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diff --git a/src/qhull/html/qhull.txt b/src/qhull/html/qhull.txt new file mode 100644 index 000000000..03753547e --- /dev/null +++ b/src/qhull/html/qhull.txt @@ -0,0 +1,1263 @@ + + + +qhull(1) qhull(1) + + +NAME + qhull - convex hull, Delaunay triangulation, Voronoi dia- + gram, halfspace intersection about a point, hull volume, facet area + +SYNOPSIS + qhull- compute convex hulls and related structures + input (stdin): dimension, #points, point coordinates + first comment (non-numeric) is listed in the summary + halfspace: use dim plus one with offsets after coefficients + + options (qh-quick.htm): + d - Delaunay triangulation by lifting points to a paraboloid + v - Voronoi diagram via the Delaunay triangulation + H1,1 - Halfspace intersection about [1,1,0,...] + d Qu - Furthest-site Delaunay triangulation (upper convex hull) + v Qu - Furthest-site Voronoi diagram + QJ - Joggle the input to avoid precision problems + . - concise list of all options + - - one-line description of all options + + Output options (subset): + FA - compute total area and volume + Fx - extreme points (convex hull vertices) + G - Geomview output (2-d, 3-d and 4-d) + Fp - halfspace intersection coordinates + m - Mathematica output (2-d and 3-d) + n - normals with offsets + o - OFF file format (if Voronoi, outputs regions) + TO file- output results to file, may be enclosed in single quotes + f - print all fields of all facets + s - summary of results (default) + Tv - verify result: structure, convexity, and point inclusion + p - vertex coordinates + i - vertices incident to each facet + + example: + rbox 1000 s | qhull Tv s FA + + - html manual: index.htm + - installation: README.txt + - see also: COPYING.txt, REGISTER.txt, Changes.txt + - WWW: <http://www.qhull.org> + - GIT: <git@github.com:qhull/qhull.git> + - mirror: <http://www6.uniovi.es/ftp/pub/mirrors/geom.umn.edu/software/ghindex.html> + - news: <http://www.qhull.org/news> + - Geomview: <http://www.geomview.org> + - news group: <news:comp.graphics.algorithms> + - FAQ: <http://www.faqs.org/faqs/graphics/algorithms-faq/> + - email: qhull@qhull.org + - bug reports: qhull_bug@qhull.org + + + + +Geometry Center 2003/12/30 1 + + + + + +qhull(1) qhull(1) + + + The sections are: + - INTRODUCTION + - DESCRIPTION, a description of Qhull + - IMPRECISION, how Qhull handles imprecision + - OPTIONS + - Input and output options + - Additional input/output formats + - Precision options + - Geomview options + - Print options + - Qhull options + - Trace options + - BUGS + - E-MAIL + - SEE ALSO + - AUTHORS + - ACKNOWLEGEMENTS + + This man page briefly describes all Qhull options. Please + report any mismatches with Qhull's html manual (qh- + man.htm). + + + +INTRODUCTION + Qhull is a general dimension code for computing convex + hulls, Delaunay triangulations, Voronoi diagram, furthest- + site Voronoi diagram, furthest-site Delaunay triangula- + tions, and halfspace intersections about a point. It + implements the Quickhull algorithm for computing the con- + vex hull. Qhull handles round-off errors from floating + point arithmetic. It can approximate a convex hull. + + The program includes options for hull volume, facet area, + partial hulls, input transformations, randomization, trac- + ing, multiple output formats, and execution statistics. + The program can be called from within your application. + You can view the results in 2-d, 3-d and 4-d with + Geomview. + + +DESCRIPTION + The format of input is the following: first line contains + the dimension, second line contains the number of input + points, and point coordinates follow. The dimension and + number of points can be reversed. Comments and line + breaks are ignored. A comment starts with a non-numeric + character and continues to the end of line. The first + comment is reported in summaries and statistics. Error + reporting is better if there is one point per line. + + The default printout option is a short summary. There are + many other output formats. + + + + +Geometry Center 2003/12/30 2 + + + + + +qhull(1) qhull(1) + + + Qhull implements the Quickhull algorithm for convex hull. + This algorithm combines the 2-d Quickhull algorithm with + the n-d beneath-beyond algorithm [c.f., Preparata & Shamos + '85]. It is similar to the randomized algorithms of + Clarkson and others [Clarkson et al. '93]. The main + advantages of Quickhull are output sensitive performance, + reduced space requirements, and automatic handling of pre- + cision problems. + + The data structure produced by Qhull consists of vertices, + ridges, and facets. A vertex is a point of the input set. + A ridge is a set of d vertices and two neighboring facets. + For example in 3-d, a ridge is an edge of the polyhedron. + A facet is a set of ridges, a set of neighboring facets, a + set of incident vertices, and