Find all points of a grid within a circle, ordered by norm

Question

How would you solve the problem of finding the points of a (integer) grid within a circle centered on the origin of the axis, with the results ordered by norm, as in distance from the centre, in C++?

I wrote an implementation that works (yeah, I know, it is extremely inefficient, but for my problem anything more would be overkill). I'm extremely new to C++, so my biggest problem was finding a data structure capable of

1. being sort-able;
2. being able to save an array in one of its elements,

rather than the implementation of the algorithm. My code is as follows. Thanks in advance, everyone!

typedef std::pair<int, int[2]> norm_vec2d;

bool norm_vec2d_cmp (norm_vec2d a, norm_vec2d b)
{
    bool bo;
    bo = (a.first < b.first ? true: false);
    return bo;
}

int energy_to_momenta_2D (int energy, std::list<norm_vec2d> *momenta)
{
    int i, j, norm, n=0;
    int energy_root = (int) std::sqrt(energy);

    norm_vec2d temp;

    for (i=-energy_root; i<=energy_root; i++)
    {
        for (j =-energy_root; j<=energy_root; j++)
        {
            norm = i*i + j*j;
            if (norm <= energy)
            {
                temp.first = norm;
                temp.second[0] = i;
                temp.second[1] = j;
                (*momenta).push_back (temp);
                n++;
            }
        }
    }
    (*momenta).sort(norm_vec2d_cmp);
    return n;
}

Answer 1

How would you solve the problem of finding the points of a (integer) grid within a circle centered on the origin of the axis, with the results ordered by norm, as in distance from the centre, in C++?

I wouldn't use a std::pair to hold the points. I'd create my own more descriptive type.

struct Point {
  int x;
  int y;
  int square() const { return x*x + y*y; }
  Point(int x = 0, int y = 0)
    : x(x), y(y) {}
  bool operator<(const Point& pt) const {
    if( square() < pt.square() )
      return true;
    if( pt.square() < square() )
      return false;
    if( x < pt.x )
      return true;
    if( pt.x < x)
      return false;
    return y < pt.y;
  }
  friend std::ostream& operator<<(std::ostream& os, const Point& pt) {
    return os << "(" << pt.x << "," << pt.y << ")";
  }
};

This data structure is (probably) exactly the same size as two ints, it is less-than comparable, it is assignable, and it is easily printable.

The algorithm walks through all of the valid points that satisfy x=[0,radius] && y=[0,x] && (x,y) inside circle:

std::set<Point>
GetListOfPointsInsideCircle(double radius = 1) {
  std::set<Point> result;

  // Only examine bottom half of quadrant 1, then
  // apply symmetry 8 ways
  for(Point pt(0,0); pt.x <= radius; pt.x++, pt.y = 0) {
    for(; pt.y <= pt.x && pt.square()<=radius*radius; pt.y++) {
      result.insert(pt);
      result.insert(Point(-pt.x, pt.y));
      result.insert(Point(pt.x, -pt.y));
      result.insert(Point(-pt.x, -pt.y));
      result.insert(Point(pt.y, pt.x));
      result.insert(Point(-pt.y, pt.x));
      result.insert(Point(pt.y, -pt.x));
      result.insert(Point(-pt.y, -pt.x));
    }
  }
  return result;
}

I chose a std::set to hold the data for two reasons:

• It is stored is sorted order, so I don't have to std::sort it, and
• It rejects duplicates, so I don't have to worry about points whose reflection are identical

Finally, using this algorithm is dead simple:

int main () {
  std::set<Point> vp = GetListOfPointsInsideCircle(2);
  std::copy(vp.begin(), vp.end(),
    std::ostream_iterator<Point>(std::cout, "\n"));
}

Answer 2

It's always worth it to add a point class for such geometric problem, since usually you have more than one to solve. But I don't think it's a good idea to overload the 'less' operator to satisfy the first need encountered. Because:

• Specifying the comparator where you sort will make it clear what order you want there.
• Specifying the comparator will allow to easily change it without affecting your generic point class.
• Distance to origin is not a bad order, but for a grid but it's probably better to use row and columns (sort by x first then y).
• Such comparator is slower and will thus slow any other set of points where you don't even care about norm.

Anyway, here is a simple solution using a specific comparator and trying to optimize a bit:

struct v2i{
    int x,y;
    v2i(int px, int py) : x(px), y(py) {}
    int norm() const {return x*x+y*y;}
};

bool r_comp(const v2i& a, const v2i& b)
    { return a.norm() < b.norm(); }

std::vector<v2i> result;
for(int x = -r; x <= r; ++x) {
    int my = r*r - x*x;
    for(int y = 0; y*y <= my; ++y) {
        result.push_back(v2i(x,y));
        if(y > 0)
            result.push_back(v2i(x,-y));
    }
}

std::sort(result.begin(), result.end(), r_comp);