GeomUtil.cpp

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/*
 * MOAB, a Mesh-Oriented datABase, is a software component for creating,
 * storing and accessing finite element mesh data.
 *
 * Copyright 2004 Sandia Corporation.  Under the terms of Contract
 * DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government
 * retains certain rights in this software.
 *
 * This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 */

/**\file Geometry.cpp
 *\author Jason Kraftcheck ([email protected])
 *\date 2006-07-27
 */

#include "moab/CartVect.hpp"
#include "moab/CN.hpp"
#include "moab/GeomUtil.hpp"
#include "moab/Matrix3.hpp"
#include "moab/Util.hpp"
#include <cmath>
#include <algorithm>
#include <cassert>
#include <iostream>
#include <limits>

namespace moab
{

namespace GeomUtil
{

    static inline void min_max_3( double a, double b, double c, double& min, double& max )
    {
        if( a < b )
        {
            if( a < c )
            {
                min = a;
                max = b > c ? b : c;
            }
            else
            {
                min = c;
                max = b;
            }
        }
        else if( b < c )
        {
            min = b;
            max = a > c ? a : c;
        }
        else
        {
            min = c;
            max = a;
        }
    }

    static inline double dot_abs( const CartVect& u, const CartVect& v )
    {
        return fabs( u[0] * v[0] ) + fabs( u[1] * v[1] ) + fabs( u[2] * v[2] );
    }

    bool segment_box_intersect( CartVect box_min,
                                CartVect box_max,
                                const CartVect& seg_pt,
                                const CartVect& seg_unit_dir,
                                double& seg_start,
                                double& seg_end )
    {
        // translate so that seg_pt is at origin
        box_min -= seg_pt;
        box_max -= seg_pt;

        for( unsigned i = 0; i < 3; ++i )
        {  // X, Y, and Z slabs

            // intersect line with slab planes
            const double t_min = box_min[i] / seg_unit_dir[i];
            const double t_max = box_max[i] / seg_unit_dir[i];

            // check if line is parallel to planes
            if( !Util::is_finite( t_min ) )
            {
                if( box_min[i] > 0.0 || box_max[i] < 0.0 ) return false;
                continue;
            }

            if( seg_unit_dir[i] < 0 )
            {
                if( t_min < seg_end ) seg_end = t_min;
                if( t_max > seg_start ) seg_start = t_max;
            }
            else
            {  // seg_unit_dir[i] > 0
                if( t_min > seg_start ) seg_start = t_min;
                if( t_max < seg_end ) seg_end = t_max;
            }
        }

        return seg_start <= seg_end;
    }

    /* Function to return the vertex with the lowest coordinates. To force the same
       ray-edge computation, the Plücker test needs to use consistent edge
       representation. This would be more simple with MOAB handles instead of
       coordinates... */
    inline bool first( const CartVect& a, const CartVect& b )
    {
        if( a[0] < b[0] )
        {
            return true;
        }
        else if( a[0] == b[0] )
        {
            if( a[1] < b[1] )
            {
                return true;
            }
            else if( a[1] == b[1] )
            {
                if( a[2] < b[2] )
                {
                    return true;
                }
                else
                {
                    return false;
                }
            }
            else
            {
                return false;
            }
        }
        else
        {
            return false;
        }
    }

    double plucker_edge_test( const CartVect& vertexa,
                              const CartVect& vertexb,
                              const CartVect& ray,
                              const CartVect& ray_normal )
    {

        double pip;
        const double near_zero = 10 * std::numeric_limits< double >::epsilon();

        if( first( vertexa, vertexb ) )
        {
            const CartVect edge        = vertexb - vertexa;
            const CartVect edge_normal = edge * vertexa;
            pip                        = ray % edge_normal + ray_normal % edge;
        }
        else
        {
            const CartVect edge        = vertexa - vertexb;
            const CartVect edge_normal = edge * vertexb;
            pip                        = ray % edge_normal + ray_normal % edge;
            pip                        = -pip;
        }

        if( near_zero > fabs( pip ) ) pip = 0.0;

        return pip;
    }

#define EXIT_EARLY           \
    if( type ) *type = NONE; \
    return false;

    /* This test uses the same edge-ray computation for adjacent triangles so that
       rays passing close to edges/nodes are handled consistently.

       Reports intersection type for post processing of special cases. Optionally
       screen by orientation and negative/nonnegative distance limits.

       If screening by orientation, substantial pruning can occur. Indicate
       desired orientation by passing 1 (forward), -1 (reverse), or 0 (no preference).
       Note that triangle orientation is not always the same as surface
       orientation due to non-manifold surfaces.

       N. Platis and T. Theoharis, "Fast Ray-Tetrahedron Intersection using Plücker
       Coordinates", Journal of Graphics Tools, Vol. 8, Part 4, Pages 37-48 (2003). */
    bool plucker_ray_tri_intersect( const CartVect vertices[3],
                                    const CartVect& origin,
                                    const CartVect& direction,
                                    double& dist_out,
                                    const double* nonneg_ray_len,
                                    const double* neg_ray_len,
                                    const int* orientation,
                                    intersection_type* type )
    {

        const CartVect raya = direction;
        const CartVect rayb = direction * origin;

        // Determine the value of the first Plucker coordinate from edge 0
        double plucker_coord0 = plucker_edge_test( vertices[0], vertices[1], raya, rayb );

        // If orientation is set, confirm that sign of plucker_coordinate indicate
        // correct orientation of intersection
        if( orientation && ( *orientation ) * plucker_coord0 > 0 )
        {
            EXIT_EARLY
        }

        // Determine the value of the second Plucker coordinate from edge 1
        double plucker_coord1 = plucker_edge_test( vertices[1], vertices[2], raya, rayb );

        // If orientation is set, confirm that sign of plucker_coordinate indicate
        // correct orientation of intersection
        if( orientation )
        {
            if( ( *orientation ) * plucker_coord1 > 0 )
            {
                EXIT_EARLY
            }
            // If the orientation is not specified, all plucker_coords must be the same sign or
            // zero.
        }
        else if( ( 0.0 < plucker_coord0 && 0.0 > plucker_coord1 ) || ( 0.0 > plucker_coord0 && 0.0 < plucker_coord1 ) )
        {
            EXIT_EARLY
        }

        // Determine the value of the second Plucker coordinate from edge 2
        double plucker_coord2 = plucker_edge_test( vertices[2], vertices[0], raya, rayb );

        // If orientation is set, confirm that sign of plucker_coordinate indicate
        // correct orientation of intersection
        if( orientation )
        {
            if( ( *orientation ) * plucker_coord2 > 0 )
            {
                EXIT_EARLY
            }
            // If the orientation is not specified, all plucker_coords must be the same sign or
            // zero.
        }
        else if( ( 0.0 < plucker_coord1 && 0.0 > plucker_coord2 ) || ( 0.0 > plucker_coord1 && 0.0 < plucker_coord2 ) ||
                 ( 0.0 < plucker_coord0 && 0.0 > plucker_coord2 ) || ( 0.0 > plucker_coord0 && 0.0 < plucker_coord2 ) )
        {
            EXIT_EARLY
        }

        // check for coplanar case to avoid dividing by zero
        if( 0.0 == plucker_coord0 && 0.0 == plucker_coord1 && 0.0 == plucker_coord2 )
        {
            EXIT_EARLY
        }

