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Geometrically, a vertex is a point with

dcoordinates and a facet is a halfspace. Ahalfspaceis defined by an oriented hyperplane through the facet's vertices. Ahyperplaneis defined bydnormalized coefficients and an offset. A point isabovea facet if its distance to the facet is positive.Qhull uses floating point coordinates for input points, vertices, halfspace equations, centrums, and an interior point.

Qhull may be configured for single precision or double precision floating point arithmetic (see realT ).

Each floating point operation may incur round-off error (see Merge). The maximum error for distance computations is determined at initialization. The roundoff error in halfspace computation is accounted for by computing the distance from vertices to the halfspace.

**Copyright © 1995-2019 C.B. Barber**

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- geometric data types and constants
- mathematical macros
- mathematical functions
- computational geometry functions
- point array functions
- geometric facet functions
- geometric roundoff functions

- coordT coordinates and coefficients are stored as realT
- pointT a point is an array
of
`DIM3`coordinates

- fabs_ returns the absolute value of a
- fmax_ returns the maximum value of a and b
- fmin_ returns the minimum value of a and b
- maximize_ maximize a value
- minimize_ minimize a value
- det2_ compute a 2-d determinate
- det3_ compute a 3-d determinate
- dX, dY, dZ compute the difference between two coordinates

- qh_backnormal solve for normal using back substitution
- qh_crossproduct compute the cross product of two 3-d vectors
- qh_determinant compute the determinant of a square matrix
- qh_gausselim Gaussian elimination with partial pivoting
- qh_gram_schmidt implements Gram-Schmidt orthogonalization by rows
- qh_maxabsval return max absolute value of a vector
- qh_minabsval return min absolute value of a dim vector
- qh_mindiff return index of min absolute difference of two vectors
- qh_normalize normalize a vector
- qh_normalize2 normalize a vector and report if too small
- qh_printmatrix print matrix given by row vectors
- qh_rand/srand generate random numbers
- qh_randomfactor return a random factor near 1.0
- qh_randommatrix generate a random dimXdim matrix in range (-1,1)

- qh_detsimplex compute determinate of a simplex of points
- qh_detvnorm determine normal for Voronoi ridge
- qh_distnorm compute distance from point to hyperplane as defined by normal and offset
- qh_facetarea_simplex return area of a simplex
- qh_getangle return cosine of angle (i.e., dot product)
- qh_getcenter return arithmetic center for a set of vertices
- qh_pointdist return distance between two points
- qh_rotatepoints rotate numpoints points by a row matrix
- qh_sethalfspace set coords to dual of halfspace relative to an interior point
- qh_sethyperplane_det return hyperplane for oriented simplex using determinates
- qh_sethyperplane_gauss return hyperplane for oriented simplex using Gaussian elimination
- qh_voronoi_center return Voronoi center for a set of points

- qh_copypoints return malloc'd copy of points
- qh_joggleinput joggle input points by qh.JOGGLEmax
- qh_maxmin return max/min points for each dimension
- qh_maxsimplex determines maximum simplex for a set of points
- qh_printpoints print ids for a set of points
- qh_projectinput project input using qh DELAUNAY and qh low_bound/high_bound
- qh_projectpoints project points along one or more dimensions
- qh_rotateinput rotate input points using row matrix
- qh_scaleinput scale input points using qh low_bound/high_bound
- qh_scalelast scale last coordinate to [0,m] for Delaunay triangulations
- qh_scalepoints scale points to new lowbound and highbound
- qh_setdelaunay project points to paraboloid for Delaunay triangulation
- qh_sethalfspace_all generate dual for halfspace intersection with interior point

- qh_distplane return distance from point to facet
- qh_facetarea return area of a facet
- qh_facetcenter return Voronoi center for a facet's vertices
- qh_findbest find visible facet or best facet for a point
- qh_findbesthorizon update best new facet with horizon facets
- qh_findbestnew find best new facet for point
- qh_furthestnewvertex return furthest new vertex for facet
- qh_furthestvertex return furthest vertex in facetA above facetB
- qh_getarea get area and volume of all facets in facetlist, collect statistics
- qh_getcentrum return centrum for a facet
- qh_getdistance returns the max and min distance of a facet's vertices to a neighboring facet
- qh_findgooddist find best good facet visible for point from facet
- qh_inthresholds return True if facet normal within 'Pdn' and 'PDn'
- qh_orientoutside
orient facet so that
`qh.interior_point`is inside - qh_projectpoint project point onto a facet
- qh_setfacetplane sets the hyperplane for a facet
- qh_sharpnewfacets true if new facets contains a sharp corner
- qh_vertex_bestdist return nearest distance between vertices, except 'skip'
- qh_vertex_isbelow true if vertexA is below vertexB's simplicial facets, or least above

- qh_detjoggle determine default joggle for points and distance roundoff error
- qh_detmaxoutside determine qh.MAXoutside target for qh_RATIO... tests
- qh_detroundoff determine maximum roundoff error and other precision constants
- qh_distround compute maximum roundoff error due to a distance computation to a normalized hyperplane
- qh_divzero divide by a number that is nearly zero
- qh_maxouter return maximum outer plane
- qh_outerinner return actual outer and inner planes

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Comments to: qhull@qhull.org

Created: May 2, 1997 --- Last modified: see top