# Name for Most Interior Point of a Plane Figure **URL:** \ **Category:** Factual Questions\ **Created:** [August 31, 2026, 10:22pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728 "2026-08-31T22:22:57Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![whc.03grady](https://avatars.discourse-cdn.com/v4/letter/w/a6a055/32.png) [@whc.03grady](https://boards.straightdope.com/u/whc.03grady)\ **Post date:** [August 31, 2026, 10:22pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/1 "2026-08-31T22:22:57Z")
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](https://en.wikipedia.org/wiki/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. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [August 31, 2026, 11:03pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/2 "2026-08-31T23:03:53Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 31, 2026, 11:11pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/3 "2026-08-31T23:11:13Z")
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. ---
**Author:** ![mixdenny](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mixdenny/32/2962_2.png) [@mixdenny](https://boards.straightdope.com/u/mixdenny)\ **Post date:** [August 31, 2026, 11:13pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/4 "2026-08-31T23:13:01Z")
For an L shaped object I would draw a center line down each leg and use their intersection. ---
**Author:** ![vbob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/vbob/32/4439_2.png) [@vbob](https://boards.straightdope.com/u/vbob)\ **Post date:** [August 31, 2026, 11:44pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/5 "2026-08-31T23:44:33Z")
> [@LSLGuy](#): > > Arbitrary shapes like Montana don’t have “sides”. They have a perimeter. Which is fractally complex and consists of an infinite number of points. ‘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. ---
**Author:** ![eschereal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/eschereal/32/18939_2.png) [@eschereal](https://boards.straightdope.com/u/eschereal)\ **Post date:** [August 31, 2026, 11:57pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/6 "2026-08-31T23:57:11Z")
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. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [September 1, 2026, 12:02am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/7 "2026-09-01T00:02:10Z")
> [@Chronos](#): > > I would call it the “incenter”, or more formally, the center of the largest inscribed circle. Or, the “excenter”, the closest point to the center of the smallest exscribed circle. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 1, 2026, 1:02am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/9 "2026-09-01T01:02:24Z")
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”. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [September 1, 2026, 3:43am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/10 "2026-09-01T03:43:01Z")
> [@Chronos](#): > > Hardly the “most interior”. 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. ---
**Author:** ![Arjuna34](https://avatars.discourse-cdn.com/v4/letter/a/eb8c5e/32.png) [@Arjuna34](https://boards.straightdope.com/u/Arjuna34)\ **Post date:** [September 1, 2026, 3:47am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/11 "2026-09-01T03:47:15Z")
two more: geography (e.g. mapbox polylabel algorithm): pole of inaccessibility also seems related: chebyshev center ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [September 1, 2026, 9:18am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/12 "2026-09-01T09:18:10Z")
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 ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 1, 2026, 10:51am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/13 "2026-09-01T10:51:32Z")
> [@Arjuna34](#): > > geography (e.g. mapbox polylabel algorithm): pole of inaccessibility Mentioned in the OP. 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. ---
**Author:** ![dcbwhaley](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dcbwhaley/32/21437_2.png) [@dcbwhaley](https://boards.straightdope.com/u/dcbwhaley)\ **Post date:** [September 1, 2026, 11:03am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/14 "2026-09-01T11:03:10Z")
Even if the boundary is fractal it can be approximated to an irregular polygon of many sides ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [September 1, 2026, 11:37am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/15 "2026-09-01T11:37:50Z")
> [@whc.03grady](#): > > 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. 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. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [September 1, 2026, 12:16pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/16 "2026-09-01T12:16:50Z")
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. ---
**Author:** ![3AxisCtrl](https://avatars.discourse-cdn.com/v4/letter/3/ecd19e/32.png) [@3AxisCtrl](https://boards.straightdope.com/u/3AxisCtrl)\ **Post date:** [September 1, 2026, 12:37pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/17 "2026-09-01T12:37:57Z")
> [@vbob](#): > > Colorado, Wyoming and Utah have borders composed entirely of straight lines geometrically defined. Broadly true at a coarse scale - but the devil’s in the details. If you look closely, Colorado has [not 4 sides, but 697](https://bigthink.com/strange-maps/colorado-is-not-a-rectangle/): those straight lines wiggle a bit, in some cases by hundreds of feet. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [September 1, 2026, 2:15pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/18 "2026-09-01T14:15:16Z")
> [@dcbwhaley](#): > > Even if the boundary is fractal it can be approximated to an irregular polygon of many sides 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. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [September 1, 2026, 3:32pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/19 "2026-09-01T15:32:03Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 1, 2026, 11:53pm UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/20 "2026-09-01T23:53:54Z")
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). ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [September 2, 2026, 12:24am UTC](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728/21 "2026-09-02T00:24:33Z")
> [@Chronos](#): > > I’m not convinced, though, that the centroid of that triangle would always be interior to the original shape (even if it’s convex). Agreed. I can imagine some convex but highly skew shapes where the centroid of the circumscribed triangle is outside of the perimeter. [Next page](https://boards.straightdope.com/t/name-for-most-interior-point-of-a-plane-figure/1032728.md?page=2)