Delaunay triangulation and Dirichlet tessellation
This function computes the Delaunay triangulation (and hence the
Dirichlet or Voronoi tesselation) of a planar point set according
to the second (iterative) algorithm of Lee and Schacter ---
see REFERENCES. The triangulation is made to be with respect to
the whole plane by
suspending it from so-called ideal points
(-Inf,-Inf), (Inf,-Inf) (Inf,Inf), and (-Inf,Inf). The triangulation
is also enclosed in a finite rectangular window. A set of dummy
points may be added, in various ways, to the set of data points
deldir(x, y, dpl=NULL, rw=NULL, eps=1e-09, sort=TRUE, plotit=FALSE, digits=6, z=NULL, zdum=NULL, ...)
- The coordinates of the point set being triangulated. These can be
given by two arguments x and y which are vectors or by a single
argument x which is a list with components
y, and possibly
z(which would consis
- A list describing the structure of the dummy points to be added to
the data being triangulated. The addition of these dummy points
is effected by the auxiliary function dumpts(). The list may
ndx: The x-dimension of
- The coordinates of the corners of the rectangular window enclosing the triangulation, in the order (xmin, xmax, ymin, ymax). Any data points (including dummy points) outside this window are discarded. If this argument is omitted, it defaults to values gi
- A value of epsilon used in testing whether a quantity is zero, mainly in the context of whether points are collinear. If anomalous errors arise, it is possible that these may averted by adjusting the value of eps upward or downward.
- Logical argument; if
TRUE(the default) the data (including dummy points) are sorted into a sequence of
binsprior to triangulation; this makes the algorithm slightly more efficient. Normally one would set sort equal to
- Logical argument; if
TRUEa plot is produced. The nature of the plot may be controlled by using the
...argument to pass appropriate arguments to
further instructiona plot o
- The number of decimal places to which all numeric values in the returned list should be rounded. Defaults to 6.
- An optional vector of
auxiliaryvalues or weightsassociated with the respective points.
- Values of
zto be associated with any dummy points that are created. See Warnings.
- Auxiliary arguments add, wlines, wpoints, number, nex, col, lty,
pch, xlim, and ylim (and possibly other plotting parameters) may be
passed to plot.deldir through
This package is a (straightforward) adaptation of the Splus library section ``delaunay'' to R. That library section is an implementation of the Lee-Schacter algorithm, which was originally written as a stand-alone Fortran program in 1987/88 by Rolf Turner, while with the Division of Mathematics and Statistics, CSIRO, Sydney, Australia. It was re-written as an Splus function (using dynamically loaded Fortran code), by Rolf Turner while visiting the University of Western Australia, May, 1995.
Further revisions were made December 1996. The author gratefully acknowledges the contributions, assistance, and guidance of Mark Berman, of D.M.S., CSIRO, in collaboration with whom this project was originally undertaken. The author also acknowledges much useful advice from Adrian Baddeley, formerly of D.M.S., CSIRO (now of CMIS, CSIRO and Adjunct Professor of Statistics at the University of Western Australia). Daryl Tingley of the Department of Mathematics and Statistics, University of New Brunswick provided some helpful insight. Special thanks are extended to Alan Johnson, of the Alaska Fisheries Science Centre, who supplied two data sets which were extremely valuable in tracking down some errors in the code.
Don MacQueen, of Lawrence Livermore National Lab, wrote an Splus driver function for the old stand-alone version of this software. That driver, which was available on Statlib, is now deprecated in favour of the current package ``delaunay'' package. Don also collaborated in the preparation of that package.
