So I'm trying to make this texture. I need to be able to make a perfect slice around any part of the sphere for it to look good (Imagine taking a sphere, and cutting a part off either left or right of the center with a katana). Anyone know how to do this?

Before anyone suggests it. I don't have access to any 3d modeling programs. Polar transformations don't make a perfectly correct slice. And using a 1:2 ratio also doesn't work.

To give an example of what I'm looking for look at the head templates. I just need to be able to make those sorts of lines anywhere I need to (Preferably without trial and error).

As an aside. For those of you who are familiar with spherical coordinates. If I wanted make a slice from a sphere... I think all that needs to be done is set: ρcos(θ)sin(φ) or ρsin(θ)sin(φ) equal to some constant less than 'ρ' and greater than zero? Dunno about that for sure though, or how I could use that to help out in this situation >_>