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.
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.
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.
‘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.
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.
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”.
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
And the issue with sufficiently-fractal shapes is that, for some of them, every point will be zero distance from the boundary. But I think that, for such a shape, you could just say that every interior point has equal claim to being the “center”. But most of what we refer to as “fractal shapes” still have a 2-dimensional interior of nonzero area containing nonzero neighborhoods, and it’s just the boundary that’s fractal.
For some figures there might be more than one candidate point - for example consider an hourglass shape - the point most distant from the sides certainly isn’t in the waist, but it could be in either of the two rounded parts.
ISTM your distance function ought to have some sort of feature to penalize long distances more than short. E.g. use RMS distance not raw distance.
Here’s a concrete example: Consider a rectangle two units by four. For concreteness assume the long dimension is horizontal = left/right in your diagram.
Clearly the most interior point lies somewhere on the centerline of the long axis. And at least 1 unit in from either end. So on a line superimposed on the long axis, two units long, and centered along that axis.
For any point on that line the distance to the left and right ends totals the same number: 4. But ISTM that 2 units left and 2 units right is “better” than 1 unit left and 3 units right. Even though both distances total to 4. The way to get that outcome is with a non-linear distance function.
Which corresponds roughly to @Francis_Vaughan’s algorithmic approach where once we collapse the distance to a set of lines, we shrink the lines from the ends towards a point.
I say “roughly” because for irregular figures I don’t know that uniformly eroding both ends of a line at the same rate is actually correct. Imagine a shape like a baster or spoon. A long rectangular region with a bulb at one end. I’m thinking maybe / probably the center ought to end up closer to the bulb. Which means the rectangular end of the centerline ought to erode faster than the other one.
This is all armwaving; I sure don’t have rigorous proof of any of this. But it’s fun.
Broadly true at a coarse scale - but the devil’s in the details. If you look closely, Colorado has not 4 sides, but 697: those straight lines wiggle a bit, in some cases by hundreds of feet.
Every finite plane shape will have at least one polygon of at least three sides that contains that shape. While there may not be a polygon that is entirely within that shape.
Which leads to a couple of new center definitions.
The centroid of the circumscribed polygon of least area.
The centroid of the circumscribed polygon of least number of sides.
Of course there’s no requirement that either of those circumscribed polygons are unique. There might be a few different shapes that are all of least area. And a horrendous number of them with the least side count.
You can always circumscribe a triangle. Maybe you want the circumscribing triangle of least area? I’m not convinced, though, that the centroid of that triangle would always be interior to the original shape (even if it’s convex).