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Odd Means Inside

Experiment № 104 · October 29, 2026

Is a point inside a shape? Shimrat's 1962 answer: shoot a ray from it in any direction and count the edges it crosses. Every crossing switches you between inside and outside, so odd means inside. Count with signs instead, +1 for an edge going one way and −1 for the other, and you get the winding number. The two rules agree until a shape crosses itself.

Colour the plane by

Shape

edges crossed by the ray— even–odd says— winding number— nonzero rule says—
edges in the shape0 inside by even–odd— inside by nonzero— where they disagree—

The whole test, for probe p

inside = false; w = 0for each edge a → b: if (a.y > p.y) != (b.y > p.y): // straddles x = a.x + (p.y−a.y)·(b.x−a.x)/(b.y−a.y) if x > p.x: // on the ray inside = !inside w += (b.y > a.y) ? +1 : −1

Coordinates are turned so the ray points along +x. The strict > on both ends is the whole trick for vertices that sit exactly on the ray: each one belongs to the edge above it, never to both.

Share of the shape, by winding number

In Disagree, pink marks points wound around an even number of times: inside by the nonzero rule, outside by even–odd. Arrows on the outline show which way each edge runs.

Move to aim the probe · drag a corner to reshape · W walks the edges · space toggles drift · H hides panels

Drag one closed loop on the canvas

Experiment № 104 · The story

Odd Means Inside

Why I picked it

Two days ago, for One Wall at a Time, I needed to know whether your finger had landed on the shape so you could drag it. I wrote a seven-line function to check, the same seven lines I have written dozens of times without thinking, and moved on. It bothered me afterwards that the most-used piece of geometry in that page was the one I never showed you. The idea is old: M. Shimrat published it as Algorithm 112 in Communications of the ACM in 1962, and a correction followed in the same journal within months. W. Randolph Franklin's compact version, PNPOLY, is the one most people have copied since, me included. Every map that tells you which country you clicked, and every game that checks whether you stood inside a zone, runs something like it.

What I enjoy about it

It turns a question about the whole shape into a count along one line. You don't need to know what the shape looks like, only how many times its boundary passes in front of you in one direction. Turn the ray-direction slider and the crossings jump around, sometimes one, sometimes five, but the parity never changes. I also like the strict comparison on line three. When the ray runs exactly through a corner, a careless test counts it twice or not at all. Asking "is this end above the ray?" for both ends means each corner belongs to the edge on one side of it, so a path that passes through a corner is counted once and a path that only touches a corner and turns back is counted zero or two times. Both answers come out right without a single special case. Then there is the pentagram. Its centre is wound around twice, so it has two crossings: even-odd says outside, the winding number says inside. Neither is wrong. They are two definitions, and SVG, canvas and every font renderer make you choose one with a fill-rule setting. Switch to Spirograph and Winding for the prettiest version of that disagreement.

Something you might take away

Many questions about a whole thing can be answered by looking in one direction and counting. Am I enclosed? Count the walls between me and infinity. Have I made a full loop? Add up the turns. And when a simple definition like "inside" stops being obvious, as it does for a shape that crosses itself, that is usually the moment to admit it was a choice all along and write down which one you made.