ChangeLog for information about further revisions
- A list (of class
deldir), invisible if plotit=
TRUE, with components:
delsgs a data frame with 6 columns. The first 4 entries of each row are the coordinates of the points joined by an edge of a Delaunay triangle, in the order (x1,y1,x2,y2). The last two entries are the indices of the two points which are joined. dirsgs a data frame with 8 columns. The first 4 entries of each row are the coordinates of the endpoints of one the edges of a Dirichlet tile, in the order (x1,y1,x2,y2). The fifth and sixth entries are the indices of the two points, in the set being triangulated, which are separated by that edge. The seventh and eighth entries are logical values. The seventh indicates whether the first endpoint of the corresponding edge of a Dirichlet tile is a boundary point (a point on the boundary of the rectangular window). Likewise for the eighth entry and the second endpoint of the edge. summary a data frame with 9 or 10 columns and
n.data + n.dumrows (see below). The rows correspond to the points in the set being triangulated. The column names are
x(the x-coordinate of the point),
z(the auxiliary values or weights if these were specified),
n.tri(the number of Delaunay triangles emanating from the point),
del.area(1/3 of the total area of all the Delaunay triangles emanating from the point),
del.wts(the corresponding entry of the
del.areacolumn divided by the sum of this column);
n.tside(the number of sides --- within the rectangular window --- of the Dirichlet tile surrounding the point),
nbpt(the number of points in which the Dirichlet tile intersects the boundary of the rectangular window),
dir.area(the area of the Dirichlet tile surrounding the point), and
dir.wts(the corresponding entry of the
dir.areacolumn divided by the sum of this column).
Note that the factor of 1/3 associated with the del.area column arises because each triangle occurs three times --- once for each corner.
n.data the number of real (as opposed to dummy) points in the set which was triangulated, with any duplicate points eliminated. The first n.data rows of
summarycorrespond to real points.
n.dum the number of dummy points which were added to the set being triangulated, with any duplicate points (including any which duplicate real points) eliminated. The last n.dum rows of
summarycorrespond to dummy points.
del.area the area of the convex hull of the set of points being triangulated, as formed by summing the
dir.area the area of the rectangular window enclosing the points being triangulated, as formed by summing the
rw the specification of the corners of the rectangular window enclosing the data, in the order (xmin, xmax, ymin, ymax).
If ndx >= 2 and ndy >= 2, then the rectangular window IS the convex
hull, and so the values of del.area and dir.area (if the latter is
NULL) are identical.
TRUE a plot of the triangulation and/or tessellation is produced
or added to an existing plot.
- The process for determining if points are duplicated
changed between versions 0.1-9 and 0.1-10. Previously there
was an argument
fracfor this function, which defaulted to 0.0001. Points were deemed to be duplicates if the difference in
x-coordinates was less than
fractimes the width of
y-coordinates was less than
fractimes the height of
rw. This process has been changed to one which uses
duplicated()on the data frame whose columns are
As a result it may happen that points which were previously eliminated as duplicates will no longer be eliminated. (And possibly vice-versa.)
- If any dummy points are created, and if a vector
auxiliaryvalues or weightsassociated with the points being triangulated, is supplied, then it is up to the user to supply the corresponding auxiliary values or weights associated with the dummy points. These values should be supplied as
zdumis not supplied then the auxiliary values or weights associated with the dummy points are all taken to be missing values (i.e.
- The components
summaryof the value returned by
deldir()are nowdata framesrather than matrices. The component
summarywas changed to allow the
zto be of arbitrary mode (i.e. not necessarily numeric). The component
delsgswas then changed for consistency. Note that the other
dirsgshas been a data frame since time immemorial.
Lee, D. T., and Schacter, B. J. Two algorithms for constructing a Delaunay triangulation, Int. J. Computer and Information Sciences, Vol. 9, No. 3, 1980, pp. 219 -- 242.
Ahuja, N. and Schacter, B. J. (1983). Pattern Models. New York: Wiley.
# Puts dummy points at the corners of the rectangular # window, i.e. at (0,0), (10,0), (10,10), and (0,10) x <- c(2.3,3.0,7.0,1.0,3.0,8.0) y <- c(2.3,3.0,2.0,5.0,8.0,9.0) tv <- deldir(x,y,list(ndx=2,ndy=2),c(0,10,0,10)) # Plots the triangulation which was created (but not the tesselation). tv <- deldir(x,y,list(ndx=2,ndy=2),c(0,10,0,10),plot=TRUE,wl='tr') # Auxiliary values associated with points; 4 dummy points to be # added so 4 dummy "z-values" provided. z <- sample(1:100,6) zdum <- rep(-99,4) tv <- deldir(x,y,list(ndx=2,ndy=2),c(0,10,0,10),z=z,zdum=zdum)