a hyperplane equation. For + simplicial facets, the ridges are defined by the vertices + and neighboring facets. When Qhull merges two facets, it + produces a non-simplicial facet. A non-simplicial facet + has more than d neighbors and may share more than one + ridge with a neighbor. + + +IMPRECISION + Since Qhull uses floating point arithmetic, roundoff error + may occur for each calculation. This causes problems for + most geometric algorithms. + + Qhull automatically sets option 'C-0' in 2-d, 3-d, and + 4-d, or option 'Qx' in 5-d and higher. These options han- + dle precision problems by merging facets. Alternatively, + use option 'QJ' to joggle the input. + + With 'C-0', Qhull merges non-convex facets while con- + structing the hull. The remaining facets are clearly con- + vex. With 'Qx', Qhull merges coplanar horizon facets, + flipped facets, concave facets and duplicated ridges. It + merges coplanar facets after constructing the hull. With + 'Qx', coplanar points may be missed, but it appears to be + unlikely. + + To guarantee triangular output, joggle the input with + option 'QJ'. Facet merging will not occur. + +OPTIONS + To get a list of the most important options, execute + 'qhull' by itself. To get a complete list of options, + execute 'qhull -'. To get a complete, concise list of + options, execute 'qhull .'. + + Options can be in any order. Capitalized options take an + argument (except 'PG' and 'F' options). Single letters + are used for output formats and precision constants. The + other options are grouped into menus for other output for- + mats ('F'), Geomview output ('G'), printing ('P'), Qhull + + + +Geometry Center 2003/12/30 3 + + + + + +qhull(1) qhull(1) + + + control ('Q'), and tracing ('T'). + + Main options: + + default + Compute the convex hull of the input points. + Report a summary of the result. + + d Compute the Delaunay triangulation by lifting the + input points to a paraboloid. The 'o' option + prints the input points and facets. The 'QJ' + option guarantees triangular output. The 'Ft' + option prints a triangulation. It adds points (the + centrums) to non-simplicial facets. + + v Compute the Voronoi diagram from the Delaunay tri- + angulation. The 'p' option prints the Voronoi ver- + tices. The 'o' option prints the Voronoi vertices + and the vertices in each Voronoi region. It lists + regions in site id order. The 'Fv' option prints + each ridge of the Voronoi diagram. The first or + zero'th vertex indicates the infinity vertex. Its + coordinates are qh_INFINITE (-10.101). It indi- + cates unbounded Voronoi regions or degenerate + Delaunay triangles. + + Hn,n,... + Compute halfspace intersection about [n,n,0,...]. + The input is a set of halfspaces defined in the + same format as 'n', 'Fo', and 'Fi'. Use 'Fp' to + print the intersection points. Use 'Fv' to list + the intersection points for each halfspace. The + other output formats display the dual convex hull. + + The point [n,n,n,...] is a feasible point for the + halfspaces, i.e., a point that is inside all of the + halfspaces (Hx+b <= 0). The default coordinate + value is 0. + + The input may start with a feasible point. If so, + use 'H' by itself. The input starts with a feasi- + ble point when the first number is the dimension, + the second number is "1", and the coordinates com- + plete a line. The 'FV' option produces a feasible + point for a convex hull. + + d Qu Compute the furthest-site Delaunay triangulation + from the upper convex hull. The 'o' option prints + the input points and facets. The 'QJ' option guar- + antees triangular otuput. You can also use facets. + + v Qu Compute the furthest-site Voronoi diagram. The 'p' + option prints the Voronoi vertices. The 'o' option + prints the Voronoi vertices and the vertices in + + + +Geometry Center 2003/12/30 4 + + + + + +qhull(1) qhull(1) + + + each Voronoi region. The 'Fv' option prints each + ridge of the Voronoi diagram. The first or zero'th + vertex indicates the infinity vertex at infinity. + Its coordinates are qh_INFINITE (-10.101). It + indicates unbounded Voronoi regions and degenerate + Delaunay triangles. + + Qt Triangulated output. + + + Input/Output options: + + f Print out all facets and all fields of each facet. + + G Output the hull in Geomview format. For imprecise + hulls, Geomview displays the inner and outer hull. + Geomview can also display points, ridges, vertices, + coplanar points, and facet intersections. See + below for a list of options. + + For Delaunay triangulations, 'G' displays the cor- + responding paraboloid. For halfspace intersection, + 'G' displays the dual polytope. + + i Output the incident vertices for each facet. Qhull + prints the number of facets followed by the ver- + tices of each facet. One facet is printed per + line. The numbers are the 0-relative indices of + the corresponding input points. The