        // get the distance to intersection
        const double inverse_sum = 1.0 / ( plucker_coord0 + plucker_coord1 + plucker_coord2 );
        assert( 0.0 != inverse_sum );
        const CartVect intersection( plucker_coord0 * inverse_sum * vertices[2] +
                                     plucker_coord1 * inverse_sum * vertices[0] +
                                     plucker_coord2 * inverse_sum * vertices[1] );

        // To minimize numerical error, get index of largest magnitude direction.
        int idx            = 0;
        double max_abs_dir = 0;
        for( unsigned int i = 0; i < 3; ++i )
        {
            if( fabs( direction[i] ) > max_abs_dir )
            {
                idx         = i;
                max_abs_dir = fabs( direction[i] );
            }
        }
        const double dist = ( intersection[idx] - origin[idx] ) / direction[idx];

        // is the intersection within distance limits?
        if( ( nonneg_ray_len && *nonneg_ray_len < dist ) ||  // intersection is beyond positive limit
            ( neg_ray_len && *neg_ray_len >= dist ) ||       // intersection is behind negative limit
            ( !neg_ray_len && 0 > dist ) )
        {  // Unless a neg_ray_len is used, don't return negative distances
            EXIT_EARLY
        }

        dist_out = dist;

        if( type )
            *type = type_list[( ( 0.0 == plucker_coord2 ) << 2 ) + ( ( 0.0 == plucker_coord1 ) << 1 ) +
                              ( 0.0 == plucker_coord0 )];

        return true;
    }

    /* Implementation copied from cgmMC ray_tri_contact (overlap.C) */
    bool ray_tri_intersect( const CartVect vertices[3],
                            const CartVect& b,
                            const CartVect& v,
                            double& t_out,
                            const double* ray_length )
    {
        const CartVect p0 = vertices[0] - vertices[1];  // abc
        const CartVect p1 = vertices[0] - vertices[2];  // def
                                                        // ghi<-v
        const CartVect p   = vertices[0] - b;           // jkl
        const CartVect c   = p1 * v;                    // eiMinushf,gfMinusdi,dhMinuseg
        const double mP    = p0 % c;
        const double betaP = p % c;
        if( mP > 0 )
        {
            if( betaP < 0 ) return false;
        }
        else if( mP < 0 )
        {
            if( betaP > 0 ) return false;
        }
        else
        {
            return false;
        }

        const CartVect d = p0 * p;  // jcMinusal,blMinuskc,akMinusjb
        double gammaP    = v % d;
        if( mP > 0 )
        {
            if( gammaP < 0 || betaP + gammaP > mP ) return false;
        }
        else if( betaP + gammaP < mP || gammaP > 0 )
            return false;

        const double tP    = p1 % d;
        const double m     = 1.0 / mP;
        const double beta  = betaP * m;
        const double gamma = gammaP * m;
        const double t     = -tP * m;
        if( ray_length && t > *ray_length ) return false;

        if( beta < 0 || gamma < 0 || beta + gamma > 1 || t < 0.0 ) return false;

        t_out = t;
        return true;
    }

    bool ray_box_intersect( const CartVect& box_min,
                            const CartVect& box_max,
                            const CartVect& ray_pt,
                            const CartVect& ray_dir,
                            double& t_enter,
                            double& t_exit )
    {
        const double epsilon = 1e-12;
        double t1, t2;

        // Use 'slabs' method from 13.6.1 of Akenine-Moller
        t_enter = 0.0;
        t_exit  = std::numeric_limits< double >::infinity();

        // Intersect with each pair of axis-aligned planes bounding
        // opposite faces of the leaf box
        bool ray_is_valid = false;  // is ray direction vector zero?
        for( int axis = 0; axis < 3; ++axis )
        {
            if( fabs( ray_dir[axis] ) < epsilon )
            {  // ray parallel to planes
                if( ray_pt[axis] >= box_min[axis] && ray_pt[axis] <= box_max[axis] )
                    continue;
                else
                    return false;
            }

            // find t values at which ray intersects each plane
            ray_is_valid = true;
            t1           = ( box_min[axis] - ray_pt[axis] ) / ray_dir[axis];
            t2           = ( box_max[axis] - ray_pt[axis] ) / ray_dir[axis];

            // t_enter = max( t_enter_x, t_enter_y, t_enter_z )
            // t_exit  = min( t_exit_x, t_exit_y, t_exit_z )
            //   where
            // t_enter_x = min( t1_x, t2_x );
            // t_exit_x  = max( t1_x, t2_x )
            if( t1 < t2 )
            {
                if( t_enter < t1 ) t_enter = t1;
                if( t_exit > t2 ) t_exit = t2;
            }
            else
            {
                if( t_enter < t2 ) t_enter = t2;
                if( t_exit > t1 ) t_exit = t1;
            }
        }

        return ray_is_valid && ( t_enter <= t_exit );
    }

    bool box_plane_overlap( const CartVect& normal, double d, CartVect min, CartVect max )
    {
        if( normal[0] < 0.0 ) std::swap( min[0], max[0] );
        if( normal[1] < 0.0 ) std::swap( min[1], max[1] );
        if( normal[2] < 0.0 ) std::swap( min[2], max[2] );

        return ( normal % min <= -d ) && ( normal % max >= -d );
    }

#define CHECK_RANGE( A, B, R )                              \
    if( ( A ) < ( B ) )                                     \
    {                                                       \
        if( ( A ) > ( R ) || ( B ) < -( R ) ) return false; \
    }                                                       \
    else if( ( B ) > ( R ) || ( A ) < -( R ) )              \
    return false

    /* Adapted from: http://jgt.akpeters.com/papers/AkenineMoller01/tribox.html
     * Use separating axis theorem to test for overlap between triangle
     * and axis-aligned box.
     *
     * Test for overlap in these directions:
     * 1) {x,y,z}-directions
     * 2) normal of triangle
     * 3) crossprod of triangle edge with {x,y,z}-direction
     */
    bool box_tri_overlap( const CartVect vertices[3], const CartVect& box_center, const CartVect& box_dims )
    {
        // translate everything such that box is centered at origin
        const CartVect v0( vertices[0] - box_center );
        const CartVect v1( vertices[1] - box_center );
        const CartVect v2( vertices[2] - box_center );

        // do case 1) tests
        if( v0[0] > box_dims[0] && v1[0] > box_dims[0] && v2[0] > box_dims[0] ) return false;
        if( v0[1] > box_dims[1] && v1[1] > box_dims[1] && v2[1] > box_dims[1] ) return false;
        if( v0[2] > box_dims[2] && v1[2] > box_dims[2] && v2[2] > box_dims[2] ) return false;
        if( v0[0] < -box_dims[0] && v1[0] < -box_dims[0] && v2[0] < -box_dims[0] ) return false;
        if( v0[1] < -box_dims[1] && v1[1] < -box_dims[1] && v2[1] < -box_dims[1] ) return false;
        if( v0[2] < -box_dims[2] && v1[2] < -box_dims[2] && v2[2] < -box_dims[2] ) return false;

        // compute triangle edge vectors
        const CartVect e0( vertices[1] - vertices[0] );
        const CartVect e1( vertices[2] - vertices[1] );
        const CartVect e2( vertices[0] - vertices[2] );

