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Outer Billiards. How to create a test to skip triangle corner in for loop?

Posted 15 days ago
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Hey Everyone! I'm currently working on a problem in a course on mathematical modeling concerning Outer Billiards. I'm supposed to write a program that does the following:

Start with a ball (a point particle) somewhere outside an equilateral triangle with side length equal to 1. You have two possibilities here and you select one of them. When it arrives at the corner it has traveled a distance d1. Then the particle continues in the same direction as before the same distance d1 There it changes direction momentarily and moves towards the other corner. Then the procedure is repeated, at the second corner it has traveled a distance d2 and it continues in the same direction the same distance d2, etc.

One problem that I've encountered is that for some points the trajectory of the point particle crosses the interior of the triangle, which is not allowed. Therefore I would like to create a test inside of my for loop which says that: "IF the trajectory towards a corner crosses the interior of the triangle, move instead to the next corner." Now, my lecturer gave me a hint that one can use determinants in order to make a pretty easy test, but I find it somewhat hard to understand intuitively. So I would like to make another test, but I don't know how exactly. Here is the program that I'm working on:

corner = {{1/2, Sqrt[3] /2}, {0, 0}, {1, 0}};


ourtriangle = Triangle[{corner[[1]], corner[[2]], corner[[3]]}];


p0 = {2, 2};
plotpoints = {p0};
cornerindex = 1;
n = 3;

For[i = 1, i < n, i++,
 p1 = 2*corner[[cornerindex]] - p0;
 p0 = p1;
 AppendTo[plotpoints, p1];
 cornerindex = Mod[i, 3] + 1;
 ]

traj = Table[plotpoints[[i]], {i, n}];
plot1 = Graphics[{Dashed, Line[traj]}];

Show[plot1, Graphics[ourtriangle], Axes -> False]

This yields the following graph: enter image description here

So, in this case I would like the trajectory to instead move towards the right-most corner but I really dont know how. Could someone please give me at least a hint?

Thank you all.

4 Replies

RegionMember can test if a point is inside a region https://reference.wolfram.com/language/ref/RegionMember.html

I tried writing some code using Regionmember[] but I couldn't come up with a good test. Thanks!

If your straight line passes through the triangle (or any other convex polygon) it must cross at least two edges.

I made a short program to detect those crossings. See the notebook attached. If the length of xlist is greater or equal to 2 the line passes through the polygon.

You may want to play around with the number of points and the direction of the line.

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Looks really neat! I will take a look at this. Seems like its a much more intuitive test to do in this case, rather than using the determinant. Thanks!

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