# Finding area of intersection regions

I want to be able to find the area of the intersection between a hexagon and a slice. I have tried RegionMeasure or Area but I seem to get incorrect answers. RegionMeasure gives length when the intersection is a line, so I switched to Area but sometimes it still gives weird answers, when the intersection happens to be a line.

One reason why I think this might occur is that I build the hexagon using Polygon, but the "slice" is with Disk. Is that the problem? How to I make a region as a slice, so that Area[RegionIntersection[hexagon, slice]] between the two regions will always give $0$ for a line intersection and no intersection, i.e. return a nonzero result only for 2D intersection.

See picture. I want to know the area of intersection of the black hexagon will each of the 6 (colored for visualization) slices below. The answer should be non-zero for the 2 upper left slices, and 0 for every other slice, even though some have "line intersections".

Thanks.

EDIT: The problem is that edges get non-zero measure. Why is the below code giving .7, and not 0?

DiskR = 1.5;
Show[
Graphics[{Orange, Opacity[0.35],
Disk[{0, 0}, DiskR, {Pi/6, 3 Pi/6}]}],
Graphics[Polygon[{{0, 0}, {0, .7}, {-1, 1}}]]]


and

Area[
RegionIntersection[
Polygon[{{0, 0}, {0, .7}, {-1, 1}}],
Disk[{0, 0}, DiskR, {Pi/6, 3 Pi/6}]
]
]


• This certainly works: Polygon[CirclePoints[{-(Sqrt[3]/2), 1/2}, {1, π/6}, 6]] ~RegionIntersection~ Disk[{0, 0}, 3, {π/2, π/2 + π/3}] // Area Jun 14 '16 at 15:52
• But why does a line give non-zero Area? See my edited question. Jun 14 '16 at 17:05
• Please post the code as text. Nobody wants to copy it out manually from the image. Jun 14 '16 at 18:08
• Sorry, new here. But I see your point. I edited and pasted the code. Jun 14 '16 at 18:15
• You didn't include the value of DiskR, but replacing DiskR with 3/2 seems to give something similar to your last figure. However, I can't reproduce your last result: Area[ RegionIntersection[ Polygon[{{0, 0}, {0, .7}, {-1, 1}}], Disk[{0, 0}, 3/2, {Pi/6, 3 Pi/6}] ] ] returns $0$ as you expected, not $0.7$ as you reported (results). I am using MMA 10.4 on Win7-64. What version are you using? Jun 15 '16 at 17:22

## 1 Answer

In this case, the intersection between two 2D regions is a 1D region - a line. It makes sense that the functions that are called by RegionMeasure and RegionIntersection might be influenced by roundoff errors associated with using floating point numbers rather than exact numbers.

So a workaround is to convert your expressions to use exact numbers prior to taking the area. I had originally used Rationalize to do this, and it works in this case, but it doesn't always. So we can borrow a function from illian that is a bit more robust

sp = Function[p, SetPrecision[p, Infinity]];
DiskR = 1.0;
Area[
RegionIntersection[sp@Polygon[{{0, 0}, {0, .7}, {-1, 1}}],
sp@Disk[{0, 0}, DiskR, {Pi/6, 3 Pi/6}]]]

(* 0 *)


While this behavior is understandable, I'd still say it's a bug