        // do case 3) tests
        double fex, fey, fez, p0, p1, p2, rad;
        fex = fabs( e0[0] );
        fey = fabs( e0[1] );
        fez = fabs( e0[2] );

        p0  = e0[2] * v0[1] - e0[1] * v0[2];
        p2  = e0[2] * v2[1] - e0[1] * v2[2];
        rad = fez * box_dims[1] + fey * box_dims[2];
        CHECK_RANGE( p0, p2, rad );

        p0  = -e0[2] * v0[0] + e0[0] * v0[2];
        p2  = -e0[2] * v2[0] + e0[0] * v2[2];
        rad = fez * box_dims[0] + fex * box_dims[2];
        CHECK_RANGE( p0, p2, rad );

        p1  = e0[1] * v1[0] - e0[0] * v1[1];
        p2  = e0[1] * v2[0] - e0[0] * v2[1];
        rad = fey * box_dims[0] + fex * box_dims[1];
        CHECK_RANGE( p1, p2, rad );

        fex = fabs( e1[0] );
        fey = fabs( e1[1] );
        fez = fabs( e1[2] );

        p0  = e1[2] * v0[1] - e1[1] * v0[2];
        p2  = e1[2] * v2[1] - e1[1] * v2[2];
        rad = fez * box_dims[1] + fey * box_dims[2];
        CHECK_RANGE( p0, p2, rad );

        p0  = -e1[2] * v0[0] + e1[0] * v0[2];
        p2  = -e1[2] * v2[0] + e1[0] * v2[2];
        rad = fez * box_dims[0] + fex * box_dims[2];
        CHECK_RANGE( p0, p2, rad );

        p0  = e1[1] * v0[0] - e1[0] * v0[1];
        p1  = e1[1] * v1[0] - e1[0] * v1[1];
        rad = fey * box_dims[0] + fex * box_dims[1];
        CHECK_RANGE( p0, p1, rad );

        fex = fabs( e2[0] );
        fey = fabs( e2[1] );
        fez = fabs( e2[2] );

        p0  = e2[2] * v0[1] - e2[1] * v0[2];
        p1  = e2[2] * v1[1] - e2[1] * v1[2];
        rad = fez * box_dims[1] + fey * box_dims[2];
        CHECK_RANGE( p0, p1, rad );

        p0  = -e2[2] * v0[0] + e2[0] * v0[2];
        p1  = -e2[2] * v1[0] + e2[0] * v1[2];
        rad = fez * box_dims[0] + fex * box_dims[2];
        CHECK_RANGE( p0, p1, rad );

        p1  = e2[1] * v1[0] - e2[0] * v1[1];
        p2  = e2[1] * v2[0] - e2[0] * v2[1];
        rad = fey * box_dims[0] + fex * box_dims[1];
        CHECK_RANGE( p1, p2, rad );

        // do case 2) test
        CartVect n = e0 * e1;
        return box_plane_overlap( n, -( n % v0 ), -box_dims, box_dims );
    }

    bool box_tri_overlap( const CartVect triangle_corners[3],
                          const CartVect& box_min_corner,
                          const CartVect& box_max_corner,
                          double tolerance )
    {
        const CartVect box_center = 0.5 * ( box_max_corner + box_min_corner );
        const CartVect box_hf_dim = 0.5 * ( box_max_corner - box_min_corner );
        return box_tri_overlap( triangle_corners, box_center, box_hf_dim + CartVect( tolerance ) );
    }

    bool box_elem_overlap( const CartVect* elem_corners,
                           EntityType elem_type,
                           const CartVect& center,
                           const CartVect& dims,
                           int nodecount )
    {

        switch( elem_type )
        {
            case MBTRI:
                return box_tri_overlap( elem_corners, center, dims );
            case MBTET:
                return box_tet_overlap( elem_corners, center, dims );
            case MBHEX:
                return box_hex_overlap( elem_corners, center, dims );
            case MBPOLYGON: {
                CartVect vt[3];
                vt[0] = elem_corners[0];
                vt[1] = elem_corners[1];
                for( int j = 2; j < nodecount; j++ )
                {
                    vt[2] = elem_corners[j];
                    if( box_tri_overlap( vt, center, dims ) ) return true;
                }
                // none of the triangles overlap, then we return false
                return false;
            }
            case MBPOLYHEDRON:
                assert( false );
                return false;
            default:
                return box_linear_elem_overlap( elem_corners, elem_type, center, dims );
        }
    }

    static inline CartVect quad_norm( const CartVect& v1, const CartVect& v2, const CartVect& v3, const CartVect& v4 )
    {
        return ( -v1 + v2 + v3 - v4 ) * ( -v1 - v2 + v3 + v4 );
    }

    static inline CartVect tri_norm( const CartVect& v1, const CartVect& v2, const CartVect& v3 )
    {
        return ( v2 - v1 ) * ( v3 - v1 );
    }

    bool box_linear_elem_overlap( const CartVect* elem_corners,
                                  EntityType type,
                                  const CartVect& box_center,
                                  const CartVect& box_halfdims )
    {
        CartVect corners[8];
        const unsigned num_corner = CN::VerticesPerEntity( type );
        assert( num_corner <= sizeof( corners ) / sizeof( corners[0] ) );
        for( unsigned i = 0; i < num_corner; ++i )
            corners[i] = elem_corners[i] - box_center;
        return box_linear_elem_overlap( corners, type, box_halfdims );
    }

    bool box_linear_elem_overlap( const CartVect* elem_corners, EntityType type, const CartVect& dims )
    {
        // Do Separating Axis Theorem:
        // If the element and the box overlap, then the 1D projections
        // onto at least one of the axes in the following three sets
        // must overlap (assuming convex polyhedral element).
        // 1) The normals of the faces of the box (the principal axes)
        // 2) The crossproduct of each element edge with each box edge
        //    (crossproduct of each edge with each principal axis)
        // 3) The normals of the faces of the element

        int e, f;  // loop counters
        int i;
        double dot, cross[2], tmp;
        CartVect norm;
        int indices[4];  // element edge/face vertex indices

        // test box face normals (principal axes)
        const int num_corner = CN::VerticesPerEntity( type );
        int not_less[3]      = { num_corner, num_corner, num_corner };
        int not_greater[3]   = { num_corner, num_corner, num_corner };
        int not_inside;
        for( i = 0; i < num_corner; ++i )
        {  // for each element corner
            not_inside = 3;

            if( elem_corners[i][0] < -dims[0] )
                --not_less[0];
            else if( elem_corners[i][0] > dims[0] )
                --not_greater[0];
            else
                --not_inside;

            if( elem_corners[i][1] < -dims[1] )
                --not_less[1];
            else if( elem_corners[i][1] > dims[1] )
                --not_greater[1];
            else
                --not_inside;

            if( elem_corners[i][2] < -dims[2] )
                --not_less[2];
            else if( elem_corners[i][2] > dims[2] )
                --not_greater[2];
            else
                --not_inside;

            if( !not_inside ) return true;
        }
        // If all points less than min_x of box, then
        // not_less[0] == 0, and therefore
        // the following product is zero.
        if( not_greater[0] * not_greater[1] * not_greater[2] * not_less[0] * not_less[1] * not_less[2] == 0 )
            return false;

