Question

I’m just wondering if there is any simple/efficient way to check if square falls inside a triangle. or at least one corner falls inside or some overlaps. e.g. considering the figure below, I should be able to tell that the 3 squares fall inside. i.e. square 1 is obviously inside, one corner of square 2 is inside, and 3 overlaps.

Answer 1

I’m checking out this nice tutorial. It explains how to test if a point is inside a triangle using various techniques. It seems it would help when a square corner falls inside.

And I liked the Barycentric technique, here I re-implemented it for matlab:

function d = isinside(p,a,b,c)

    % Test if a point p(x,y) is inside a triangle
    % with vertices a(x,y), b(x,y) and c(x,y)

    v0 = c - a;
    v1 = b - a;
    v2 = p - a;

    A = [dot(v0,v0) dot(v1,v0);dot(v0,v1) dot(v1,v1)];
    b = [dot(v2,v0); dot(v2,v1)];

    x  = A\b;

    % Check if point is in triangle
    if (x(1) > 0) && (x(2) > 0) && (sum(x) < 1)
        d = true;
    else
        d = false;
    end

Then I would test each vertex of the square, and if it happens one of them falls inside I would return. Quite lot of computation but it worths a try.

For an overlap, I would test for intersections, as discussed in this thread, for every combination of lines from both a triangle and a square.