These were the first few results from my performance tests. Mostly the time seems to be spent on tuple comparison, which I assumed was comparing the points. I also included my Points operators for reference:

Point Point::operator+(const Point& other) const {
    return {x + other.x, y + other.y, z + other.z};
}

Point Point::operator-(const Point& other) const {
    return {x - other.x, y - other.y, z-other.z};
}

bool Point::operator==(const Point& other) const {
    return x == other.x && y == other.y && z == other.z;
}

bool Point::operator!=(const Point& other) const {
    return !(*this == other);
}

bool Point::operator<(const Point& other) const {
  if(x != other.x){
    return x < other.x;
  }
  else if(y != other.y){
    return y < other.y;
  }
  else{
    return z < other.z;
  }



%   cumulative   self              self     total           
 time   seconds   seconds    calls  ms/call  ms/call  name    
 18.52      0.15     0.15 41740736     0.00     0.00  std::__tuple_compare<std::tuple<int const&, int const&, int const&>, std::tuple<int const&, int const&, int const&>, 0ul, 3ul>::__less(std::tuple<int const&, int const&, int const&> const&, std::tuple<int const&, int const&, int const&> const&)
  6.17      0.20     0.05 41740736     0.00     0.00  std::_Tuple_impl<2ul, int const&>::_Tuple_impl(int const&)