        // Test edge-edge crossproducts

        // Edge directions for box are principal axis, so
        // for each element edge, check along the cross-product
        // of that edge with each of the tree principal axes.
        const int num_edge = CN::NumSubEntities( type, 1 );
        for( e = 0; e < num_edge; ++e )
        {  // for each element edge
            // get which element vertices bound the edge
            CN::SubEntityVertexIndices( type, 1, e, indices );

            // X-Axis

            // calculate crossproduct: axis x (v1 - v0),
            // where v1 and v0 are edge vertices.
            cross[0] = elem_corners[indices[0]][2] - elem_corners[indices[1]][2];
            cross[1] = elem_corners[indices[1]][1] - elem_corners[indices[0]][1];
            // skip if parallel
            if( ( cross[0] * cross[0] + cross[1] * cross[1] ) >= std::numeric_limits< double >::epsilon() )
            {
                dot         = fabs( cross[0] * dims[1] ) + fabs( cross[1] * dims[2] );
                not_less[0] = not_greater[0] = num_corner - 1;
                for( i = ( indices[0] + 1 ) % num_corner; i != indices[0]; i = ( i + 1 ) % num_corner )
                {  // for each element corner
                    tmp = cross[0] * elem_corners[i][1] + cross[1] * elem_corners[i][2];
                    not_less[0] -= ( tmp < -dot );
                    not_greater[0] -= ( tmp > dot );
                }

                if( not_less[0] * not_greater[0] == 0 ) return false;
            }

            // Y-Axis

            // calculate crossproduct: axis x (v1 - v0),
            // where v1 and v0 are edge vertices.
            cross[0] = elem_corners[indices[0]][0] - elem_corners[indices[1]][0];
            cross[1] = elem_corners[indices[1]][2] - elem_corners[indices[0]][2];
            // skip if parallel
            if( ( cross[0] * cross[0] + cross[1] * cross[1] ) >= std::numeric_limits< double >::epsilon() )
            {
                dot         = fabs( cross[0] * dims[2] ) + fabs( cross[1] * dims[0] );
                not_less[0] = not_greater[0] = num_corner - 1;
                for( i = ( indices[0] + 1 ) % num_corner; i != indices[0]; i = ( i + 1 ) % num_corner )
                {  // for each element corner
                    tmp = cross[0] * elem_corners[i][2] + cross[1] * elem_corners[i][0];
                    not_less[0] -= ( tmp < -dot );
                    not_greater[0] -= ( tmp > dot );
                }

                if( not_less[0] * not_greater[0] == 0 ) return false;
            }

            // Z-Axis

            // calculate crossproduct: axis x (v1 - v0),
            // where v1 and v0 are edge vertices.
            cross[0] = elem_corners[indices[0]][1] - elem_corners[indices[1]][1];
            cross[1] = elem_corners[indices[1]][0] - elem_corners[indices[0]][0];
            // skip if parallel
            if( ( cross[0] * cross[0] + cross[1] * cross[1] ) >= std::numeric_limits< double >::epsilon() )
            {
                dot         = fabs( cross[0] * dims[0] ) + fabs( cross[1] * dims[1] );
                not_less[0] = not_greater[0] = num_corner - 1;
                for( i = ( indices[0] + 1 ) % num_corner; i != indices[0]; i = ( i + 1 ) % num_corner )
                {  // for each element corner
                    tmp = cross[0] * elem_corners[i][0] + cross[1] * elem_corners[i][1];
                    not_less[0] -= ( tmp < -dot );
                    not_greater[0] -= ( tmp > dot );
                }

                if( not_less[0] * not_greater[0] == 0 ) return false;
            }
        }

        // test element face normals
        const int num_face = CN::NumSubEntities( type, 2 );
        for( f = 0; f < num_face; ++f )
        {
            CN::SubEntityVertexIndices( type, 2, f, indices );
            switch( CN::SubEntityType( type, 2, f ) )
            {
                case MBTRI:
                    norm = tri_norm( elem_corners[indices[0]], elem_corners[indices[1]], elem_corners[indices[2]] );
                    break;
                case MBQUAD:
                    norm = quad_norm( elem_corners[indices[0]], elem_corners[indices[1]], elem_corners[indices[2]],
                                      elem_corners[indices[3]] );
                    break;
                default:
                    assert( false );
                    continue;
            }

            dot = dot_abs( norm, dims );

            // for each element vertex
            not_less[0] = not_greater[0] = num_corner;
            for( i = 0; i < num_corner; ++i )
            {
                tmp = norm % elem_corners[i];
                not_less[0] -= ( tmp < -dot );
                not_greater[0] -= ( tmp > dot );
            }

            if( not_less[0] * not_greater[0] == 0 ) return false;
        }

        // Overlap on all tested axes.
        return true;
    }

    bool box_hex_overlap( const CartVect* elem_corners, const CartVect& center, const CartVect& dims )
    {
        // Do Separating Axis Theorem:
        // If the element and the box overlap, then the 1D projections
        // onto at least one of the axes in the following three sets
        // must overlap (assuming convex polyhedral element).
        // 1) The normals of the faces of the box (the principal axes)
        // 2) The crossproduct of each element edge with each box edge
        //    (crossproduct of each edge with each principal axis)
        // 3) The normals of the faces of the element

        unsigned i, e, f;  // loop counters
        double dot, cross[2], tmp;
        CartVect norm;
        const CartVect corners[8] = { elem_corners[0] - center, elem_corners[1] - center, elem_corners[2] - center,
                                      elem_corners[3] - center, elem_corners[4] - center, elem_corners[5] - center,
                                      elem_corners[6] - center, elem_corners[7] - center };

        // test box face normals (principal axes)
        int not_less[3]    = { 8, 8, 8 };
        int not_greater[3] = { 8, 8, 8 };
        int not_inside;
        for( i = 0; i < 8; ++i )
        {  // for each element corner
            not_inside = 3;

            if( corners[i][0] < -dims[0] )
                --not_less[0];
            else if( corners[i][0] > dims[0] )
                --not_greater[0];
            else
                --not_inside;

            if( corners[i][1] < -dims[1] )
                --not_less[1];
            else if( corners[i][1] > dims[1] )
                --not_greater[1];
            else
                --not_inside;

            if( corners[i][2] < -dims[2] )
                --not_less[2];
            else if( corners[i][2] > dims[2] )
                --not_greater[2];
            else
                --not_inside;

            if( !not_inside ) return true;
        }
        // If all points less than min_x of box, then
        // not_less[0] == 0, and therefore
        // the following product is zero.
        if( not_greater[0] * not_greater[1] * not_greater[2] * not_less[0] * not_less[1] * not_less[2] == 0 )
            return false;

        // Test edge-edge crossproducts
        const unsigned edges[12][2] = { { 0, 1 }, { 0, 4 }, { 0, 3 }, { 2, 3 }, { 2, 1 }, { 2, 6 },
                                        { 5, 6 }, { 5, 1 }, { 5, 4 }, { 7, 4 }, { 7, 3 }, { 7, 6 } };

        // Edge directions for box are principal axis, so
        // for each element edge, check along the cross-product
        // of that edge with each of the tree principal axes.
        for( e = 0; e < 12; ++e )
        {  // for each element edge
            // get which element vertices bound the edge
            const CartVect& v0 = corners[edges[e][0]];
            const CartVect& v1 = corners[edges[e][1]];

