This picture of an 8:12 & 10:12 irregular hip roof model shows how the hip rafter backing angle aligns the hip rafter with the square tail fascia.

This picture shows how the hip rafter backing angle aligns the hip with the roof surface.

This picture shows the valley rafter backing angle aligning with the plan angle.

This picture shows how the hip rafter backing angle aligns the hip rafter with the plan angle.

Valley rafters backed out to keep the top of the valley rafters aligned with the roof surface.

The hip rafter backing angle is developed from the dihedral angle triangle. The dihedral angle is the angle between two planes. In roof framing geometry the dihedral angle is the angle between the two roof surfaces on each side of the hip rafter.

This drawing shows the dihedral angle drawn in plan view as well as a 3D view of the dihedral angle.

In this drawing , in plan view, the dihedral angle triangle is drawn from the eave line (AB) to the opposing eave line (AF). Note, the eave line is also referenced as the gutter line. To draw the dihedral angle triangle start with a line at point W. The point W can be located anywhere on the eave line. The line WX is always drawn perpendicular to the hip rafter run line AE. At point X draw a line to point Y that is perpendicular to the hip rafter length line AG. Then swing an arc from point X with the radius equal to the line length XY until it intersects the hip rafter run line AE. Angle XWZ is the hip rafter backing angle and angle XFZ is the hip rafter backing angle for the other side of the hip rafter. For an equal pitched roof these angles will be the same. For irregular pitched roofs the hip rafter backing angle will not be the same angle.

The dihedral angle is angle WZF. The angle between the roof planes.

Here are a couple more drawings showing the dihedral angle in 3D.

Here's a link to the Google SketchUp files showing the Basic Roof Framing Kernel with the hip rafter backing angle and the location of the dihedral angle.

Hip Rafter Backing Angle-1.skp

Hip Rafter Backing Angle-3.skp

Hip Rafter Backing Angle-4.skp

Hip Rafter Backing Angle-6.skp