How to check if a point (int - Coordinates) is within the hypotenuse of a triangle

Question

I've got a right triangle and I want to check if a given point is on the Hypotenuse of that triangle. All points are plain integers, not floating-point variables.

Edit: All green squares would be on the hypotenuse, the white squares not. I know x, y, the Coordinates of the Corners and the Coordinates of the Point I want to test. All Coordinates are whole numbers (y is a little bit off in the drawing, sorry).

Answer 1

I'll start doing this:

points = an array;
delta=y/x
acc = 0
j = 0
for (i=0;i<x;i++){
  points.push(i, j)
  acc+=delta
  while (acc > 1){
     acc-=1
     j++
     points.push(i,j)
  }
}

And then you have all the points in the hypotenuse. There are better algorithms for drawing lines, but this could be a start.

Answer 2

This could work:

You know the triangle, so just create the function for the hypothenuse, in your example it would be y = 5x/12. If you now get a point, say x = 6, y = 3, you can use these variables to see if it comes out right: 3 = roundup(5*6/12). If it does, the point is on the triangle, if not - then not.

Answer 3

There are two cases to handle: one where the hypotenuse is vertical and the other where it is not.

For the vertical case, you just check if the point in question has a y value in the range of the hypotenuse.

For the non-vertical case, derive the equation of the hypotenuse using its endpoints. The equation of a line is y = mx + b where m is the slope which is dx/dy. Then b = y - mx.

Now that you have m and b, see if a candidate point's x and y satisfy the equation (does the point's y equal m * x + b ?).

However, practically, you should check for nearness instead of exact equality so check if the point's y is within some small delta of (m * x + b).