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

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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.

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