Principles Du Devers Traite A LA Sauterelle

I was checking my Roof Framing Geometry Proposition – Axioms against Patrick Moore's Principles Du Devers Traite A LA Sauterelle drawing on this page at Historical Carpentry . My Roof Framing Geometry Proposition – Axioms work with his techniques for Devers De Pas, but mind require a rafter depth to determine some of the lines. Patrick's techniques are in theory only in that drawing, showing the principles of the art of line. No width or depth of the rafters are required for his drawing.
Drawings showing the difference between the Principles Du Devers Traite A LA Sauterelle techniques and the techniques for using the Roof Framing Geometry Proposition – Axioms.

Sauterelle means using a bevel square to transfer the angles from the geometric layout to the timber.

Drawing showing the basic geometric layout of the roof. In this example I used a 90° eave angel with equal profile rafter slopes of 45°.

 Adding the DP lines for the rotated hip rafter.

Establishing the vertical plane tilt of the hip rafter.

Showing the triangle that represents the plane of the rotated hip rafter.

3D drawing of the rotated hip rafter with the vertical plane tilt drawn in 2D and 3D.

Developing the triangle that represents the plane of the rotated hip rafter for both sides of the hip rafter.

Showing the location of the plumb to the earth jack rafter and purlin rafter parallel to the plate line.

Drawing showing the theory used by Patrick for the upper and lower claw angles of the jack rafter.

3D drawing of the hip rafter with the jack rafter upper and lower claw angles dimensioned.

Drawing showing my Roof Framing Geometry Proposition – Axioms versus the theoretical way of laying out the geometry for the jack rafter upper and lower claw angles.

Drawing showing my Roof Framing Geometry Proposition – Axioms for laying out the geometry for the jack rafter upper and lower claw angles.

 Drawing showing the theoretical way of laying out the geometry for the jack rafter upper and lower claw angles.

 3D drawing of the purlin rafter. 

Drawing showing my Roof Framing Geometry Proposition – Axioms versus the theoretical way of laying out the geometry for the purlin rafter upper and lower claw angles.

Drawing showing my Roof Framing Geometry Proposition – Axioms for laying out the geometry for the purlin rafter upper and lower claw angles.

 Drawing showing the theoretical way of laying out the geometry for the jack rafter and purlin rafter upper and lower claw angles.

Using Trigonometry to check the geometry for Sauterelle Angles
Initial Given Values:
Hip Rafter Section Profile = Square

Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 45.00000
S = Adjacent Rafter Slope Angle = 45.00000
DD = Main Plan Angle = 45.00000
D = Adjacent Plan Angle = 45.00000
R1 = Hip Rafter Slope Angle = 35.26439
C5m Main Hip Rafter Backing Angle = 30.00000
C5a Adjacent Hip Rafter Backing Angle = 30.00000
P2m = Main Jack Rafter Side Cut Angle = 35.26439
90° - P2m = Main Roof Sheathing Angle = 54.73561
P2a = Adjacent Jack Rafter Side Cut Angle = 35.26439
90° - P2a = Adjacent Roof Sheathing Angle = 54.73561

Hip Rafter Section Profile View Rotation Angle = 45° 

Hip Rafter Foot Calculations by Joe Bartok

FA-DP = Hip Rafter Footprint Angle in Plan View = arctan (sin 35.26439° × tan 45°) = 30.00°

R1-DP = Hip Rafter Slope Angle on DP Shoulder = arccos (cos 35.26439° × cos
30.00°) = 45.00°

Vertical Plane Rotation Angle from Plumb for Tilted Hip Rafter on DP Line
R1V-DP = arctan(1 ÷ (tan(R1) ÷ sin(FA-DP)))
R1V-DP = arctan(1 ÷ (tan(35.26439°) ÷ sin(30.00°))) = 35.26439°