Having trouble searching the interwebs because I'm unsure of the terms. Lets say I have a vector that can rotate around the Y and X axis, but only to a limited degree, and I want to draw the shape formed by that. In 2d, it would be a single arc, and a piece of a pie, in 3d it would be like a piece of a sphere.

I believe what you're looking for is Solid Angle. EDIT2: Actually, maybe not. See Promit's comment below. EDIT: A solid angle What you describe is a cone, but with a semi-spherical base. If you're thinking more of a rectangular shape, as if a globe were sliced to the center from a rectangle on its surface, I've seen that called a "rectangular solid angle," though, as I remember from my solid geometry days, that may also mean the solid angle subtended by a truncated rectangular pyramid. Anyway - some phrases to consider.

I believe what you're looking for is solid angle.

Well, no. That's an angle. I'm thinking spherical segment or spherical lune, maybe? Neither seems quite correct though. I think we need a spherical segment that's been cut again by another plane.

Well, no. That's an angle. I'm thinking spherical segment or spherical lune, maybe? Neither seems quite correct though.

You're correct. As you mention, segment just doesn't seem to fit the bill, and lune is a different concept. Perhaps there just ain't no name for it.

EDIT: Spherical wedge?

Spherical cap maybe ? With or without the associated cone to the center.

I wonder if it's a spherical polygon? https://en.wikipedia.org/wiki/Spherical_trigonometry#Spherical_polygons

Solid of revolution? I.e., a pie-shape rotated about it's symmetric axis?

Can we call this bike-shedding yet? Or is there a minimum number of posts, and/or minimum time elapsed for that?

Do you mean http://en.wikipedia.org/wiki/Spherical_sector ?

To summarize, if you want the arc (the portion of the CIRCLE that is "in between" the two angles, as a 1D curve) then the equivalent is spherical cap (as a 2D surface), if you want the sector/"piece of a pie" (the portion of the DISK that is "in between" the two angles, as a 2D area) then the equivalent is a spherical sector (as a 3D volume).

generalized to non-circular ellipsoid shapes, i've also heard them referred to as spheroid surfaces or spheroid surface sections, or spheroid volumes or spheroid volume sections.

sections of surfaces of rotation, or sections of volumes of rotation is another possibility.