            // X-Axis

            // calculate crossproduct: axis x (v1 - v0),
            // where v1 and v0 are edge vertices.
            cross[0] = v0[2] - v1[2];
            cross[1] = v1[1] - v0[1];
            // skip if parallel
            if( ( cross[0] * cross[0] + cross[1] * cross[1] ) >= std::numeric_limits< double >::epsilon() )
            {
                dot         = fabs( cross[0] * dims[1] ) + fabs( cross[1] * dims[2] );
                not_less[0] = not_greater[0] = 7;
                for( i = ( edges[e][0] + 1 ) % 8; i != edges[e][0]; i = ( i + 1 ) % 8 )
                {  // for each element corner
                    tmp = cross[0] * corners[i][1] + cross[1] * corners[i][2];
                    not_less[0] -= ( tmp < -dot );
                    not_greater[0] -= ( tmp > dot );
                }

                if( not_less[0] * not_greater[0] == 0 ) return false;
            }

            // Y-Axis

            // calculate crossproduct: axis x (v1 - v0),
            // where v1 and v0 are edge vertices.
            cross[0] = v0[0] - v1[0];
            cross[1] = v1[2] - v0[2];
            // skip if parallel
            if( ( cross[0] * cross[0] + cross[1] * cross[1] ) >= std::numeric_limits< double >::epsilon() )
            {
                dot         = fabs( cross[0] * dims[2] ) + fabs( cross[1] * dims[0] );
                not_less[0] = not_greater[0] = 7;
                for( i = ( edges[e][0] + 1 ) % 8; i != edges[e][0]; i = ( i + 1 ) % 8 )
                {  // for each element corner
                    tmp = cross[0] * corners[i][2] + cross[1] * corners[i][0];
                    not_less[0] -= ( tmp < -dot );
                    not_greater[0] -= ( tmp > dot );
                }

                if( not_less[0] * not_greater[0] == 0 ) return false;
            }

            // Z-Axis

            // calculate crossproduct: axis x (v1 - v0),
            // where v1 and v0 are edge vertices.
            cross[0] = v0[1] - v1[1];
            cross[1] = v1[0] - v0[0];
            // skip if parallel
            if( ( cross[0] * cross[0] + cross[1] * cross[1] ) >= std::numeric_limits< double >::epsilon() )
            {
                dot         = fabs( cross[0] * dims[0] ) + fabs( cross[1] * dims[1] );
                not_less[0] = not_greater[0] = 7;
                for( i = ( edges[e][0] + 1 ) % 8; i != edges[e][0]; i = ( i + 1 ) % 8 )
                {  // for each element corner
                    tmp = cross[0] * corners[i][0] + cross[1] * corners[i][1];
                    not_less[0] -= ( tmp < -dot );
                    not_greater[0] -= ( tmp > dot );
                }

                if( not_less[0] * not_greater[0] == 0 ) return false;
            }
        }

        // test element face normals
        const unsigned faces[6][4] = { { 0, 1, 2, 3 }, { 0, 1, 5, 4 }, { 1, 2, 6, 5 },
                                       { 2, 6, 7, 3 }, { 3, 7, 4, 0 }, { 7, 4, 5, 6 } };
        for( f = 0; f < 6; ++f )
        {
            norm = quad_norm( corners[faces[f][0]], corners[faces[f][1]], corners[faces[f][2]], corners[faces[f][3]] );

            dot = dot_abs( norm, dims );

            // for each element vertex
            not_less[0] = not_greater[0] = 8;
            for( i = 0; i < 8; ++i )
            {
                tmp = norm % corners[i];
                not_less[0] -= ( tmp < -dot );
                not_greater[0] -= ( tmp > dot );
            }

            if( not_less[0] * not_greater[0] == 0 ) return false;
        }

        // Overlap on all tested axes.
        return true;
    }

    static inline bool box_tet_overlap_edge( const CartVect& dims,
                                             const CartVect& edge,
                                             const CartVect& ve,
                                             const CartVect& v1,
                                             const CartVect& v2 )
    {
        double dot, dot1, dot2, dot3, min, max;

        // edge x X
        if( fabs( edge[1] * edge[2] ) > std::numeric_limits< double >::epsilon() )
        {
            dot  = fabs( edge[2] ) * dims[1] + fabs( edge[1] ) * dims[2];
            dot1 = edge[2] * ve[1] - edge[1] * ve[2];
            dot2 = edge[2] * v1[1] - edge[1] * v1[2];
            dot3 = edge[2] * v2[1] - edge[1] * v2[2];
            min_max_3( dot1, dot2, dot3, min, max );
            if( max < -dot || min > dot ) return false;
        }

        // edge x Y
        if( fabs( edge[1] * edge[2] ) > std::numeric_limits< double >::epsilon() )
        {
            dot  = fabs( edge[2] ) * dims[0] + fabs( edge[0] ) * dims[2];
            dot1 = -edge[2] * ve[0] + edge[0] * ve[2];
            dot2 = -edge[2] * v1[0] + edge[0] * v1[2];
            dot3 = -edge[2] * v2[0] + edge[0] * v2[2];
            min_max_3( dot1, dot2, dot3, min, max );
            if( max < -dot || min > dot ) return false;
        }

        // edge x Z
        if( fabs( edge[1] * edge[2] ) > std::numeric_limits< double >::epsilon() )
        {
            dot  = fabs( edge[1] ) * dims[0] + fabs( edge[0] ) * dims[1];
            dot1 = edge[1] * ve[0] - edge[0] * ve[1];
            dot2 = edge[1] * v1[0] - edge[0] * v1[1];
            dot3 = edge[1] * v2[0] - edge[0] * v2[1];
            min_max_3( dot1, dot2, dot3, min, max );
            if( max < -dot || min > dot ) return false;
        }

        return true;
    }

    bool box_tet_overlap( const CartVect* corners_in, const CartVect& center, const CartVect& dims )
    {
        // Do Separating Axis Theorem:
        // If the element and the box overlap, then the 1D projections
        // onto at least one of the axes in the following three sets
        // must overlap (assuming convex polyhedral element).
        // 1) The normals of the faces of the box (the principal axes)
        // 2) The crossproduct of each element edge with each box edge
        //    (crossproduct of each edge with each principal axis)
        // 3) The normals of the faces of the element

        // Translate problem such that box center is at origin.
        const CartVect corners[4] = { corners_in[0] - center, corners_in[1] - center, corners_in[2] - center,
                                      corners_in[3] - center };

        // 0) Check if any vertex is within the box
        if( fabs( corners[0][0] ) <= dims[0] && fabs( corners[0][1] ) <= dims[1] && fabs( corners[0][2] ) <= dims[2] )
            return true;
        if( fabs( corners[1][0] ) <= dims[0] && fabs( corners[1][1] ) <= dims[1] && fabs( corners[1][2] ) <= dims[2] )
            return true;
        if( fabs( corners[2][0] ) <= dims[0] && fabs( corners[2][1] ) <= dims[1] && fabs( corners[2][2] ) <= dims[2] )
            return true;
        if( fabs( corners[3][0] ) <= dims[0] && fabs( corners[3][1] ) <= dims[1] && fabs( corners[3][2] ) <= dims[2] )
            return true;

