Images of 3D Model ... Interlocking Tetrahedra extracted from a Hip Roof

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


Right Triangles and Tetrahedra modeling the 7/12 Side Angles

Layout of Lines and Angles in Plan View
Tetrahedron modeling the Hip Roof Angles positioned with respect to the Plan Angles, viewed from triangle representing the Common Slope
Tetrahedron modeling the Hip Roof Angles positioned with respect to the Plan Angles, viewed from triangle representing the Hip Slope
Sheathing Angle Triangle on surface of roof ... Top View
Backing Angle created by Sheathing Angle Triangle on surface of roof ... Oblique View from side of Common Slope Triangle

Geometry Notes ... Lines and Planes defining the Backing Angle
(1) The plumb plane of the triangle representing the Hip Slope Angle and the level plane of the Plan Angle are perpendicular to one another.
(2) The line perpendicular to the Hip Run is perpendicular to the plumb plane of the triangle representing the Hip Slope Angle.
(3) Any line drawn on the triangle representing the Hip Slope Angle will be at a right angle to the line perpendicular to the Hip Run.
(4) Therefore the line at right angles to the Hip Length on the plane of the triangle representing the Hip Slope Angle, and the line perpendicular to the Hip Run, form a right angle.

(1) The line of the Hip Length defines the intersection of the plane of the surface of the roof and the plane of the Hip Slope Triangle.
(2) One line at a right angle to the Hip Length lies on the plane following the surface of the roof.
(3) The other line at a right angle to the Hip Length lies on the plumb plane of the Hip Slope Angle.
(4) The angle between these two lines is the measure of the Dihedral Angle between the plane following the surface of the roof and the plumb plane of the Hip Slope Angle. This Dihedral Angle is the complement of the angle defined as the Backing Angle.
Three lines and two angles composing the triangle have been defined above. The remaining angle in the right triangle is the Backing Angle. Note how the three lines form a right triangle in the image of the model below.
Lines defining the triangle of the Backing Angle
Lines defining the triangle of the Backing Angle ... Oblique View from side of Hip Slope Triangle

Joe Bartok