How to Tell if a Point Is Inside a Shape

As I continue to approach readiness for the Little Mathematics A-to-Z, let me share another piece you might have missed. Back in 2016 somehow two A-to-Z’s wasn’t enough for me. I also did a string of “Theorem Thursdays”, trying to explain some interesting piece of mathematics. The Jordan Curve Theorem is one of them.

The theorem, at heart, seems too simple to even be mathematics. It says that a simple closed curve on the plane divides the plane into an inside and an outside. There are similar versions for surfaces in three-dimensional spaces. Or volumes in four-dimensional spaces and so on. Proving the theorem turns out to be more complicated than I could fit into an essay. But proving a simplified version, where the curve is a polygon? That’s doable. Easy, even.

And as a sideline you get an easy way to test whether a point is inside a shape. It’s obvious, yeah, if a point is inside a square. But inside a complicated shape, some labyrinthine shape? Then it’s not obvious, and it’s nice to have an easy test.

This is even mathematics with practical application. A few months ago in my day job I needed an automated way to place a label inside a potentially complicated polygon. The midpoint of the polygon’s vertices wouldn’t do. The shapes could be L- or U- shaped, so that the midpoint wasn’t inside, or was too close to the edge of another shape. Starting from the midpoint, though, and finding the largest part of the polygon near to it? That’s doable, and that’s the Jordan Curve Theorem coming to help me.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

3 thoughts on “How to Tell if a Point Is Inside a Shape”

  1. I know that theory! It’s a great one to show my students too – “Here’s a maze, but I haven’t coloured the path or the walls. Is this cross on a wall or a path?”

    And the one about tracing a line without taking your pencil off the paper. It means I can watch those ads for a game involving that and go “Should have started at the 3-way intersection. Nope, that shape isn’t solvable.” Smug noises


    1. They’re both wonderful ones, and the kinds of theorem that look like magic. Especially about counting the number of three-way intersections, for tracing a figure. When you don’t know the trick, being told the steps to follow sounds like an irrelevant bit of nonsense and then suddenly … you’re there!


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