Timber Porn

Billy Dillon called my copy of "Traité Théorique et Pratique de Charpente" by Louis Mazerolle Timber Porn and it pretty much stayed a porn book on Stereotomy geometry until I read through Tim Moore's blog on Stereotomy. The Trapezoidal Ground Plan on page 52 was meant to be an exercise in Theorie des  dévers  de pas et des niveaux dévers.  level cant ?? or level slope?? 


The end view of a hip rafter or common rafter without elevation? Draw the end view of a rafter without any slope?
http://compagnonnage.info/blog/blogs/blog1.php/2011/11/21/remarques-sur-le-cachet-des-indiens-de-buenos-aires


I thought the exercise was pretty straight forward until I got to the skewed rafter in the Trapezoidal Ground Plan. The angles that the skewed rafter had for the miter angle at the foot and the miter angle at the peak of the skewed rafter didn't make any since mathematically. So I extracted a tetrahedron from the peak and foot of the skewed rafter to find out where these angles came from.


This first drawing is from the Google Sketchup plugin I'm writing for the exercise. I know it doesn't have much to do with drawing a 3D object in 2D space, but it helps me understand the geometry  better.

Here's a screen shot the of the Google Sketchup plugin.

To understand the miter and bevel angles at the peak of the skewed rafter I extracted a tetrahedron from the peak of the skewed rafter. 

The tetrahedron extracted from the peak of the rafter does not have an right angle corner in each of the faces of the tetrahedron so I needed to treat it like a regular polyhedron and find the footprint of the polyhedron like we do for the platonic solid  tetrahedron with 3 equilateral triangles at each vertex. 

http://www.sbebuilders.com/tools/geometry/tetrahedron_platonic_soild.php

By drawing a line perpendicular to the line EA through C-B2 it develops the common rafter run that we can use for the slope of the extracted tetrahedron. The slope angle of the extracted tetrahedron is the same slope  angle as the slope angle from the point G to K. So, now it's beginning to make since mathematically.

The angle at B2 is similar to  a jack rafter side cut angle. The miter angle at the skewed rafter peak can be calculated as the Purlin rafter slope angle.

Skewed Rafter Slope =     43.70889 °

Skewed Rafter Plane Tilt =   8.68257°

Purlin Rafter Slope Angle = arcsin ( sin ( Pitch Angle ) × cos ( Jack Rafter Side Cut Angle ))

Miter Angle at Peak of Skewed Rafter = arcsin ( sin ( 43.70889° ) × cos ( 8.68257° )) = 43.08446°

To find the bevel angle at the peak of the skewed rafter 

Plan Angle at Peak of Rafter = arctan( sin ( Pitch Angle  ) ÷ tan( Plan Angle))

Plan Angle at Peak of Rafter = arctan( sin ( 43.70889 °  ) ÷ tan( 73°)) = 11.9288°


Purlin Bevel Angle = arctan( tan ( Pitch Angle ) × sin ( Plan Angle At Peak of Rafter ))
Purlin Bevel Angle = arctan( tan ( 43.70889 ° ) × sin ( 11.9288° )) = 11.17679°

A similar process can be used to fine the miter and bevel angles at the skewed rafter foot.

Skewed Rafter Slope =     90° - 43.70889 ° = 46.29111°

Skewed Rafter Horizontal Rotation Angle =   8.30528°

Purlin Rafter Slope Angle = arcsin ( sin ( Pitch Angle ) × cos ( Jack Rafter Side Cut Angle ))

Slope Angle at Foot of Skewed Rafter =  arcsin ( sin (  46.29111° ) × cos ( 8.30528° )) = 45.66606° 

Miter Angle at Foot of Skewed Rafter = 90° - 45.66606°  = 44.33393°

To find the bevel angle at the foot of the skewed rafter 

Purlin Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Purlin Bevel Angle  = arcsin( cos ( 43.7088 )  *  cos( 73 )) = 12.20111°

Here are some of the Google SketchUp files that were generated by the Google SketchUp Plugin

Trapezoidal Timber

Trapezoidal Ground Plan

Trapezoidal Timber Planes

Two Google SketchUp files with the tetrahedrons extracted.

Tetrahedron Skewed Rafter Foot

Tetrahedron At Skewed Rafter Peak