The angles on the developed faces of the rafter or purlin are laid out to scale using a blend of geometry and trigonometry. The dihedral angles may be double checked after the model is constructed.
When I first studied compound joinery geometry I made simple models of the triangles and tetrahedra. Sorry, no pics of these ... I outgrew them and passed them along to other folks who are just starting to learn the math.
The models in the pages below are "working models" for use in the field. They take no longer to make than orthographic, isometric or perspective drawings and have the advantage of not suffering from the angles and planes being distorted. It's nice to see exactly what you are shooting for in 3D when doing the layout on a log or timber.
Models of Roof Member IntersectionsModels of Square Hung Fascia intersects foot of Hip RafterModel of Three-Branch Compound Joint ... scroll down the page
Curved intersections can be developed and modeled as well ...
Barrel Arch intersects Conical RoofBarrel Arch intersects Barrel ArchFlashing intersects slope of RoofI don't have any photos of the models in their developed (laid out flat) form but there is a sketch drawn to scale near the bottom of this page. Just lay out the angles and dimensions on the faces, cut, fold, and tape and/or glue ...
Square Hung Fascia at an Irregular Plan Angle
