Overview of Geometric Solids modeling the Hip Roof

Main Slope = 10/12     Adjoining Slope = 7/12
Corner Angle measured between Eaves in Plan View = 90°

Wireframe Sketckes of Tetrahedra modeling the
backed and unbacked Upper Shoulder of a Hip Rafter

Developments of the Tetrahedra
modeling the Fundamental Roof Angles

Tetrahedra modeling the Fundamental Hip Roof Angles

Juxtaposed Tetrahedra in position on Plan Angles

Wireframe Sketch of Post-Type Model
superimposed on the Juxtaposed Tetrahedra

Development of Post Type Model

Oblique View of the Post Type Model

Tetrahedron relating the Hip Side Cut Angle at the Hip Peak (R4P)
the Backing Angle (C5), and the Jack Rafter Side Cut Angle (P2)

tan R4P = tan P2 ÷ cos C5

Comparison of nested Tetrahedra (right)
extracted from the Post Type Model (left)

Development of the Tetrahedron extracted from the
Post Type Model of the Hip Rafter Side Cut Angles

(Trigonometric Scale)

Nested Tetrahedra ... Triangles of Common Slope and P6

Nested Tetrahedra ... Triangle of Backing Angle

Nested Tetrahedra ... Triangle of Hip Side Cut Angle at the Hip Peak

Tetrahedron relating the Principal Purlin Housing Angle (P3)
the Backing Angle (C5), and the Jack Rafter Side Cut Angle (P2)

tan P3 = tan C5 × cos P2

Development of the Tetrahedron extracted from the
Post Type Model of the Hip Rafter Side Cut Angles

(Trigonometric Scale)

Nested Tetrahedra juxtaposed along the Triangle of Backing Angle
Upper Shoulder of unbacked Hip Rafter ... Triangle of R3

Nested Tetrahedra ... Triangle of P3

Proof of Formula for Angle P3

3D models ... Underpurlin intersects Valley Rafter

Main Slope = 16/12     Adjoining Slope = 12/12
Corner Angle measured between Eaves in Plan View = 90°

Valley Rafter viewed from 12/12 (Adjoining) Side

Valley Rafter viewed from 16/12 (Main) Side

Bottom View of intersecting Valley Rafter and Underpurlin

Bottom View of Valley Rafter, notched to accommodated the Underpurlin

Traces created on the Underpurlin by the intersecting Valley Rafter