facets are + oriented. + + In 4-d and higher, Qhull triangulates non-simpli- + cial facets. Each apex (the first vertex) is a + created point that corresponds to the facet's cen- + trum. Its index is greater than the indices of the + input points. Each base corresponds to a simpli- + cial ridge between two facets. To print the ver- + tices without triangulation, use option 'Fv'. + + m Output the hull in Mathematica format. Qhull + writes a Mathematica file for 2-d and 3-d convex + hulls and for 2-d Delaunay triangulations. Qhull + produces a list of objects that you can assign to a + variable in Mathematica, for example: "list= << + <outputfilename> ". If the object is 2-d, it can be + visualized by "Show[Graphics[list]] ". For 3-d + objects the command is "Show[Graphics3D[list]]". + + n Output the normal equation for each facet. Qhull + prints the dimension (plus one), the number of + facets, and the normals for each facet. The + facet's offset follows its normal coefficients. + + o Output the facets in OFF file format. Qhull prints + the dimension, number of points, number of facets, + and number of ridges. Then it prints the + + + +Geometry Center 2003/12/30 5 + + + + + +qhull(1) qhull(1) + + + coordinates of the input points and the vertices + for each facet. Each facet is on a separate line. + The first number is the number of vertices. The + remainder are the indices of the corresponding + points. The vertices are oriented in 2-d, 3-d, and + in simplicial facets. + + For 2-d Voronoi diagrams, the vertices are sorted + by adjacency, but not oriented. In 3-d and higher, + the Voronoi vertices are sorted by index. See the + 'v' option for more information. + + p Output the coordinates of each vertex point. Qhull + prints the dimension, the number of points, and the + coordinates for each vertex. With the 'Gc' and + 'Gi' options, it also prints coplanar and interior + points. For Voronoi diagrams, it prints the coor- + dinates of each Voronoi vertex. + + s Print a summary to stderr. If no output options + are specified at all, a summary goes to stdout. + The summary lists the number of input points, the + dimension, the number of vertices in the convex + hull, the number of facets in the convex hull, the + number of good facets (if 'Pg'), and statistics. + + The last two statistics (if needed) measure the + maximum distance from a point or vertex to a facet. + The number in parenthesis (e.g., 2.1x) is the ratio + between the maximum distance and the worst-case + distance due to merging two simplicial facets. + + + Precision options + + An Maximum angle given as a cosine. If the angle + between a pair of facet normals is greater than n, Qhull + merges one of the facets into a neighbor. If 'n' + is negative, Qhull tests angles after adding each + point to the hull (pre-merging). If 'n' is posi- + tive, Qhull tests angles after constructing the + hull (post-merging). Both pre- and post-merging + can be defined. + + Option 'C0' or 'C-0' is set if the corresponding + 'Cn' or 'C-n' is not set. If 'Qx' is set, then 'A- + n' and 'C-n' are checked after the hull is con- + structed and before 'An' and 'Cn' are checked. + + Cn Centrum radius. If a centrum is less than n below + a neighboring facet, Qhull merges one of the + facets. If 'n' is negative or '-0', Qhull tests + and merges facets after adding each point to the + hull. This is called "pre-merging". If 'n' is + + + +Geometry Center 2003/12/30 6 + + + + + +qhull(1) qhull(1) + + + positive, Qhull tests for convexity after con- + structing the hull ("post-merging"). Both pre- and + post-merging can be defined. + + For 5-d and higher, 'Qx' should be used instead of + 'C-n'. Otherwise, most or all facets may be merged + together. + + En Maximum roundoff error for distance computations. + + Rn Randomly perturb distance computations up to +/- n + * max_coord. This option perturbs every distance, + hyperplane, and angle computation. To use time as + the random number seed, use option 'QR-1'. + + Vn Minimum distance for a facet to be visible. A + facet is visible if the distance from the point to + the facet is greater than 'Vn'. + + Without merging, the default value for 'Vn' is the + round-off error ('En'). With merging, the default + value is the pre-merge centrum ('C-n') in 2-d or + 3--d, or three times that in other dimensions. If + the outside width is specified ('Wn'), the maximum, + default value for 'Vn' is 'Wn'. + + Un Maximum distance below a facet for a point to be + coplanar to the facet. The default value is 'Vn'. + + Wn Minimum outside width of the hull. Points are + added to the convex hull only if they are clearly + outside of a facet. A point is outside of a facet + if its distance to the facet is greater than 'Wn'. + The normal value for 'Wn' is 'En'. If the user + specifies pre-merging and does not set 'Wn', than + 'Wn' is set to