Name for Most Interior Point of a Plane Figure

Factual Questions
1

For any finitely large plane figure, there has to be a name for the point most distant from all its sides, right?

I’d’ve thought ‘centroid’ did it, until I saw how to determine the centroid of an L-shaped object. Maybe ‘centroid’ works for all convex shapes?

I’m thinking of something like a continental pole of inaccessibility, where the sides are equivalent to the ocean. Like, how can I figure the most interior point of, say, Montana?

Edited to add: It looks like for some concave shapes, the point most distant from all its sides might not be a point at all, but a line (I think?). But I’m still interested if the concept has a name.

2

Arbitrary shapes like Montana don’t have “sides”. They have a perimeter. Which is fractally complex and consists of an infinite number of points.

You’ll need a precise mathematical definition of what you consider “Most distant from the entire perimeter of an arbitrary fractal shape” before we can actually answer your question.

But for any generally convex shape, the centroid is going to be more or less what you’re looking for.


As a thought experiment, try to explain what you’d consider the best answer for an L-shaped region with appreciable thickness to the legs.

3

I would call it the “incenter”, or more formally, the center of the largest inscribed circle. Though it’s not necessarily uniquely defined: For a non-square rectangle, for instance, there will be a line segment in its middle, any point of which can be called that.

4

For an L shaped object I would draw a center line down each leg and use their intersection.

5

‘side’ jack - Montana shares boundaries with 4 states and 3 provinces - the boundary with Idaho may appear to be fractally complex, but that with Canada is a line rigorously defined by length and direction. Colorado, Wyoming and Utah have borders composed entirely of straight lines geometrically defined.

6

Several of the western states have full or partial borders with each other that are rigorously defined as straight lines, which are poorly surveyed, varying sporadically by 2" or more along their length. Given that some of these borders traverse mountain ranges, that is hardly surpising.

7

Or, the “excenter”, the closest point to the center of the smallest exscribed circle.

9

I don’t think that’s what the OP’s describing, because for a non-convex shape, that point could itself be exterior to the shape, and even for a convex shape (such as a right or obtuse triangle), it could be exactly on the boundary. Hardly the “most interior”.

10

Maybe. Are we maximizing the closest distance to a boundary, or minimizing the farthest distance to a boundary?

Of course, we could get away from circles, and instead integrate over the area or the boundary. And what weighting function?

It’s an interesting question because there’s lots of ways of approaching it, which give different answers.

11

two more:

geography (e.g. mapbox polylabel algorithm): pole of inaccessibility

also seems related: chebyshev center

12

You might define the point algorithmically. If you erode the perimeter equally the body will shrink down to yield one or more lines or points. Lines will erode from the ends until they form a point. You can still end up with more than one point.

Still not sure what such points would be called but the pole of inaccessibility as mentioned above might be a good fit.

Bodies with fractal like boundaries should still erode sensibly. Creating a precise definition for erosion might prove troublesome for true fractals