How to find whether a point belongs to a shaded area on a plane?

Question

Mathematics

coordinates are only positive (it turns out this is only the first quarter of the entire plane), the plane itself is infinite.

Answer 1

Separately, check that the dot is in the first quarter.
Further, the figure is repeated in 2x2 blocks, i.e. you don't care about the coordinates themselves, only their position in the block.
To calculate a coordinate in a block, you can either subtract 2 while the coordinate is greater than 2, or subtract from x (floor(x) - floor(x)%2). floor(x) will give integers. Then we need to make this number even, it is possible to subtract 1. floor(x)%2 - just gives us what to subtract.
And then - a set of if-s. you only need to describe that the point belongs to one of the two quadruples of the circle in the lower left corner.

Answer 2

In, this is already cool)
squares - 1x1?
offhand like this:

if (Math.ceil(y) % 2 && !(Math.floor(x) % 2)) {
  return ((x - Math.floor(x)) ^ 2 + (y - Math.ceil(y)) ^ 2 <= 1);
}
if (!(Math.ceil(x) % 2) && Math.floor(y) % 2) {
  return ((x - Math.ceil(x)) ^ 2 + (y - Math.floor(y)) ^ 2 <= 1);
}
return false;

Math.ceil - round up, Math.floor - down, ^2 - squared

Answer 3

Here, as it were, filled circles, with step 2:
with centers at ( 2 * i, 1 + 2 * j )where iand jintegers.
Odd quarters were cut off from the circles.
I do not write in C ++, an example in JS:

function isInArea(x, y) {
  centerX = 2 * Math.round(x/2);
  centerY = 1 + 2 * Math.round((y - 1)/2);
  if (! ((x >= centerX) ^ (y >= centerY))) { // НЕ (XOR)
    return false;  // не та четверть
  }
  // проверить расстояние
  return 1 >= (x - centerX) * (x - centerX) + (y - centerY) * (y - centerY);
}