First develop the sloping/raking construction lines that define the direction of the averaging/hip rafter intersecting lines.
Draw the averaging/hip rafter intersecting lines and draw the hip rafter run lines through the intersecting lines.
Draw the construction lines for the Sectional Area of the Intersection of the Roof Planes. This area represents a level plane of the intersection of the roof planes.
Drawings showing the Sectional Area of the Intersection of the Roof Planes.
Theories: I'm still not sure about all the geometrical theories for developing the geometric roof design with sloping/raking eave, but this last drawing showing the eave angles, roof slope angles, hip rafter slope angles suggest that this particular ground plan can be calculated using some trigonometry. The only real theory I can could up with is using the 1.5 rise tangent to the corner opposite of the raking eave line develops a level plane of the intersecting roof planes. That represents the eave angles to use with trigonometry to calculate the rest of the roof. I just need a way to find plan angles without knowing the true slope angle of the roof plane on the raking eaves(44.99930°).
- Use the sloping eave rise to establish the eave angles. In this drawing the sloping eave line has a rise of 1.5 and the length of the ground plan is 10.00.
- arcsin(1.5 ÷ 10.00) = 8.62692°
- Eave angle = 90° + 8.62692° = 98.62692°
- Eave angle = 90° - (2 x 8.65692°) = 72.74614°
The eave lines are all suppose to be sloping at 40°, however the sloping/raking eave lines do not. With the raking eave lines not at a 40° roof slope angle it's hard to take the trigonometry any future. So, once again this theoretical roof design can only be calculated using geometry at this time.