley rafter and the surface of the roof.

P1 ... Miter angle for jack purlin intersection with the side (plumb) face of the Hip-Valley rafter. Measured on the face of the jack purlin perpendicular to the surface of the roof.

R2 ... Housing Layout Angle traced on the side (plumb) face of Hip rafter to accommodate an intersecting jack purlin. Also the miter angle at the Hip rafter foot to accommodate square fascia. Measured from a line square to the Hip-Valley rafter length.

Tetrahedra describing the intersection of a Jack Purlin with the side (plumb) face of the Hip-Valley Rafter

Exploded view of component triangles describing the intersection of a Jack Purlin with the side (plumb) face of the Hip-Valley Rafter
C1 ... Saw Blade Bevel Angle to make the cut on the face of the jack purlin perpendicular to the surface of the roof, at the purlin intersection with the side (plumb) face of the Hip-Valley rafter.

Q3 ... equal to R1 + R2, the Housing Layout Angle measured from a plumb line on the side face of the Hip-Valley rafter, to accommodate a jack purlin (or square fascia).

Intersection of upper shoulder of Hip-Valley Rafter with face of Purlin perpendicular to surface of roof and plumb plane following a Wall

Exploded view of component triangles modeling the intersection of upper shoulder of Hip-Valley Rafter with face of Principal Purlin perpendicular to surface of roof
R3 ... Traced on the unbacked shoulder of a Valley rafter, where the Valley intersects a supporting or principal purlin. Also the miter angle on the upper shoulder of a Hip rafter to accommodate square fascia, 90 R3 is the Square Fascia Side Cut Angle.

P3 ... Housing Layout Angle on supporting or principal purlin, to accommodate the unbacked shoulder of the intersecting Valley rafter. Tenon Layout Angle on the jack purlin face perpendicular to the surface of the roof to produce a tenon plane parallel to the unbacked Hip-Valley shoulder. Also the angle created on the jack purlin face perpendicular to the surface of the roof by the shelf cut to match the housing on the Hip-Valley rafter (chamfer or square cut parallel to the unbacked Hip-Valley shoulder).

Exploded view of component triangles modeling the intersection of upper shoulder of Hip-Valley Rafter with face of a Principal Rafter following a Wall

Tetrahedron modeling the intersection of the Hip-Valley Rafter with plumb face of Center Post

Exploded view of component triangles modeling the intersection of the Hip-Valley Rafter with plumb face of Center Post
A7 ... Backing Angle viewed in section on a backed Hip-Valley rafter if the cutting plane is rotated to plumb (angle R1).

P5 ... Tenon Layout Angle on the jack rafter shoulder set in the surface of the roof to produce a tenon face perpendicular to the side (plumb) face of the Hip-Valley rafter. Also the angle created on the jack rafter by the housing chamfer cut (square cut) for the plumb line (the chamfer to remove the feather edge along miter angle SS).

Tetrahedra modeling Chamfer (Square) Cuts for housings

Exploded view of component triangles modeling the Chamfer (Square) Cuts for housings
Q1 ... Backing Angle viewed in section on a backed Hip-Valley rafter if the cutting plane is rotated to angle R2.

P4 ... Tenon Layout Angle on the jack purlin shoulder set in the surface of the roof to produce a tenon face perpendicular to the side (plumb) face of the Hip-Valley rafter. Also the angle created on the jack purlin by the housing chamfer cut (square cut) for the face perpendicular to the surface of the roof (the chamfer to remove the feather edge along miter angle P1).

A8 ... Angle produced on the backed shoulder of Hip-Valley rafter, by the housing chamfer cut or square cut along the line of R2 to accommodate the jack purlin tenon. Cognate to cut that forms angle P4 on the jack purlin.

A9 ... Angle produced on the backed shoulder of Hip-Valley rafter, by the housing chamfer cut or square cut along the plumb line to accommodate the jack rafter tenon. Cognate to cut that forms angle P5 on the jack rafter.

Joe Bartok