Intersecting Hip Rafters, by Joe Bartok , March 22, 2014
This is not about mathematics, technical drawing or complex roof framing.
This is art.
Intersecting Hip Rafters
Study of French Drawing
Page 2: Known Lengths and Angles
Page 3: Development of Upper Shoulder Angles – Part 1
Page 4: Development of Upper Shoulder Angles – Part 2
Page 5: Plan View – Intersection Points of DP Lines Page 6: Hip A Lower Right Shoulder Trace Angle
Page 7: Hip B Upper Right Shoulder Angle
Page 8: Hip B Lower Right Shoulder Angle
Page 9: Plan View – Hip A Working Points
Page 10: Hip A Working Points on the 3D Model
Page 11: Hip A Working Point Rise – an example Trigonometric Solution
Page 12: Development of Hip A Lower Right Shoulder Trace Angles
Page 13: Developed 3D Model of Hip Rafter A Page 14: Developed 3D Model of Hip Rafter B
Page 15: Definition of Hip Shoulders and associated Dihedral Angles
Page 16: Table of Direction Cosines perpendicular to Hip Shoulders
Page 17: Developments of Warlock Cut Compound Angle given the Angles on the Stick
Study of Tetrahedra (application of the Pythagorean Theorem)
Page 20: Tetrahedron Developmental Geometry Preliminary Trig Calculations
Page 21: Developments of Tréteaux Angles
Page 23: Tetrahedron Developmental Geometry Trig Calculations continued
Page 26: Developments of Warlock Cut Compound Angle given the Dihedral Angles
Page 29: 3D Model of Intersecting Hip Rafters
Page 34: Negative Profile 3D Models of the Warlock Cut
As it was difficult for many to cut, the carpenters in bygone days dubbed the Hip rafter compound angles to accommodate square fascia the Witch Cut. The Intersecting Hip Rafters presented a similar challenge so it seems appropriate to name the compound angle the Warlock Cut.
Note: I've been studying the French compagnons charpentiers art of line, trait de charpenterie, that was developed over the last 700 years. A lot of the construction lines in the book Traité théorique et pratique de charpente by Louis Mazerolle are misleading or not understandable for developing the intersection of timbers. This treatise by Joe Bartok on the geometry and trigonometry of the Warlock Cut presents the intersection of tilted hip rafters, when hips collide, in such a way that we can now unfold a intersecting joint of tilted, devers de pas, rafters precisely. Philibert De l’Orme, Mathurin Jousse, François Derand and even Girard Desargues would be impressed by this study.