Pondering complex hip roofs this afternoon and considering the graphic below:

A few rules seem to emerge:

1.) An outside corner will always create a hip that is 45 deg. from each leg of the corner.

2.) An inside corner will likewise always create a valley 135 deg. from each leg of the corner.

3.) Where two hips or flying hips meet a ridge will extend 135 deg. from each hip.

4.) When two valleys meet at 90 deg. they will terminate and a ridge will extend at 45 deg. from each valley.

5.) With a building with walls only running north-south or east-west all ridge lines will always be north-south or east-west.

6.) Likewise all hips, valleys and flying hips will be oriented northwest, northeast, southwest, or southeast.

7.) When a valley meets a ridge, (they will always meet at 45 deg) a flying hip is generated that is 90 deg. from the valley and 135 deg from the ridge.

8.) When two valleys meet at 180 deg. from each other, the result will be either to flying hips perp. to the valleys or the degenerate case of 4 valleys and 4 ridges.

9.) When a valley and a hip meet each other at 180 deg. then two ridges that are 45 deg from the valley will be the result (typical L shaped roof).

10.) When 4 hips meet the result is a pyramid.

There may be a few other degenerate cases I'm missing but I think that covers it.