        // 1) Check for overlap on each principal axis (box face normal)
        // X
        if( corners[0][0] < -dims[0] && corners[1][0] < -dims[0] && corners[2][0] < -dims[0] &&
            corners[3][0] < -dims[0] )
            return false;
        if( corners[0][0] > dims[0] && corners[1][0] > dims[0] && corners[2][0] > dims[0] && corners[3][0] > dims[0] )
            return false;
        // Y
        if( corners[0][1] < -dims[1] && corners[1][1] < -dims[1] && corners[2][1] < -dims[1] &&
            corners[3][1] < -dims[1] )
            return false;
        if( corners[0][1] > dims[1] && corners[1][1] > dims[1] && corners[2][1] > dims[1] && corners[3][1] > dims[1] )
            return false;
        // Z
        if( corners[0][2] < -dims[2] && corners[1][2] < -dims[2] && corners[2][2] < -dims[2] &&
            corners[3][2] < -dims[2] )
            return false;
        if( corners[0][2] > dims[2] && corners[1][2] > dims[2] && corners[2][2] > dims[2] && corners[3][2] > dims[2] )
            return false;

        // 3) test element face normals
        CartVect norm;
        double dot, dot1, dot2;

        const CartVect v01 = corners[1] - corners[0];
        const CartVect v02 = corners[2] - corners[0];
        norm               = v01 * v02;
        dot                = dot_abs( norm, dims );
        dot1               = norm % corners[0];
        dot2               = norm % corners[3];
        if( dot1 > dot2 ) std::swap( dot1, dot2 );
        if( dot2 < -dot || dot1 > dot ) return false;

        const CartVect v03 = corners[3] - corners[0];
        norm               = v03 * v01;
        dot                = dot_abs( norm, dims );
        dot1               = norm % corners[0];
        dot2               = norm % corners[2];
        if( dot1 > dot2 ) std::swap( dot1, dot2 );
        if( dot2 < -dot || dot1 > dot ) return false;

        norm = v02 * v03;
        dot  = dot_abs( norm, dims );
        dot1 = norm % corners[0];
        dot2 = norm % corners[1];
        if( dot1 > dot2 ) std::swap( dot1, dot2 );
        if( dot2 < -dot || dot1 > dot ) return false;

        const CartVect v12 = corners[2] - corners[1];
        const CartVect v13 = corners[3] - corners[1];
        norm               = v13 * v12;
        dot                = dot_abs( norm, dims );
        dot1               = norm % corners[0];
        dot2               = norm % corners[1];
        if( dot1 > dot2 ) std::swap( dot1, dot2 );
        if( dot2 < -dot || dot1 > dot ) return false;

        // 2) test edge-edge cross products

        const CartVect v23 = corners[3] - corners[2];
        return box_tet_overlap_edge( dims, v01, corners[0], corners[2], corners[3] ) &&
               box_tet_overlap_edge( dims, v02, corners[0], corners[1], corners[3] ) &&
               box_tet_overlap_edge( dims, v03, corners[0], corners[1], corners[2] ) &&
               box_tet_overlap_edge( dims, v12, corners[1], corners[0], corners[3] ) &&
               box_tet_overlap_edge( dims, v13, corners[1], corners[0], corners[2] ) &&
               box_tet_overlap_edge( dims, v23, corners[2], corners[0], corners[1] );
    }

    // from:
    // http://www.geometrictools.com/Documentation/DistancePoint3Triangle3.pdf#search=%22closest%20point%20on%20triangle%22
    /*       t
     *   \(2)^
     *    \  |
     *     \ |
     *      \|
     *       \
     *       |\
     *       | \
     *       |  \  (1)
     *  (3)  tv  \
     *       |    \
     *       | (0) \
     *       |      \
     *-------+---sv--\----> s
     *       |        \ (6)
     *  (4)  |   (5)   \
     */
    // Worst case is either 61 flops and 5 compares or 53 flops and 6 compares,
    // depending on relative costs.  For all paths that do not return one of the
    // corner vertices, exactly one of the flops is a divide.
    void closest_location_on_tri( const CartVect& location, const CartVect* vertices, CartVect& closest_out )
    {                                                    // ops      comparisons
        const CartVect sv( vertices[1] - vertices[0] );  // +3 = 3
        const CartVect tv( vertices[2] - vertices[0] );  // +3 = 6
        const CartVect pv( vertices[0] - location );     // +3 = 9
        const double ss  = sv % sv;                      // +5 = 14
        const double st  = sv % tv;                      // +5 = 19
        const double tt  = tv % tv;                      // +5 = 24
        const double sp  = sv % pv;                      // +5 = 29
        const double tp  = tv % pv;                      // +5 = 34
        const double det = ss * tt - st * st;            // +3 = 37
        double s         = st * tp - tt * sp;            // +3 = 40
        double t         = st * sp - ss * tp;            // +3 = 43
        if( s + t < det )
        {  // +1 = 44, +1 = 1
            if( s < 0 )
            {  //          +1 = 2
                if( t < 0 )
                {  //          +1 = 3
                    // region 4
                    if( sp < 0 )
                    {                                   //          +1 = 4
                        if( -sp > ss )                  //          +1 = 5
                            closest_out = vertices[1];  //      44       5
                        else
                            closest_out = vertices[0] - ( sp / ss ) * sv;  // +7 = 51,      5
                    }
                    else if( tp < 0 )
                    {                                   //          +1 = 5
                        if( -tp > tt )                  //          +1 = 6
                            closest_out = vertices[2];  //      44,      6
                        else
                            closest_out = vertices[0] - ( tp / tt ) * tv;  // +7 = 51,      6
                    }
                    else
                    {
                        closest_out = vertices[0];  //      44,      5
                    }
                }
                else
                {
                    // region 3
                    if( tp >= 0 )                   //          +1 = 4
                        closest_out = vertices[0];  //      44,      4
                    else if( -tp >= tt )            //          +1 = 5
                        closest_out = vertices[2];  //      44,      5
                    else
                        closest_out = vertices[0] - ( tp / tt ) * tv;  // +7 = 51,      5
                }
            }
            else if( t < 0 )
            {  //          +1 = 3
                // region 5;
                if( sp >= 0.0 )                 //          +1 = 4
                    closest_out = vertices[0];  //      44,      4
                else if( -sp >= ss )            //          +1 = 5
                    closest_out = vertices[1];  //      44       5
                else
                    closest_out = vertices[0] - ( sp / ss ) * sv;  // +7 = 51,      5
            }
            else
            {
                // region 0
                const double inv_det = 1.0 / det;             // +1 = 45
                s *= inv_det;                                 // +1 = 46
                t *= inv_det;                                 // +1 = 47
                closest_out = vertices[0] + s * sv + t * tv;  //+12 = 59,      3
            }
        }
        else
        {
            if( s < 0 )
            {  //          +1 = 2
                // region 2
                s = st + sp;  // +1 = 45
                t = tt + tp;  // +1 = 46
                if( t > s )
                {                                         //          +1 = 3
                    const double num = t - s;             // +1 = 47
                    const double den = ss - 2 * st + tt;  // +3 = 50
                    if( num > den )                       //          +1 = 4
                        closest_out = vertices[1];        //      50,      4
                    else
                    {
                        s           = num / den;                          // +1 = 51
                        t           = 1 - s;                              // +1 = 52
                        closest_out = s * vertices[1] + t * vertices[2];  // +9 = 61,      4
                    }
                }
                else if( t <= 0 )               //          +1 = 4
                    closest_out = vertices[2];  //      46,      4
                else if( tp >= 0 )              //          +1 = 5
                    closest_out = vertices[0];  //      46,      5
                else
                    closest_out = vertices[0] - ( tp / tt ) * tv;  // +7 = 53,      5
            }
            else if( t < 0 )
            {  //          +1 = 3
                // region 6
                t = st + tp;  // +1 = 45
                s = ss + sp;  // +1 = 46
                if( s > t )
                {                                         //          +1 = 4
                    const double num = t - s;             // +1 = 47
                    const double den = tt - 2 * st + ss;  // +3 = 50
                    if( num > den )                       //          +1 = 5
                        closest_out = vertices[2];        //      50,      5
                    else
                    {
                        t           = num / den;                          // +1 = 51
                        s           = 1 - t;                              // +1 = 52
                        closest_out = s * vertices[1] + t * vertices[2];  // +9 = 61,      5
                    }
                }
                else if( s <= 0 )               //          +1 = 5
                    closest_out = vertices[1];  //      46,      5
                else if( sp >= 0 )              //          +1 = 6
                    closest_out = vertices[0];  //      46,      6
                else
                    closest_out = vertices[0] - ( sp / ss ) * sv;  // +7 = 53,      6
            }
            else
            {
                // region 1
                const double num = tt + tp - st - sp;  // +3 = 47
                if( num <= 0 )
                {                               //          +1 = 4
                    closest_out = vertices[2];  //      47,      4
                }
                else
                {
                    const double den = ss - 2 * st + tt;  // +3 = 50
                    if( num >= den )                      //          +1 = 5
                        closest_out = vertices[1];        //      50,      5
                    else
                    {
                        s           = num / den;                          // +1 = 51
                        t           = 1 - s;                              // +1 = 52
                        closest_out = s * vertices[1] + t * vertices[2];  // +9 = 61,      5
                    }
                }
            }
        }
    }

