# Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point inside of the body to another point inside of it is completely contained in the body.

But what would you call the attribute that the shape of the body can be completely determined by rotating it and looking at it's 2D projection on a surface?

For example, imagine a cylinder that you curve into the shape of the letter $C$. The body that you get isn't convex, but if you imagine how it's silhouette would look you could completely describe it.

Ok, so to give another point of view and hopefully better describe what I mean, imagine this. You have a block of clay. In front of the clay is a surface. On the surface, there's a projection of the body under some rotation. What you do is, from the current point of view, cut out all the clay that's "not in the silhouette", cookie cutter style. Then you rotate the block of clay around itself but keep the surface between you and it. The surface doesn't move, but the projection changes as if you were rotating the body that's projected. For each rotation you repeat the cutting out.

Now, the bodies that I'm describing would be identical to the end result of the clay cutting, and the bodies that don't have this attribute would have less volume than the thing you get from cutting the clay. A red blood cell, for example, is something that you couldn't make like this because it has that dimple in the middle that will simply have no effect on the silhouette no matter how you rotate it.

– lhf
Jun 18, 2014 at 22:57
• @lhf That actually seems spot on. Any idea what the name for my thing could be? Jun 18, 2014 at 23:10
• they're called "complements of reunion of straight lines" or maybe "laser-of-doom sculptures" Sep 13, 2016 at 0:36
• Possibly related: mathoverflow.net/questions/39127 Sep 13, 2016 at 14:02
• Every point in the complement is part of a straight line contained in the complement, so "complements of (re)union of straight lines" seems to be adequate. Note that connected sets that are complements of unions of planes are exactly convex sets in $\Bbb R^3$.
– san
Sep 16, 2016 at 21:05