Plumb Hip Rafter or Inclined Rafter with Edge Bevel

In the last task model in Bernd Kupper's book, The new Book on the ancient knowledge of Roof Framing, Pent roof with Pointed Dormer, Bernd shows how the lower miter angle is developed for the plumb inclined rafter with edge bevel when it intersects the canted rafter. He uses the canted rafter's bottom edge in ground plan to develop the lines for the inclined rafter's lower miter angle. This is one of those situations when it helps to know how to develop the hip rafter trace angles on the hip rafter to check your geometric line development.

I never could get the canted rafters ground lines to work correctly, so I used the hip rafter trace angle geometry to find the lower miter angle on the inclined rafter with edge bevel.

Tried using the canted rafter DP tilt triangle for the canted rafter lines in ground plan, but the lines still would not line up correctly.

Tried rotating the canted rafter block in ground plan and still couldn't get the bottom edge of the canted rafter to align correctly. I'm probably drawing something wrong, just not sure where the mistake is in the drawing.

Here's couple of drawings from Bernd. These should correct my mistakes.

Apparently I had redrawn the plumb hip rafter too many times, without checking my construction lines. Bernd saw that they were not parallel lines and sure enough after redrawing the entire file the miter cut on the bottom of the plumb hip rafter that intersects the canted rafter is now correct.

One of the problems I had was drawing the bottom edge of the canted rafter in ground plan correctly. So, I used some trigonometry to rotate the rafter block precisely to locate the bottom edge of the canted rafter in ground plan.

Rafter block, 20mm x 30mm

Profile Slope Angle = 45°

Plan Angle = 30°

Hip Rafter Backing Angle  = arcsin (sin Profile Rafter Slope Angle x cos Plan Angle)

Hip Rafter Backing Angle = arcsin(sin 45 x cos 30) = 37.761235 

Width of top edge lines in ground plan 

20mm x cos(37.761235) = 15.811388

Width of line for bottom edge of canted rafter in ground plan
30mm x sin(37.761235) = 18.371169