    void closest_location_on_tri( const CartVect& location,
                                  const CartVect* vertices,
                                  double tolerance,
                                  CartVect& closest_out,
                                  int& closest_topo )
    {
        const double tsqr = tolerance * tolerance;
        int i;
        CartVect pv[3], ev, ep;
        double t;

        closest_location_on_tri( location, vertices, closest_out );

        for( i = 0; i < 3; ++i )
        {
            pv[i] = vertices[i] - closest_out;
            if( ( pv[i] % pv[i] ) <= tsqr )
            {
                closest_topo = i;
                return;
            }
        }

        for( i = 0; i < 3; ++i )
        {
            ev = vertices[( i + 1 ) % 3] - vertices[i];
            t  = ( ev % pv[i] ) / ( ev % ev );
            ep = closest_out - ( vertices[i] + t * ev );
            if( ( ep % ep ) <= tsqr )
            {
                closest_topo = i + 3;
                return;
            }
        }

        closest_topo = 6;
    }

    // We assume polygon is *convex*, but *not* planar.
    void closest_location_on_polygon( const CartVect& location,
                                      const CartVect* vertices,
                                      int num_vertices,
                                      CartVect& closest_out )
    {
        const int n = num_vertices;
        CartVect d, p, v;
        double shortest_sqr, dist_sqr, t_closest, t;
        int i, e;

        // Find closest edge of polygon.
        e         = n - 1;
        v         = vertices[0] - vertices[e];
        t_closest = ( v % ( location - vertices[e] ) ) / ( v % v );
        if( t_closest < 0.0 )
            d = location - vertices[e];
        else if( t_closest > 1.0 )
            d = location - vertices[0];
        else
            d = location - vertices[e] - t_closest * v;
        shortest_sqr = d % d;
        for( i = 0; i < n - 1; ++i )
        {
            v = vertices[i + 1] - vertices[i];
            t = ( v % ( location - vertices[i] ) ) / ( v % v );
            if( t < 0.0 )
                d = location - vertices[i];
            else if( t > 1.0 )
                d = location - vertices[i + 1];
            else
                d = location - vertices[i] - t * v;
            dist_sqr = d % d;
            if( dist_sqr < shortest_sqr )
            {
                e            = i;
                shortest_sqr = dist_sqr;
                t_closest    = t;
            }
        }

        // If we are beyond the bounds of the edge, then
        // the point is outside and closest to a vertex
        if( t_closest <= 0.0 )
        {
            closest_out = vertices[e];
            return;
        }
        else if( t_closest >= 1.0 )
        {
            closest_out = vertices[( e + 1 ) % n];
            return;
        }

        // Now check which side of the edge we are one
        const CartVect v0   = vertices[e] - vertices[( e + n - 1 ) % n];
        const CartVect v1   = vertices[( e + 1 ) % n] - vertices[e];
        const CartVect v2   = vertices[( e + 2 ) % n] - vertices[( e + 1 ) % n];
        const CartVect norm = ( 1.0 - t_closest ) * ( v0 * v1 ) + t_closest * ( v1 * v2 );
        // if on outside of edge, result is closest point on edge
        if( ( norm % ( ( vertices[e] - location ) * v1 ) ) <= 0.0 )
        {
            closest_out = vertices[e] + t_closest * v1;
            return;
        }

        // Inside.  Project to plane defined by point and normal at
        // closest edge
        const double D = -( norm % ( vertices[e] + t_closest * v1 ) );
        closest_out    = ( location - ( norm % location + D ) * norm ) / ( norm % norm );
    }

    void closest_location_on_box( const CartVect& min, const CartVect& max, const CartVect& point, CartVect& closest )
    {
        closest[0] = point[0] < min[0] ? min[0] : point[0] > max[0] ? max[0] : point[0];
        closest[1] = point[1] < min[1] ? min[1] : point[1] > max[1] ? max[1] : point[1];
        closest[2] = point[2] < min[2] ? min[2] : point[2] > max[2] ? max[2] : point[2];
    }

    bool box_point_overlap( const CartVect& box_min_corner,
                            const CartVect& box_max_corner,
                            const CartVect& point,
                            double tolerance )
    {
        CartVect closest;
        closest_location_on_box( box_min_corner, box_max_corner, point, closest );
        closest -= point;
        return closest % closest < tolerance * tolerance;
    }

    bool boxes_overlap( const CartVect& box_min1,
                        const CartVect& box_max1,
                        const CartVect& box_min2,
                        const CartVect& box_max2,
                        double tolerance )
    {

        for( int k = 0; k < 3; k++ )
        {
            double b1min = box_min1[k], b1max = box_max1[k];
            double b2min = box_min2[k], b2max = box_max2[k];
            if( b1min - tolerance > b2max ) return false;
            if( b2min - tolerance > b1max ) return false;
        }
        return true;
    }

