Octagon Overlay Sleeper Math Notes

I thought I could break down the different parts to an Octagonal roof overlay, so you would only need to use some trigonometry to calculate the different dimensions to precisely frame the Octagonal Roof Overlay, California sleepers. However, I don't think it's possible. To precisely locate the sleepers on the main roof you have to geometrically draw out the main roof surface showing the location of the valleys sleepers developed from the location of the valleys sleeper in plan view.

In this drawing I developed the roof surface of the Octagonal Roof from the Octagon roof in plan view and elevation view. You can developed the valley sleepers on the octagonal roof surfaces, but it does not give the location of the sleepers on the main roof surface.

 To develop the Octagonal Roof sleepers on the main roof surface you need to rotate the Octagon in plan view, so it's perpendicular to the main roof slope in the Octagon in elevation view. Dropping perpendicular from the main roof slope in elevation view develops the location of the sleepers on the main roof surface.

After you've developed  the sleepers on the main roof surface when you can get the sleeper miter angles you still need a saw blade bevel angle to miter the two sleepers together. You can use this formula for sleeper #1, Saw Blade Bevel Angle = arcsin(sin main roof slope) x cos(67.5)). For the saw blade bevel angle for the miter at sleepers #2 & #3 use  Saw Blade Bevel Angle = arcsin(sin main roof slope) x cos(22.5)). Or you can use the RafterTools+ app on the iPhone. Enter 135° eave angle and use the hip rafter backing angle for the Saw Blade Bevel Angle. Use an eave angle of 45° for the saw blade bevel angle for sleepers #2 & #3.

Use the same saw blade bevel angle to cut the miter on the bottom of the Octagonal Hip Rafters that frame into the sleepers. For the miter angle cut at the hip rafters that frame into the sleeper you could also use Layover Cut Angle = Layover Rafter Slope Angle + arctan (tan Main Roof Slope Angle x sin Layover Rotation Angle in Plan View) ....where the Rotation Angle in Plan View is  67.5° and 22.5°.