Roof Framing Geometry Propositions – Axioms

I should have called this article "The Mazerolle Code", because I feel like I'm cracking the secret code hidden in the French book "Traité Théorique et Pratique de Charpente" by Louis Mazerolle published in 1850. Over the last couple of months Joe Bartok and I have looked at Dever De Pas from both sides of the drawing using geometry and trigonometry that define the parallelogram planes in timbers used to develop the upper and lower claw angles.  We also don't think in terms of miter angles on the sides of rafters anymore. It's either an upper claw angle, aka miter angle, or lower claw angles, aka lip angles.

English speaking carpenters are handicapped right from the start when studying Roof Framing Geometry, because we reference roof framing angles using a framing square. To understand the secret code of Roof Framing Geometry that was developed in the 13th century or earlier by the European carpenters you have to think in terms of angles on the stick... Or angles on the timers. As an example  an 8:12 pitched roof  has a slope angle of 33.69007°, but to develop the geometry layout used on the stick, timber, you have to think in terms of 90° - 33.69007°.

In this article we'll try to cover the Roof Framing Propositions – Axioms that define all claw angles for rafters that are plumb to the earth, rotated into the roof surface plane, rotated in any plane or skewed from the plate line.

Roof Framing Geometry Proposition – Axioms  # 1:
All rafters have a DP Line, Devers De Pas Line. The DP line is the line in plan view that follows the vertical plane tilt of the rafter. The DP line can also be the rafter run line for rafters that are plumb to the earth. that have vertical plane tilt of zero degrees. For all rafters that are plumb to the earth, the DP Line is the same as the rafter run line.

Roof Framing Geometry Proposition – Axioms  # 2:
All rafters have a vertical plane tilt line. The vertical plane tilt of the rafter can be zero, like a plumb hip rafter or a plumb common rafter. All purlin rafters will have a vertical plane tilt greater than zero.

Roof Framing Geometry Proposition – Axioms  # 3:

The intersection of the DP Lines of two rafters define the line for the upper claw angle on the rafter.

Roof Framing Geometry Proposition – Axioms  # 4:

The intersection of the rafter DP Line and the hip rafter foot print line define the line for the lower claw angle on the rafter.

Roof Framing Geometry Proposition – Axioms  # 5:

The intersection of the rafter run line and the hip rafter claw line defines the line for the secret line that is used for the lower claw angle on the rafter.

Roof Framing Geometry Proposition – Axioms  # 6:

The secret line is perpendicular to the DP line of the rafter.

Roof Framing Geometry Proposition – Axioms # 7:
A line from the intersection of the DP lines of two Hip Rafters, "a", to the hip rafter slope line "b" following the DP line, with a length equal to an arc length of the side face of the hip rafter lower claw length will determine point "c", for the angle on the bottom of the rotated hip rafter.

Roof Framing Geometry Proposition – Axioms # 8:
The intersection of the Rafter Run Line & TC Line locates Hip Rafter DP Line.

Roof Framing Geometry Proposition – Axioms # 9:
If the hip rafter foot print line is parallel to the jack rafter run line, then the lower claw line for the jack rafter is parallel to the jack rafter run line.

Roof Framing Geometry Proposition – Axioms # 10:
The intersections of the valley rafter roof surface line, at the foot of the valley rafter, and the profile rafter foot print line defines the valley rafter claw line for the upper claw lines on jack rafters.

Roof Framing Geometry Proposition – Axioms # 11:
The intersections of the valley rafter foot print line and the top of the purlin rafter top edge plane line defines the valley rafter claw line for the lower claw line on purlin rafters.

Here's a couple of drawings showing the DP line of a standard purlin rafter and the upper and lower claw angles developed from the Roof Framing Geometry Proposition – Axioms .

Here's a drawing showing the Roof Framing Geometry Proposition – Axioms on a standard roof. Where the hip rafter is plumb and the jack rafter is also plumb to the earth.

Drawing with axioms for a rotated hip rafter and a jack rafter that is plumb to the earth. For jack rafters that are plumb to the earth the vertical plane tilt is always equal to zero and the jack rafter run line is also the DP line of the jack rafter.

Here's a couple of drawing showing how the Axioms work with Joe Bartok's Warlock Rhombic study. The Warlock Rhombic study has two hip rafters square in cross section, rotated 45° to the hip rafter plane and one hip rafter that's has a parallelogram cross section, 60°x 120° rotated 45° to the hip rafter plane. These drawings show the axioms that develop the upper and lower claw angles on both sides of the  parallelogram cross section hip rafter.

There's more Roof Framing Geometry Proposition – Axioms to come...develop.....