# existing name for name for a "castable" 3d object?

For our purposes, consider an object castable (as in 'cast in a mold') if there exists a plane such that every line orthogonal to that plane passes through the surface of the object exactly zero times or exactly twice.

We can also have a variation: A plane exists such that every ray orthogonal to this plane originating from this plane passes through the surface of the object exactly 0 times or exactly once. This is stricter because it eliminates certain types of overhangs, and is equivalent to saying that there exists a plane dividing the object such every point of the object's surface projects onto this plane without that projection ray passing through any other part of the surface.

For example: A solid sphere is castable, for instance by bisecting it. A torus is also castable by both definitions (the plane cuts it like you cut a bagel). A partially hollowed sphere (like a soccer ball - remember it's filled with air and has a thickness) with an inner and outer surface, is NOT castable by either definition, since some line perpendicular to any plane will pass through the surface 4 times. If you cut this hollowed sphere in half then that half is castable by definition 1, but not by definition 2.

• convex, probably? Commented Aug 29, 2018 at 20:45
• hmm, no, convex is a stronger requirement. Commented Aug 29, 2018 at 21:20
• Your description might be clearer if you gave an illustration, as for example sphere. I'm confused about whether the definition was meant to apply (for example) to a hollow sphere or a solid ball. Commented Aug 29, 2018 at 21:21
• Clarifying that the question is about the surface of the object. Commented Aug 29, 2018 at 21:28
• Shouldn't you allow lines that pass the surface once, namely those lines that intersect the plane where the object also intersects the plane. With the sphere for example, the lines tangent to the sphere and orthognal to the plane only intersect the sphere once Commented Aug 29, 2018 at 21:50

For example star convexity. Here we only look at lines trough a certain point, but the rest remains the same: A set is star convex iff every intersection with a line trough a fixed point $p$ is connected.
In Restricted-Orientation Convexity by Eugene Fink and Derick Wood, this is further generalised to $\mathcal{O}$-convexity. The lines with which we will consider the intersections are the translations of lines in some set $\mathcal{O}$. If you choose $\mathcal{O}$ to be the set of the 3 coordinate axis, then you get ordinary ortho-convexity.
If you chose $\mathcal{O}$ to be a single line orthogonal to the plane, then a set is $\mathcal{O}$-convex iff and only iff every intersecion with a line orthogonal to the plane is connected.
If I understood your condition correctly, then a castable object is an object such that its union with a plane is $\mathcal{O}$-convex, where $\mathcal{O}$ is the set containing one line orthogonal to the plane.