Monday, September 2, 2013

Rotated Hip Rafters with Prismatic Foot Print

Here's a study on rotated  hip rafters with the  Prismatic Foot Print of Hip Rafter  developed from the plan view of the hip rafter rotation angle. In this study the hip rafter, 3 1/2" x 3 1/2" is simply rotated 45° from the hip rafter run line in plan view.

Draw the hip rafter in elevation view and transfer perpendicular lines from the level cut of the hip rafter in elevation view. The center line that is perpendicular to the hip rafter level cut in elevation view is at the mid point of the level cut in this example.

To develop the miter Angle on the hip rafter you use 90° - Common Rafter Slope Angle.

Saw Blade Bevel Angle Development

Plan Angle = 45°
Miter Angle = 90° - Roof Slope Angle = 19.79887°
Saw Blade Bevel Angle = arctan (sin Miter Angle x tan Plan Angle) = 18.71224°
Saw Blade Bevel Angle = arctan (sin Angle of Saw Travel  ÷ tan Angle on Adjoining Face)
Saw Blade Bevel Angle = arctan (sin 70.20112°  ÷ tan 70.20112°) = 18.71224°

Prismatic Foot Print of Hip Rafter developed as a prism. 

Prismatic Foot Print of Hip Rafter unfolded developed from prism.

Tetrahedron unfolded, showing the trigonometry developed from the geometry. 

What's really interesting here is that the hip rafter backing angle 41.70594°, when the hip rafter is not rotated, is the angle on the top of the prismatic cut when the hip rafter is rotated.

Tetrahedron Angles

D Angle = 45.0000000000
A Angle = 70.2011256662
C Angle = 63.0185900000
E Angle = 18.7122394087
B Angle = 41.7059492665 .... hip rafter backing angle

90-D Angle = 45.0000000000
90-A Angle = 19.7988743338
90-C Angle = 26.9814100000
90-E Angle = 71.2877605913
90-B Angle = 48.2940507335

Update: 02/28/2014
Drawings I developed after Study of the Tréteaux Angles. The slope angle on the side of the hip rafter is equal to the common/profile rafter slope angle.
R8-DP = Tilted Hip Rafter Slope Angle on DP Side = SS = S

In these updated drawings the DP line is developed from the TC line following the tilt of the hip rafter block in the hip rafter backing triangle.



  1. Sim,

    You may want to double check your figures. A quick glance at the footprint in the model as compared to the wire 3-D plan extension shows that the hip rafter would have to be a parallelogram in a perpendicular cross view to bring the surfaces into plane. The lumber stick in the model are obviously square, therefore the developed angles will only be close. What are the diagonal dims of the hips prismatic footprint in plan? (equal?) if so, the hip rafters are not milled square.(?)

    Am I confusing the project or missing the point?


  2. In the 3D drawing I took the prism off the bottom of the post in a 3D view. So the prism angles, as well as the miter & saw bevel angles are correct.

    The 4x4 post in the model is cut correctly, but it's only 3 5/16" wide and not square edged. The lines in the drawing were drawn for a 3 1/2" wide post. So, the sides of the oblique plane of the foot print of the hip rafter do not touch the lines in plan.

  3. Sim,

    I spent a little more time on your illustrations, got it! My first view had me wondering about the 16.98* angle shown on your P/W square. It is 26.98*. I should have caught that.

    I recently built a timber frame style swing set. This tutorial is much like the scenario I encountered. I kept on side of the irregular splayed legs in plane. I didn't draw plans and haven't taken any pics yet. Looks like a giant sawhorse.

    It would be a good project to show the exposed Hip Rafters in with your layout style. Separate the hips with a ridge beam and you have the swings "A" frame.

    Thanks for the tutorial!

  4. I do not get it , If the timber is square even if its smaller it should be on the lines .I am driving today so I will not be able to draw till tonite or tomorrow . Or am I just crazy ?

    1. Yep u are right I thought it out as I was driving and i had not thought out the planes as i should

    2. The last couple of pictures show the 63.01° hip rafter slope template touching the edge of the post. So , the prismatic foot print cut is correct. However, the post foot print should touch the lines, but the lines are drawn for a post 3 1/2" wide with square edges. The post is only 3 5/16" wide with rounded corners.