the premerge 'Cn' and maxco- + ord*(1-An). + + + Additional input/output formats + + Fa Print area for each facet. For Delaunay triangula- + tions, the area is the area of the triangle. For + Voronoi diagrams, the area is the area of the dual + facet. Use 'PAn' for printing the n largest + facets, and option 'PFn' for printing facets larger + than 'n'. + + The area for non-simplicial facets is the sum of + the areas for each ridge to the centrum. Vertices + far below the facet's hyperplane are ignored. The + reported area may be significantly less than the + actual area. + + + + +Geometry Center 2003/12/30 7 + + + + + +qhull(1) qhull(1) + + + FA Compute the total area and volume for option 's'. + It is an approximation for non-simplicial facets + (see 'Fa'). + + Fc Print coplanar points for each facet. The output + starts with the number of facets. Then each facet + is printed one per line. Each line is the number + of coplanar points followed by the point ids. + Option 'Qi' includes the interior points. Each + coplanar point (interior point) is assigned to the + facet it is furthest above (resp., least below). + + FC Print centrums for each facet. The output starts + with the dimension followed by the number of + facets. Then each facet centrum is printed, one + per line. + + Fd Read input in cdd format with homogeneous points. + The input starts with comments. The first comment + is reported in the summary. Data starts after a + "begin" line. The next line is the number of + points followed by the dimension+1 and "real" or + "integer". Then the points are listed with a + leading "1" or "1.0". The data ends with an "end" + line. + + For halfspaces ('Fd Hn,n,...'), the input format is + the same. Each halfspace starts with its offset. + The sign of the offset is the opposite of Qhull's + convention. + + FD Print normals ('n', 'Fo', 'Fi') or points ('p') in + cdd format. The first line is the command line + that invoked Qhull. Data starts with a "begin" + line. The next line is the number of normals or + points followed by the dimension+1 and "real". + Then the normals or points are listed with the + offset before the coefficients. The offset for + points is 1.0. The offset for normals has the + opposite sign. The data ends with an "end" line. + + FF Print facets (as in 'f') without printing the + ridges. + + Fi Print inner planes for each facet. The inner plane + is below all vertices. + + Fi Print separating hyperplanes for bounded, inner + regions of the Voronoi diagram. The first line is + the number of ridges. Then each hyperplane is + printed, one per line. A line starts with the num- + ber of indices and floats. The first pair lists + adjacent input sites, the next d floats are the + normalized coefficients for the hyperplane, and the + + + +Geometry Center 2003/12/30 8 + + + + + +qhull(1) qhull(1) + + + last float is the offset. The hyperplane is ori- + ented toward verify that the hyperplanes are per- + pendicular bisectors. Use 'Fo' for unbounded + regions, and 'Fv' for the corresponding Voronoi + vertices. + + FI Print facet identifiers. + + Fm Print number of merges for each facet. At most 511 + merges are reported for a facet. See 'PMn' for + printing the facets with the most merges. + + FM Output the hull in Maple format. See 'm' + + Fn Print neighbors for each facet. The output starts + with the number of facets. Then each facet is + printed one per line. Each line is the number of + neighbors followed by an index for each neighbor. + The indices match the other facet output formats. + + A negative index indicates an unprinted facet due + to printing only good facets ('Pg'). It is the + negation of the facet's id (option 'FI'). For + example, negative indices are used for facets "at + infinity" in the Delaunay triangulation. + + FN Print vertex neighbors or coplanar facet for each + point. The first line is the number of points. + Then each point is printed, one per line. If the + point is coplanar, the line is "1" followed by the + facet's id. If the point is not a selected vertex, + the line is "0". Otherwise, each line is the num- + ber of neighbors followed by the corresponding + facet indices (see 'Fn'). + + Fo Print outer planes for each facet in the same for- + mat as 'n'. The outer plane is above all points. + + Fo Print separating hyperplanes for unbounded, outer + regions of the Voronoi diagram. The first line is + the number of ridges. Then each hyperplane is + printed, one per line. A line starts with the num- + ber of indices and floats. The first pair lists + adjacent input sites, the next d floats are the + normalized coefficients for the hyperplane, and the + last float is the offset. The hyperplane is ori- + ented toward verify that the hyperplanes are per- + pendicular bisectors. Use 'Fi' for bounded + regions, and 'Fv' for the corresponding Voronoi + vertices. + + FO