s.geom(1)	       Grass User's Manual		s.geom(1)



NNAAMMEE
       ss..ggeeoomm	-  Computes  Delaunay triangulation, MinMax-Angle
       triangulation, MinMax-Slope  triangulation,  MaxMin-Height
       triangulation,  Regular triangulation, planesweep triangu-
       lation, Voronoi diagram, and convex hull of sites in 2 and
       2 1/2 dimensions.

SSYYNNOOPPSSIISS
       ss..ggeeoomm
       ss..ggeeoomm hheellpp
       ss..ggeeoomm  iinnppuutt==_n_a_m_e  oouuttppuutt==_n_a_m_e	[pprreecciissiioonn==_v_a_l_u_e] [ooppeerraa--
       ttiioonn==_n_a_m_e]

DDEESSCCRRIIPPTTIIOONN
       _s_._g_e_o_m takes a sites file as input  and	computes  various
       triangulations, the Voronoi diagram, or the convex hull of
       the sites. The z-coordinate is read from  the  description
       field  if  it is specified, otherwise 0 is assumed. The z-
       coordinate is used for the MinMax-slope triangulation  and
       for  the regular triangulation (where it is interpreted as
       the weight of the site). For all other computations the z-
       coordinate is ignored.


       The  MinMax-angle  triangulation  is the triangulation for
       the sites which minmizes  (lexicographically)  the  sorted
       vector  of  all	the angles of triangles in the triangula-
       tion. The MaxMin-height	and  MinMax-slope  triangulations
       are similar.  The algorithms used for the computations are
       not heuristics, they actually achieve the optimum.


       The regular triangulation is the weighted version  of  the
       delaunay triangulation (weights are assigned to the sites,
       the delaunay triangulation corresponds to the regular tri-
       angulation where all the sites have identical weights).


       The output is saved in vector file format.

OOPPTTIIOONNSS
   PPaarraammeetteerrss::
       iinnppuutt==_n_a_m_e
	      Input vector (level 2) file.

       oouuttppuutt==_n_a_m_e
	      Output vector file.

       pprreecciissiioonn==_v_a_l_u_e
	      Number  of  significant positions after the decimal
	      point. (default is 0).

       ooppeerraattiioonn==_n_a_m_e
	      One  of  the  following:	_s_w_e_e_p,	_d_e_l_a_u_n_a_y,  _a_n_g_l_e,
	      _h_e_i_g_h_t,	_s_l_o_p_e,	_h_u_l_l.  These  correspond  to  the
	      planesweep triangulation,  Delaunay  triangulation,
	      MinMax-angle  triangulation, MaxMin-height triangu-
	      lation, MinMax-slope triangulation, regular  trian-
	      gulation, Voronoi diagram, and convex hull, respec-
	      tively.  (default is Delaunay triangulation).


NNOOTTEE
       Only the sites which fall into the current region are used
       for the computations.


       The  computation  times for the various operations depends
       strongly on the algorithm used.


       The plansweep triangulation and	convex	hull  computation
       require _O_(_n _l_o_g _n_) operations in the worst case [Ed].  The
       Delaunay heuristic needs _O_(_n_^_2_) time in	the  worst  case,
       however	it  performs much faster in practice. The MinMax-
       angle and MaxMin-height triangulations need _O_(_n_^_2  _l_o_g  _n_)
       operations  [EdTa],  and  the  MinMax-slope  triangulation
       needs _O_(_n_^_3_) operations [BeEd].


       Internally, the coordinates of the  sites  are  stored  in
       fix-point format.  Therefore, the number of decimal digits
       cannot exceed 64 bit (or apprx.	16 decimal digits).

BBUUGGSS
       Some fields of the header in the output file are not prop-
       erly set.

RREEFFEERREENNCCEESS
       [BeEd]  M.Bern,	H. Edelsbrunner, D. Eppstein, S. Mitchel,
       T.S. Tan.  Edge Insertion for Optimal Triangulations.   _I_n
       _P_r_o_c_.  _1_s_t  _L_a_t_i_n  _A_m_e_r_i_c_a_n  _S_y_m_p_o_s_. _T_h_e_o_r_e_t_.  _I_n_f_o_r_m_a_t_i_c_s
       _1_9_9_2, 46--60.


       [Ed] H. Edelsbrunner.  _A_l_g_o_r_i_t_h_m_s _i_n _C_o_m_b_i_n_a_t_o_r_i_a_l  _G_e_o_m_e_-
       _t_r_y_. Springer-Verlag, Heidelberg, Germany, 1987.


       [EdSh]  H. Edelsbrunner, N. R. Shah.  Incremental Flipping
       Works for Regular Triangulations.  _I_n _P_r_o_c_. _8_t_h _A_n_n_.  _S_y_m_-
       _p_o_s_. _C_o_m_p_u_t_. _G_e_o_m_. _1_9_9_2, 43-52.


       [EdTa]  H.  Edelsbrunner, T.S. Tan and R. Waupotitsch.  An
       O(n^2 log n) Time Algorithm for the MinMax Angle  Triangu-
       lation.	 _S_I_A_M _J_. _S_c_i_. _S_t_a_t_i_s_t_. _C_o_m_p_u_t_. _1_3 _1_9_9_2, 994-1008.

SSEEEE AALLSSOO
       _v_._g_e_o_m

AAUUTTHHOORR
       Roman Waupotitsch

       _L_a_s_t _c_h_a_n_g_e_d_: _$_D_a_t_e_: _2_0_0_2_/_0_1_/_2_5 _0_5_:_4_5_:_3_5 _$



GRASS 5.0						s.geom(1)