    // see if boxes formed by 2 lists of "CartVect"s overlap
    bool bounding_boxes_overlap( const CartVect* list1, int num1, const CartVect* list2, int num2, double tolerance )
    {
        assert( num1 >= 1 && num2 >= 1 );
        CartVect box_min1 = list1[0], box_max1 = list1[0];
        CartVect box_min2 = list2[0], box_max2 = list2[0];
        for( int i = 1; i < num1; i++ )
        {
            for( int k = 0; k < 3; k++ )
            {
                double val = list1[i][k];
                if( box_min1[k] > val ) box_min1[k] = val;
                if( box_max1[k] < val ) box_max1[k] = val;
            }
        }
        for( int i = 1; i < num2; i++ )
        {
            for( int k = 0; k < 3; k++ )
            {
                double val = list2[i][k];
                if( box_min2[k] > val ) box_min2[k] = val;
                if( box_max2[k] < val ) box_max2[k] = val;
            }
        }

        return boxes_overlap( box_min1, box_max1, box_min2, box_max2, tolerance );
    }

    // see if boxes formed by 2 lists of 2d coordinates overlap (num1>=3, num2>=3, do not check)
    bool bounding_boxes_overlap_2d( const double* list1, int num1, const double* list2, int num2, double tolerance )
    {
        /*
         * box1:
         *         (bmax11, bmax12)
         *      |-------|
         *      |       |
         *      |-------|
         *  (bmin11, bmin12)

         *         box2:
         *                (bmax21, bmax22)
         *             |----------|
         *             |          |
         *             |----------|
         *     (bmin21, bmin22)
         */
        double bmin11, bmax11, bmin12, bmax12;
        bmin11 = bmax11 = list1[0];
        bmin12 = bmax12 = list1[1];

        double bmin21, bmax21, bmin22, bmax22;
        bmin21 = bmax21 = list2[0];
        bmin22 = bmax22 = list2[1];

        for( int i = 1; i < num1; i++ )
        {
            if( bmin11 > list1[2 * i] ) bmin11 = list1[2 * i];
            if( bmax11 < list1[2 * i] ) bmax11 = list1[2 * i];
            if( bmin12 > list1[2 * i + 1] ) bmin12 = list1[2 * i + 1];
            if( bmax12 < list1[2 * i + 1] ) bmax12 = list1[2 * i + 1];
        }
        for( int i = 1; i < num2; i++ )
        {
            if( bmin21 > list2[2 * i] ) bmin21 = list2[2 * i];
            if( bmax21 < list2[2 * i] ) bmax21 = list2[2 * i];
            if( bmin22 > list2[2 * i + 1] ) bmin22 = list2[2 * i + 1];
            if( bmax22 < list2[2 * i + 1] ) bmax22 = list2[2 * i + 1];
        }

        if( ( bmax11 < bmin21 + tolerance ) || ( bmax21 < bmin11 + tolerance ) ) return false;

        if( ( bmax12 < bmin22 + tolerance ) || ( bmax22 < bmin12 + tolerance ) ) return false;

        return true;
    }

    /**\brief Class representing a 3-D mapping function (e.g. shape function for volume element) */
    class VolMap
    {
      public:
        /**\brief Return \f$\vec \xi\f$ corresponding to logical center of element */
        virtual CartVect center_xi() const = 0;
        /**\brief Evaluate mapping function (calculate \f$\vec x = F($\vec \xi)\f$ )*/
        virtual CartVect evaluate( const CartVect& xi ) const = 0;
        /**\brief Evaluate Jacobian of mapping function */
        virtual Matrix3 jacobian( const CartVect& xi ) const = 0;
        /**\brief Evaluate inverse of mapping function (calculate \f$\vec \xi = F^-1($\vec x)\f$ )*/
        bool solve_inverse( const CartVect& x, CartVect& xi, double tol ) const;
    };

    bool VolMap::solve_inverse( const CartVect& x, CartVect& xi, double tol ) const
    {
        const double error_tol_sqr = tol * tol;
        double det;
        xi             = center_xi();
        CartVect delta = evaluate( xi ) - x;
        Matrix3 J;
        while( delta % delta > error_tol_sqr )
        {
            J   = jacobian( xi );
            det = J.determinant();
            if( det < std::numeric_limits< double >::epsilon() ) return false;
            xi -= J.inverse() * delta;
            delta = evaluate( xi ) - x;
        }
        return true;
    }

    /**\brief Shape function for trilinear hexahedron */
    class LinearHexMap : public VolMap
    {
      public:
        LinearHexMap( const CartVect* corner_coords ) : corners( corner_coords ) {}
        virtual CartVect center_xi() const;
        virtual CartVect evaluate( const CartVect& xi ) const;
        virtual Matrix3 jacobian( const CartVect& xi ) const;

      private:
        const CartVect* corners;
        static const double corner_xi[8][3];
    };

    const double LinearHexMap::corner_xi[8][3] = { { -1, -1, -1 }, { 1, -1, -1 }, { 1, 1, -1 }, { -1, 1, -1 },
                                                   { -1, -1, 1 },  { 1, -1, 1 },  { 1, 1, 1 },  { -1, 1, 1 } };
    CartVect LinearHexMap::center_xi() const
    {
        return CartVect( 0.0 );
    }

    CartVect LinearHexMap::evaluate( const CartVect& xi ) const
    {
        CartVect x( 0.0 );
        for( unsigned i = 0; i < 8; ++i )
        {
            const double N_i =
                ( 1 + xi[0] * corner_xi[i][0] ) * ( 1 + xi[1] * corner_xi[i][1] ) * ( 1 + xi[2] * corner_xi[i][2] );
            x += N_i * corners[i];
        }
        x *= 0.125;
        return x;
    }

    Matrix3 LinearHexMap::jacobian( const CartVect& xi ) const
    {
        Matrix3 J( 0.0 );
        for( unsigned i = 0; i < 8; ++i )
        {
            const double xi_p      = 1 + xi[0] * corner_xi[i][0];
            const double eta_p     = 1 + xi[1] * corner_xi[i][1];
            const double zeta_p    = 1 + xi[2] * corner_xi[i][2];
            const double dNi_dxi   = corner_xi[i][0] * eta_p * zeta_p;
            const double dNi_deta  = corner_xi[i][1] * xi_p * zeta_p;
            const double dNi_dzeta = corner_xi[i][2] * xi_p * eta_p;
            J( 0, 0 ) += dNi_dxi * corners[i][0];
            J( 1, 0 ) += dNi_dxi * corners[i][1];
            J( 2, 0 ) += dNi_dxi * corners[i][2];
            J( 0, 1 ) += dNi_deta * corners[i][0];
            J( 1, 1 ) += dNi_deta * corners[i][1];
            J( 2, 1 ) += dNi_deta * corners[i][2];
            J( 0, 2 ) += dNi_dzeta * corners[i][0];
            J( 1, 2 ) += dNi_dzeta * corners[i][1];
            J( 2, 2 ) += dNi_dzeta * corners[i][2];
        }
        return J *= 0.125;
    }

    bool nat_coords_trilinear_hex( const CartVect* corner_coords, const CartVect& x, CartVect& xi, double tol )
    {
        return LinearHexMap( corner_coords ).solve_inverse( x, xi, tol );
    }

    bool point_in_trilinear_hex( const CartVect* hex, const CartVect& xyz, double etol )
    {
        CartVect xi;
        return nat_coords_trilinear_hex( hex, xyz, xi, etol ) && fabs( xi[0] ) - 1 < etol && fabs( xi[1] ) - 1 < etol &&
               fabs( xi[2] ) - 1 < etol;
    }

}  // namespace GeomUtil

}  // namespace moab