List all options to stderr, including the default + values. Additional 'FO's are printed to stdout. + + Fp Print points for halfspace intersections (option + 'Hn,n,...'). Each intersection corresponds to a + + + +Geometry Center 2003/12/30 9 + + + +qhull(1) qhull(1) + + + facet of the dual polytope. The "infinity" point + [-10.101,-10.101,...] indicates an unbounded + intersection. + + FP For each coplanar point ('Qc') print the point id + of the nearest vertex, the point id, the facet id, + and the distance. + + FQ Print command used for qhull and input. + + Fs Print a summary. The first line consists of the + number of integers ("7"), followed by the dimen- + sion, the number of points, the number of vertices, + the number of facets, the number of vertices + selected for output, the number of facets selected + for output, the number of coplanar points selected + for output. + + The second line consists of the number of reals + ("2"), followed by the maxmimum offset to an outer + plane and and minimum offset to an inner plane. + Roundoff is included. Later versions of Qhull may + produce additional integers or reals. + + FS Print the size of the hull. The first line con- + sists of the number of integers ("0"). The second + line consists of the number of reals ("2"), fol- + lowed by the total facet area, and the total vol- + ume. Later versions of Qhull may produce addi- + tional integers or reals. + + The total volume measures the volume of the inter- + section of the halfspaces defined by each facet. + Both area and volume are approximations for non- + simplicial facets. See option 'Fa'. + + Ft Print a triangulation with added points for non- + simplicial facets. The first line is the dimension + and the second line is the number of points and the + number of facets. The points follow, one per line, + then the facets follow as a list of point indices. + With option points include the point-at-infinity. + + Fv Print vertices for each facet. The first line is + the number of facets. Then each facet is printed, + one per line. Each line is the number of vertices + followed by the corresponding point ids. Vertices + are listed in the order they were added to the hull + (the last one is first). + + Fv Print all ridges of a Voronoi diagram. The first + line is the number of ridges. Then each ridge is + printed, one per line. A line starts with the num- + ber of indices. The first pair lists adjacent + + + +Geometry Center 2003/12/30 10 + + + + + +qhull(1) qhull(1) + + + input sites, the remaining indices list Voronoi + vertices. Vertex '0' indicates the vertex-at- + infinity (i.e., an unbounded ray). In 3-d, the + vertices are listed in order. See 'Fi' and 'Fo' + for separating hyperplanes. + + FV Print average vertex. The average vertex is a fea- + sible point for halfspace intersection. + + Fx List extreme points (vertices) of the convex hull. + The first line is the number of points. The other + lines give the indices of the corresponding points. + The first point is '0'. In 2-d, the points occur + in counter-clockwise order; otherwise they occur in + input order. For Delaunay triangulations, 'Fx' + lists the extreme points of the input sites. The + points are unordered. + + + Geomview options + + G Produce a file for viewing with Geomview. Without + other options, Qhull displays edges in 2-d, outer + planes in 3-d, and ridges in 4-d. A ridge can be + explicit or implicit. An explicit ridge is a dim-1 + dimensional simplex between two facets. In 4-d, + the explicit ridges are triangles. When displaying + a ridge in 4-d, Qhull projects the ridge's vertices + to one of its facets' hyperplanes. Use 'Gh' to + project ridges to the intersection of both hyper- + planes. + + Ga Display all input points as dots. + + Gc Display the centrum for each facet in 3-d. The + centrum is defined by a green radius sitting on a + blue plane. The plane corresponds to the facet's + hyperplane. The radius is defined by 'C-n' or + 'Cn'. + + GDn Drop dimension n in 3-d or 4-d. The result is a + 2-d or 3-d object. + + Gh Display hyperplane intersections in 3-d and 4-d. + In 3-d, the intersection is a black line. It lies + on two neighboring hyperplanes (c.f., the blue + squares associated with centrums ('Gc')). In 4-d, + the ridges are projected to the intersection of + both hyperplanes. + + Gi Display inner planes in 2-d and 3-d. The inner + plane of a facet is below all of its vertices. It + is parallel to the facet's hyperplane. The inner + plane's color is the opposite (1-r,1-g,1-b) of the + + + +Geometry Center 2003/12/30 11 + + + + + +qhull(1) qhull(1) + + + outer plane. Its edges are determined by the ver- + tices. + + Gn Do not display inner or outer planes. By default, + Geomview displays the precise plane (no merging) or + both inner and output planes (merging). Under + merging, Geomview does not display