// this captures the case that the point is not (almost) in the direction of the element.:
  // first, project the problem onto the unit sphere:
  Point p_o(p - my_origin);
  pt0_o /= pt0_o.norm();
  pt1_o /= pt1_o.norm();
  pt2_o /= pt2_o.norm();
  p_o /= p_o.norm();

  // now, check if it is in the projected face; by comparing the distance of
  // any point in the element to \p p with the largest distance between this point
  // to any other point in the element.
  if ((p_o - pt0_o).norm_sq() > std::max((pt0_o - pt1_o).norm_sq(), (pt0_o - pt2_o).norm_sq()) ||
      (p_o - pt1_o).norm_sq() > std::max((pt1_o - pt2_o).norm_sq(), (pt1_o - pt0_o).norm_sq()) ||
      (p_o - pt2_o).norm_sq() > std::max((pt2_o - pt0_o).norm_sq(), (pt2_o - pt1_o).norm_sq()) )
    {
      // the physical point is definitely not contained in the element
      return false;
    }

                              if (neighbor->p_level() < my_p_level &&
                                  neighbor->p_refinement_flag() != Elem::REFINE)
                                {
                                  neighbor->set_p_refinement_flag(Elem::REFINE);
                                  level_one_satisfied = false;
                                  compatible_with_coarsening = false;
                                }
                              if (neighbor->p_level() == my_p_level &&