---------------------------------

  geom.h 
    header file for geometric routines

   see qh-geom.htm and geom.c

   copyright (c) 1993-2003 The Geometry Center        
*/

#ifndef qhDEFgeom
#define qhDEFgeom 1

/* ============ -macros- ======================== */

/*----------------------------------
   
  fabs_(a)
    returns the absolute value of a
*/
#define fabs_( a ) ((( a ) < 0 ) ? -( a ):( a ))
               
/*----------------------------------
  
  fmax_(a,b)
    returns the maximum value of a and b
*/
#define fmax_( a,b )  ( ( a ) < ( b ) ? ( b ) : ( a ) )

/*----------------------------------

  fmin_(a,b)
    returns the minimum value of a and b
*/
#define fmin_( a,b )  ( ( a ) > ( b ) ? ( b ) : ( a ) )

/*----------------------------------

  maximize_(maxval, val)
    set maxval to val if val is greater than maxval
*/
#define maximize_( maxval, val ) {if (( maxval ) < ( val )) ( maxval )= ( val );}

/*----------------------------------

  minimize_(minval, val)
    set minval to val if val is less than minval
*/
#define minimize_( minval, val ) {if (( minval ) > ( val )) ( minval )= ( val );}

/*----------------------------------

  det2_(a1, a2,     
        b1, b2)
  
    compute a 2-d determinate
*/
#define det2_( a1,a2,b1,b2 ) (( a1 )*( b2 ) - ( a2 )*( b1 ))

/*----------------------------------
  
  det3_(a1, a2, a3,    
       b1, b2, b3,
       c1, c2, c3)
  
    compute a 3-d determinate
*/
#define det3_( a1,a2,a3,b1,b2,b3,c1,c2,c3 ) ( ( a1 )*det2_( b2,b3,c2,c3 ) \
                - ( b1 )*det2_( a2,a3,c2,c3 ) + ( c1 )*det2_( a2,a3,b2,b3 ) )

/*----------------------------------
  
  dX( p1, p2 )
  dY( p1, p2 )
  dZ( p1, p2 )
  
    given two indices into rows[],

    compute the difference between X, Y, or Z coordinates
*/
#define dX( p1,p2 )  ( *( rows[p1] ) - *( rows[p2] ))
#define dY( p1,p2 )  ( *( rows[p1]+1 ) - *( rows[p2]+1 ))
#define dZ( p1,p2 )  ( *( rows[p1]+2 ) - *( rows[p2]+2 ))
#define dW( p1,p2 )  ( *( rows[p1]+3 ) - *( rows[p2]+3 ))

/*============= prototypes in alphabetical order, infrequent at end ======= */

void    qh_backnormal (realT **rows, int numrow, int numcol, boolT sign, coordT *normal, boolT *nearzero);
void	qh_distplane (pointT *point, facetT *facet, realT *dist);
facetT *qh_findbest (pointT *point, facetT *startfacet,
		     boolT bestoutside, boolT isnewfacets, boolT noupper,
		     realT *dist, boolT *isoutside, int *numpart);
facetT *qh_findbesthorizon (boolT ischeckmax, pointT *point, 
	             facetT *startfacet, boolT noupper, realT *bestdist, int *numpart);
facetT *qh_findbestnew (pointT *point, facetT *startfacet, realT *dist, 
		     boolT bestoutside, boolT *isoutside, int *numpart);
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, int numrow, int numcol, boolT '
void 	qh_gausselim(realT **rows, i