the inner plane + if the the difference between inner and outer is + too small. + + Go Display outer planes in 2-d and 3-d. The outer + plane of a facet is above all input points. It is + parallel to the facet's hyperplane. Its color is + determined by the facet's normal, and its edges are + determined by the vertices. + + Gp Display coplanar points and vertices as radii. A + radius defines a ball which corresponds to the + imprecision of the point. The imprecision is the + maximum of the roundoff error, the centrum radius, + and maxcoord * (1-An). It is at least 1/20'th of + the maximum coordinate, and ignores post-merging if + pre-merging is done. + + Gr Display ridges in 3-d. A ridge connects the two + vertices that are shared by neighboring facets. + Ridges are always displayed in 4-d. + + Gt A 3-d Delaunay triangulation looks like a convex + hull with interior facets. Option 'Gt' removes the + outside ridges to reveal the outermost facets. It + automatically sets options 'Gr' and 'GDn'. + + Gv Display vertices as spheres. The radius of the + sphere corresponds to the imprecision of the data. + See 'Gp' for determining the radius. + + + Print options + + PAn Only the n largest facets are marked good for + printing. Unless 'PG' is set, 'Pg' is automati- + cally set. + + Pdk:n Drop facet from output if normal[k] <= n. The + option 'Pdk' uses the default value of 0 for n. + + PDk:n Drop facet from output if normal[k] >= n. The + option 'PDk' uses the default value of 0 for n. + + PFn Only facets with area at least 'n' are marked good + for printing. Unless 'PG' is set, 'Pg' is automat- + ically set. + + + + +Geometry Center 2003/12/30 12 + + + + + +qhull(1) qhull(1) + + + Pg Print only good facets. A good facet is either + visible from a point (the 'QGn' option) or includes + a point (the 'QVn' option). It also meets the + requirements of 'Pdk' and 'PDk' options. Option + 'Pg' is automatically set for options 'PAn' and + 'PFn'. + + PG Print neighbors of good facets. + + PMn Only the n facets with the most merges are marked + good for printing. Unless 'PG' is set, 'Pg' is + automatically set. + + Po Force output despite precision problems. Verify ('Tv') does not check + coplanar points. Flipped facets are reported and + concave facets are counted. If 'Po' is used, + points are not partitioned into flipped facets and + a flipped facet is always visible to a point. + Also, if an error occurs before the completion of + Qhull and tracing is not active, 'Po' outputs a + neighborhood of the erroneous facets (if any). + + Pp Do not report precision problems. + + + Qhull control options + + Qbk:0Bk:0 + Drop dimension k from the input points. This + allows the user to take convex hulls of sub-dimen- + sional objects. It happens before the Delaunay and + Voronoi transformation. + + QbB Scale the input points to fit the unit cube. After + scaling, the lower bound will be -0.5 and the upper + bound +0.5 in all dimensions. For Delaunay and + Voronoi diagrams, scaling happens after projection + to the paraboloid. Under precise arithmetic, scal- + ing does not change the topology of the convex + hull. + + Qbb Scale the last coordinate to [0, m] where m is the + maximum absolute value of the other coordinates. + For Delaunay and Voronoi diagrams, scaling happens + after projection to the paraboloid. It reduces + roundoff error for inputs with integer coordinates. + Under precise arithmetic, scaling does not change + the topology of the convex hull. + + Qbk:n Scale the k'th coordinate of the input points. + After scaling, the lower bound of the input points + will be n. 'Qbk' scales to -0.5. + + + +Geometry Center 2003/12/30 13 + + + + + +qhull(1) qhull(1) + + + QBk:n Scale the k'th coordinate of the input points. + After scaling, the upper bound will be n. 'QBk' + scales to +0.5. + + Qc Keep coplanar points with the nearest facet. Out- + put formats 'p', 'f', 'Gp', 'Fc', 'FN', and 'FP' + will print the points. + + Qf Partition points to the furthest outside facet. + + Qg Only build good facets. With the 'Qg' option, + Qhull will only build those facets that it needs to + determine the good facets in the output. See + 'QGn', 'QVn', and 'PdD' for defining good facets, + and 'Pg' and 'PG' for printing good facets and + their neighbors. + + QGn A facet is good (see 'Qg' and 'Pg') if it is visi- + ble from point n. If n < 0, a facet is good if it + is not visible from point n. Point n is not added + to the hull (unless 'TCn' or 'TPn'). With rbox, + use the 'Pn,m,r' option to define your point; it + will be point 0 (QG0). + + Qi Keep interior points with the nearest facet. Out- + put formats 'p', 'f', 'Gp', 'FN', 'FP', and 'Fc' + will print the points. + + QJn Joggle each input coordinate by adding a random + number in [-n,n]. If a precision error occurs, + then qhull increases n and tries again. It does + not increase n beyond a certain value, and it stops + after a certain number of attempts [see user.h]. + Option 'QJ' selects a default value for n. The + output will be simplicial. For Delaunay triangula- + tions, 'QJn' sets 'Qbb' to scale the last coordi- + nate (not if 'Qbk:n' or 'QBk:n' is set). 