Inscribed circle in general case

Question

EDITED

Let we have some planar figure with smooth (or poly-line) boundaries, but in general case non-convex and with holes.

The main goal is to inscribe in this figure maximum possible circle.

But what is exact definition for inscribed circle for such general figures?

Firstly, if the circle lies entirely inside the figure it is not the case.

If the circle has only one common point with the figure we can say that circle touching the figure (?)

So if the circle has two or more common points with the figure (and except these points lies inside the figure) we can say that the circle is inscribed in the figure.

Is it correct?

UPD Here I've found a very clear definitions. E.g.: The maximum inscribed circle, sometimes referred to as the plug gauge circle, is the largest circle that is totally enclosed by the profile.
But what does "totally enclosed" exactly mean?

Answer 1

Edit in response to edited question.

"Totally enclosed" means simply that the circle (interior and boundary) consists only of points that are in the figure. The intersection of the circle and the figure is the circle.

There may be several "maximum inscribed circles" (think of a dumbbell figure). I suspect that finding one (with an algorithm) is a hard problem.

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I'm not quite sure what you mean by a "general definition". You will have to decide for yourself what definition makes sense for your project.

You seem to be concerned with figures that have smooth enough boundaries (piecewise linear in your example).

Saying that an inscribed circle is one whose interior is entirely contained in the region while the boundary touches the boundary of the figure at least twice might be what you want. That would mean a small circle in the corner of a square (so it touched just two sides) would count as inscribed, even though there are larger circles that touch the same two sides once each. If that's unacceptable, consider requiring that the circle and the boundary share at least three points.

If your regions are allowed to have holes (think of the region between two concentric circles) you may have to adjust your definition.