Okay, so, say you have a convex shape. Then the centroid of that set lies somewhere inside. You also have the “pole of inaccessibility” (centre of the largest inscribed circle) the OP was talking about (and there are algorithms to find it given a map of Montana). However, even for a triangle, in general the two points will not coincide.
In fact, there are something like 50,000 different definitions for a “center of a triangle”.
71, 392 according to
Brian
Sure. But the OP wanted the “[interior] point most distant from all its sides”, and I think that has been more-or-less answered, and it was not the centroid.
Related question now that the main question has been answered: do all of those 70k ish center definitions coincide for an equilateral triangle? Equivalently, are there any “centers” that are different to the centroid for an equilateral triangle?
This article is on-point: Triangle center - Wikipedia. The third paragraph reads in part:
That article also has lots of other high level discussions about various categories of centers and their properties. The details get deep quick, but the overview is approachable enough to the layman.
Speaking just off my own bat, I would think that any proposed center function that did not equal the centroid on an equilateral triangle would be disqualified. Back to the drawing board.
So a function’s meeting the equilateral center is a necessary condition, not a consequence.
For an irregularly shaped area like a state or a country. I would define the centre as the point furthest away from any edge.
The centre of mainland UK (the point furthest from the sea) is Church Flatts Farm, located less than a mile southeast of Coton in the Elms, Derbyshire, approximately 70 miles (113 km) from the nearest coast.
Theoretically, there is no such point because there is always another point between any two points ad infinitum. It’s like calculating a limit in calculus. You never reach a terminal point, you just get infinitely closer to it.
When you say closer to “it”, what do you mean? Closer to what exactly? Sounds like you may have identified such a point. Also, just because there are infinitely many points between two points, there is always precisely one point that is half way (minimising the distance to each simultaneously). Depending on the values of your starting two points, you may need arbitrarily many digits (potentially infinitely many) to specify that point, but that doesn’t change the fact that it exists.
The theoretical mid-point in the OP. Since there is always yet another point between any 2 points, there is no finite mid-point.
That makes no sense.
Just because there are infinitely points between two points doesn’t mean there are multiple midpoints.
ETA:
While there are infinitely many definable points between (0,0) and (0,2), that doesn’t change the fact that (0,1) is the precise midpoint, and no other point meets that definition.
That’s not what I said. I said there is NO midpoint.
Why would there be no midpoint?
Because there can be no midpoint in a line segment comprised of an infinite number of points, just like a penalty that is “half the distance to the goal” can never reach the goalline even the penalty is repeated an infinite number of times.
Again, the midpoint between (0,0) and (0,2) is (0,1). For your statement to be true, you would have to be claiming that the point (0,1) does not exist on the line between (0,0) and (0,2).
ETA: You seem to be conflating the idea that any length can be cut in half with the idea that half exists at all.
But that wasn’t the OP. The following is an OP quote:
For any finitely large plane figure, there has to be a name for the point most distant from all its sides, right?
There are an infinite number of points between all of the sides of the figure. Since there are an infinite number of points, and since I can always find another point on any side of your postulated midpoint, then there is no midpoint.
Wait a second! According to my 3rd level calculus text book, an irregularly shaped figure has a single center point called a centroid, even though the shape contains an infinite number of points. Wow! See, that’s why I only got a “B” in Calculus 3. LOL
If it’s the center, then it’s that same distance from three different coasts.
What do you mean “on any side” of a point? Points don’t have sides.
I wonder if the Chebyshev centre is the answer to the question that the OP asked?
Possibly, but “Centroid” seems to satisfy his stipulations, also.