'QJn' is + deprecated for Voronoi diagrams. See also 'Qt'. + + Qm Only process points that would otherwise increase + max_outside. Other points are treated as coplanar + or interior points. + + Qr Process random outside points instead of furthest + ones. This makes Qhull equivalent to the random- + ized incremental algorithms. CPU time is not + reported since the randomization is inefficient. + + QRn Randomly rotate the input points. If n=0, use time + as the random number seed. If n>0, use n as the + random number seed. If n=-1, don't rotate but use + time as the random number seed. For Delaunay tri- + angulations ('d' and 'v'), rotate about the last + axis. + + + + +Geometry Center 2003/12/30 14 + + + + + +qhull(1) qhull(1) + + + Qs Search all points for the initial simplex. + + Qt Triangulated output. Triangulate non-simplicial + facets. 'Qt' is deprecated for Voronoi diagrams. + See also 'QJn' + + Qv Test vertex neighbors for convexity after post- + merging. To use the 'Qv' option, you also need to + set a merge option (e.g., 'Qx' or 'C-0'). + + QVn A good facet (see 'Qg' and 'Pg') includes point n. + If n<0, then a good facet does not include point n. + The point is either in the initial simplex or it is + the first point added to the hull. Option 'QVn' + may not be used with merging. + + Qx Perform exact merges while building the hull. The + "exact" merges are merging a point into a coplanar + facet (defined by 'Vn', 'Un', and 'C-n'), merging + concave facets, merging duplicate ridges, and merg- + ing flipped facets. Coplanar merges and angle + coplanar merges ('A-n') are not performed. Concav- + ity testing is delayed until a merge occurs. + + After the hull is built, all coplanar merges are + performed (defined by 'C-n' and 'A-n'), then post- + merges are performed (defined by 'Cn' and 'An'). + + Qz Add a point "at infinity" that is above the + paraboloid for Delaunay triangulations and Voronoi + diagrams. This reduces precision problems and + allows the triangulation of cospherical points. + + + Qhull experiments and speedups + + Q0 Turn off pre-merging as a default option. With + 'Q0'/'Qx' and without explicit pre-merge options, + Qhull ignores precision issues while constructing + the convex hull. This may lead to precision + errors. If so, a descriptive warning is generated. + + Q1 With 'Q1', Qhull sorts merges by type (coplanar, + angle coplanar, concave) instead of by angle. + + Q2 With 'Q2', Qhull merges all facets at once instead + of using independent sets of merges and then + retesting. + + Q3 With 'Q3', Qhull does not remove redundant ver- + tices. + + Q4 With 'Q4', Qhull avoids merges of an old facet into + a new facet. + + Q5 With 'Q5', Qhull does not correct outer planes at + the end. The maximum outer plane is used instead. + + + + +Geometry Center 2003/12/30 15 + + + + + +qhull(1) qhull(1) + + + Q6 With 'Q6', Qhull does not pre-merge concave or + coplanar facets. + + Q7 With 'Q7', Qhull processes facets in depth-first + order instead of breadth-first order. + + Q8 With 'Q8' and merging, Qhull does not retain near- + interior points for adjusting outer planes. 'Qc' + will probably retain all points that adjust outer + planes. + + Q9 With 'Q9', Qhull processes the furthest of all out- + side sets at each iteration. + + Q10 With 'Q10', Qhull does not use special processing + for narrow distributions. + + Q11 With 'Q11', Qhull copies normals and recomputes + centrums for tricoplanar facets. + + Q12 With 'Q12', Qhull does not report a very wide merge due + to a duplicated ridge with nearly coincident vertices + + Trace options + + Tn Trace at level n. Qhull includes full execution + tracing. 'T-1' traces events. 'T1' traces the + overall execution of the program. 'T2' and 'T3' + trace overall execution and geometric and topologi- + cal events. 'T4' traces the algorithm. 