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I'm looking for geometric terminology that would describe this kind of shape, if there is a term for it. Picture any arbitrary closed 2D shape. Picture the smallest circle that will completely contain that shape. Picture radial lines, like spokes, from the center of the circle to the edges. Each radial line should cross the shape once and only once; meaning every point on the shape can be defined as a sub-length of a radius of the circle, and every radius would have exactly one point.

"Convex" does not cut it, because while I think all convex shapes would qualify, some concave ones would too, but the shape cannot have twists, self-intersections, or undercuts.

For example, this heart is concave but would be OK

enter image description here

But this crescent would not.

enter image description here

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The closest existing term that is used in the mathematical literature that I can think of off the top of my head is "star-shaped." However, the formal definition of "star-shaped" differs from your description in that it suffices that there exists a point in the region from which all radial lines drawn from that point intersect the boundary exactly once. In your case, your condition is more strict in the sense that the radial origin must be the center of the smallest circumscribing circle. It is an easy exercise for you to construct a shape that is star-shaped but does not satisfy your criteria. However, it is trivial to see that shapes satisfying your criteria are star-shaped.

For more information, see: Star Domain

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  • $\begingroup$ The question seems to be whether that line about 'smallest circle' is essential to the OP's purpose. If not, then star-shaped certainly works. If it is, then some modifier like 'minimally' or 'radially' for star-shaped would seem necessary. $\endgroup$ Sep 5, 2014 at 20:39
  • $\begingroup$ Wouldn't star-shaped be limited to polygons? I'm including curves. $\endgroup$ Sep 6, 2014 at 11:52
  • $\begingroup$ Once you have such a point, you can describe your curve using polar coordinates. Besides, proving that a certain form is star-shaped is not evident. See related "art gallery problem" (en.wikipedia.org/wiki/Art_gallery_problem). $\endgroup$
    – Jean Marie
    Sep 10, 2019 at 16:12

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