## Tuesday, November 5, 2013

### Der verkantete schräger Giebelsparren

Der verkantete schräger Giebelsparren

1. The canted oblique gable rafters
2. Schräger Sparren
3. Sloping Rafter
4. Oblique Rafter
5. Slanted Rafter
6. Prow Rafter
7. Crow's Peak Gable Rafter
8. Crow's Peak Prow Rafter
Whatever you call the prow gable rafter rotated perpendicular to the roof surface it can be challenging, to find the correct miter angle and saw blade bevel angles for the ridge cut.  The article "Framing a Prow Roof(s)" by Scott Wells, published by ThisIsCarpentry.com by Gary Katz, is an excellent article on how Scott Wells solved the problem.

Prow rafters that are plumb to the earth are easy enough to figure out, but once the prow rafter is rotated perpendicular to the roof surface the ridge cut angles are not that obvious. Scott had to build both types of prow rafters on Gary Katz's new shop and Scott used full scale geometry on a piece of plywood to find the angles for the prow rafter plumb to the earth. He then used SketchUp to draw a 3D model of the prow rafter rotated perpendicular to the roof surface. His SketchUp video is worth watching for all carpenters. The protractor and tape measure tool in SketchUp are a carpenters best friend. Scott also shows you how he used the Angle Between Faces ruby sketchup plugin by Clark Bremer of the Timber Framing Guild to find the dihedral angle at the ridge cut to find the saw blade bevel angle that was used to cut the prow rafter ridge cut.

The math for the prow rafter rotated perpendicular to the roof surface peaked my interest, especially since I've been studying the different ways to draw out the geometry for the rafters for the WorldSkills Carpentry Competition. This first drawing would be the correct geometry to use in the Carpentry Competitions, because you're suppose to cut the prow rafter ridge cut with a handsaw.

If we're using a compound miter saw like Scott, then we'll use some trigonometry to calculate the saw blade bevel angle.

Profile Rafter Slope Angle = 18.43495°

Prow Plan Angle = 71.56505°

Prow Hip Rafter Slope Angle = arctan(tan(18.43495) * sin(71.56505)) = 17.54840°
Miter Angle = arctan(cos(17.54840) ÷  tan(90 -71.56505)) = 70.73022°
Miter Angle American = 90 -  arctan(cos(17.54840) ÷  tan(90 -71.56505)) =19.26977°
Saw Blade Bevel Angle = arccos( cos (Plan Angle ) ÷ cos ( Miter Angle ))
Saw Blade Bevel Angle = arccos( cos ( 71.56505 ) ÷ cos ( 70.73022 )) = 16.62096°

This model was confusing, because the prow hip rafter slope angle was the same as the jack rafter side cut angle. So I used a Tetrahedron Slice to see if I could understand the trigonometric relationships better. No, the tetrahedron slice didn't make it any clearer.

For WorldSkills Carpentry Competitions we need to draw out the roof using geometry, but for everyday framing practices I think Scott's 3D drawing in SketchUp was the best way to solve the problem of finding the prow rafter miter angle and saw blade bevel angle for the compound cut.