How to mitre exterior trim when the pieces are canted

It is not possible to have crown molding on the rake and eve that are identical in both size and profile if the spring for both is stipulated to be identical. It is geometrically impossible.

The simplest example is zero spring and 1x stock. It is impossible to use, say, 1x6 for both the rake and the eve without leaving a triangular cheek of the rake board hanging below the eve (which is typically removed with a horizontal saw cut)

That is one of the issues solved by 'returns', as in the following sketch:

The spring on the rake and on the eve necessarily must differ if the crown moldings (even 1x) are to be the same size and profile.

Try this thought experiment. Install crown moulding in a room that has an 'outside' corner. Convince yourself that if all the crown moulding has the same size and profile, then the spring must necessarily be the same on all walls. Now take down the two lengths of moulding that meet at the outside corner, then reattach them at the mitre into there previous 90 degree L configuration. Now hold that assembly up as if one length were the rake and the other length were the eve. You'll see that the spring on the rake is the same as it was in the room, and spring on the eve is much greater. No way around it.

The thought experiment shows how to figure out the angles for rake and eve crown moulding: do it the same way as if your were trimming a room. When installed, the spring of the eve will equal the spring of the rake plus the pitch of the roof.

Given that, here is a method of obtaining the correct mitre angles that does not require any 'figuring out'.

1. choose a spring angle for the rake, say 38 degrees
2. from 3/4" plywood, cut a couple of small triangles to have that angle
3. attach the triangles to a scrap of 1x4 by screwing through the 1x
4. mount the assembly in a chop saw with the triangles bearing against the backrest
5. swing the chopsaw to make a standard 45 degree cut
6. if it is a compound chopsaw, set the vertical angle to 90 degrees
7. cut the end off the 1x
8. detach the triangles from the 1x
9. measure the angles, or simply adjust the setup using the 1x as a template

Note that using a 1x4 works even if the actual crown moulding has a radically different size or shape.

An minor alternative is to subtract the spring angle from 90, then cut the small triangles to that angle, then mount the assemble on a chopsaw with the triangles bearing on the bed as opposed to the backrest. If you made right-triangles, they are one-in-the-same.