'T5' + includes information about memory allocation and + Gaussian elimination. + + Ta Annotate output with codes that identify the + corresponding qh_fprintf() statement. + + Tc Check frequently during execution. This will catch + most inconsistency errors. + + TCn Stop Qhull after building the cone of new facets + for point n. The output for 'f' includes the cone + and the old hull. See also 'TVn'. + + TFn Report progress whenever more than n facets are + created During post-merging, 'TFn' reports progress + after more than n/2 merges. + + TI file + Input data from 'file'. The filename may not include + spaces or quotes. + + TO file + Output results to 'file'. The name may be enclosed + in single quotes. + + TPn Turn on tracing when point n is added to the hull. + Trace partitions of point n. If used with TWn, turn off + tracing after adding point n to the hull. + + TRn Rerun qhull n times. Usually used with 'QJn' to + determine the probability that a given joggle will + fail. + + Ts Collect statistics and print to stderr at the end + of execution. + + Tv Verify the convex hull. This checks the topologi- + cal structure, facet convexity, and point inclu- + sion. If precision problems occurred, facet con- + vexity is tested whether or not 'Tv' is selected. + Option 'Tv' does not check point inclusion if + + + +Geometry Center 2003/12/30 16 + + + + + +qhull(1) qhull(1) + + + forcing output with 'Po', or if 'Q5' is set. + + For point inclusion testing, Qhull verifies that + all points are below all outer planes (facet->max- + outside). Point inclusion is exhaustive if merging + or if the facet-point product is small enough; oth- + erwise Qhull verifies each point with a directed + search (qh_findbest). + + Point inclusion testing occurs after producing out- + put. It prints a message to stderr unless option + 'Pp' is used. This allows the user to interrupt + Qhull without changing the output. + + TVn Stop Qhull after adding point n. If n < 0, stop + Qhull before adding point n. Output shows the hull + at this time. See also 'TCn' + + TMn Turn on tracing at n'th merge. + + TWn Trace merge facets when the width is greater than + n. + + Tz Redirect stderr to stdout. + + +BUGS + Please report bugs to Brad Barber at + qhull_bug@qhull.org. + + If Qhull does not compile, it is due to an incompatibility + between your system and ours. The first thing to check is + that your compiler is ANSI standard. If it is, check the + man page for the best options, or find someone to help + you. If you locate the cause of your problem, please send + email since it might help others. + + If Qhull compiles but crashes on the test case (rbox D4), + there's still incompatibility between your system and + ours. Typically it's been due to mem.c and memory align- + ment. You can use qh_NOmem in mem.h to turn off memory + management. Please let us know if you figure out how to + fix these problems. + + If you do find a problem, try to simplify it before + reporting the error. Try different size inputs to locate + the smallest one that causes an error. You're welcome to + hunt through the code using the execution trace as a + guide. This is especially true if you're incorporating + Qhull into your own program. + + When you do report an error, please attach a data set to + the end of your message. This allows us to see the error + for ourselves. Qhull is maintained part-time. + + + +Geometry Center 2003/12/30 17 + + + + + +qhull(1) qhull(1) + + +E-MAIL + Please send correspondence to qhull@qhull.org and + report bugs to qhull_bug@qhull.org. Let us know how + you use Qhull. If you mention it in a paper, please send + the reference and an abstract. + + If you would like to get Qhull announcements (e.g., a new + version) and news (any bugs that get fixed, etc.), let us + know and we will add you to our mailing list. If you + would like to communicate with other Qhull users, we will + add you to the qhull_users alias. For Internet news about + geometric algorithms and convex hulls, look at comp.graph- + ics.algorithms and sci.math.num-analysis + + +SEE ALSO + rbox(1) + + Barber, C. B., D.P. Dobkin, and H.T. Huhdanpaa, "The + Quickhull Algorithm for Convex Hulls," ACM Trans. on Math- + ematical Software, 22(4):469-483, Dec. 1996. + http://portal.acm.org/citation.cfm?doid=235815.235821 + http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.405 + + + Clarkson, K.L., K. Mehlhorn, and R. Seidel, "Four results + on randomized incremental construction," Computational + Geometry: Theory and Applications, vol. 3, p. 185-211, + 1993. + + Preparata, F. and M. Shamos, Computational Geometry, + Springer-Verlag, New York, 1985. + + + +AUTHORS + C. Bradford Barber Hannu Huhdanpaa + bradb@shore.net hannu@qhull.org + + + +ACKNOWLEDGEMENTS + A special thanks to Albert Marden, Victor Milenkovic, the + Geometry Center, Harvard University, and Endocardial Solu- + tions, Inc. for supporting this work. + + Qhull 1.0 and 2.0 were developed under National Science Foundation + grants NSF/DMS-8920161 and NSF-CCR-91-15793 750-7504. David Dobkin + + + +Geometry Center 2003/12/30 18 + + + + + +qhull(1) qhull(1) + + + guided the original work at Princeton University. If you find it + useful, please let us know. + + The Geometry Center was supported by grant DMS-8920161 from the National + Science Foundation, by grant DOE/DE-FG02-92ER25137 from the Department + of Energy, by the University of Minnesota, and by Minnesota Technology, Inc. + + Qhull is available from http://www.qhull.org + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Geometry Center 2003/12/30 19 + + |