R: Delaunay triangulation and Dirichlet tessellation

deldir {deldir}

R Documentation

Delaunay triangulation and Dirichlet tessellation

Description

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 being triangulated.

Usage

deldir(x, y, dpl=NULL, rw=NULL, eps=1e-09, sort=TRUE, plotit=FALSE,
       digits=6, z=NULL, zdum=NULL, suppressMsge=FALSE, ...)

Arguments

`x,y`	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 either a data frame or a generic list. If `x` is a data frame then the `x` coordinates of the points to be triangulated are taken to be the column of this data frame which is named “x” if there is one, else the first column of the data frame which is not named either “y” or “z”. The `y` coordinates are taken to be the column of this data frame which is named “y” if there is one, else the first column of the data frame which is not named either “x” or “x”. If there is a column named “z” and if the argument `z` is NULL, then this the column named “z” is taken to be the value of `z`. If `x` is a list (but not a data frame) then it must have components named `x` and `y`, and possibly a component named `z`. The `x` and `y` components give the `x` and `y` coordinates respectively of the points to be triangulated, and the component `z` if present (and if argument `z` is `NULL`) is taken to be the value of `z` (i.e. of a vector of “auxiliary” values or “weights” associated with the respective points).
`dpl`	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 have components: `ndx`: The x-dimension of a rectangular grid; if either ndx or ndy is null, no grid is constructed. `ndy`: The y-dimension of the aforementioned rectangular grid. `nrad`: The number of radii or “spokes”, emanating from each data point, along which dummy points are to be added. `nper`: The number of dummy points per spoke. `fctr`: A factor determining the length of each spoke; each spoke is of length equal to fctr times the mean nearest neighbour distance of the data. (This distance is calculated by the auxiliary function mnnd().) `x`: A vector of x-coordinates of “ad hoc” dummy points `y`: A vector of the corresponding y-coordinates of “ad hoc” dummy points
`rw`	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 given by the range of the data, plus and minus 10 percent.
`eps`	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.
`sort`	Logical argument; if `TRUE` (the default) the data (including dummy points) are sorted into a sequence of “bins” prior to triangulation; this makes the algorithm slightly more efficient. Normally one would set sort equal to `FALSE` only if one wished to observe some of the fine detail of the way in which adding a point to a data set affected the triangulation, and therefore wished to make sure that the point in question was added last. Essentially this argument would get used only in a de-bugging process.
`plotit`	Logical argument; if `TRUE` a plot is produced. The nature of the plot may be controlled by using the `...` argument to pass appropriate arguments to `plot.deldir()`. Without “further instruction” a plot of the points being triangulated and of both the triangulation and the tessellation is produced;
`digits`	The number of decimal places to which all numeric values in the returned list should be rounded. Defaults to 6.
`z`	An optional vector of “auxiliary” values or “weights” associated with the respective points. (NOTE: These “weights” are values associated with the points and hence with the tiles of the tessellation produced. They DO NOT affect the tessellation, i.e. the tessellation produced is the same as is it would be if there were no weights. The `deldir` package DOES NOT do weighted tessellation.) If this argument is left `NULL` then it is taken to be the third column of `x`, if `x` is a data frame or to be the `z` component of `x` if `x` is a generic list.
`zdum`	Values of `z` to be associated with any dummy points that are created. See Warnings.
`suppressMsge`	Logical scalar indicating whether a message (alerting the user to changes from previous versions of `deldir`) should be suppressed.
`...`	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 `...` if plotit=`TRUE`.

Details

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.

See the ChangeLog for information about further revisions and bug-fixes.

Value

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.dum`rows (see below). The rows correspond to the points in the set being triangulated. The column names are `x` (the x-coordinate of the point), `y` (the y-coordinate), `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.area` column 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.area` column 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 `summary` correspond 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 `summary` correspond to dummy points.
`del.area`	the area of the convex hull of the set of points being triangulated, as formed by summing the `del.area` column of `summary`.
`dir.area`	the area of the rectangular window enclosing the points being triangulated, as formed by summing the `dir.area` column of `summary`.
`rw`	the specification of the corners of the rectangular window enclosing the data, in the order (xmin, xmax, ymin, ymax).
`ind.orig`	A vector of the indices of the points (x,y) in the set of coordinates initially supplied to `deldir()` before duplicate points (if any) were removed. These coordinates are used by `triang.list()`.

Remark:

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 not NULL) are identical.

Side Effects

If plotit==TRUE a plot of the triangulation and/or tessellation is produced or added to an existing plot.

Warnings

The process for determining if points are duplicated changed between versions 0.1-9 and 0.1-10. Previously there was an argument frac for this function, which defaulted to 0.0001. Points were deemed to be duplicates if the difference in x-coordinates was less than frac times the width of rw and y-coordinates was less than frac times the height of rw. This process has been changed to one which uses duplicated() on the data frame whose columns are x and y.

As a result it may happen that points which were previously eliminated as duplicates will no longer be eliminated. (And possibly vice-versa.)
The components delsgs and summary of the value returned by deldir() are now data frames rather than matrices. The component summary was changed to allow the “auxiliary” values z to be of arbitrary mode (i.e. not necessarily numeric). The component delsgs was then changed for consistency. Note that the other “matrix-like” component dirsgs has been a data frame since time immemorial.

A message alerting the user to the foregoing two items is printed out the first time that deldir() is called with suppressMsge=FALSE in a given session. In succeeding calls to deldir() in the same session, no message is printed. (I.e. the “alerting” message is printed at most once in any given session.)

The “alerting” message is not produced via the warning() function, so suppressWarnings() will not suppress its appearance. To effect such suppression (necessary only on the first call to deldir() in a given session) one must set the suppressMsge argument of deldir equal to TRUE.
If any dummy points are created, and if a vector z, of “auxiliary” values or “weights” associated 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 zdum. If zdum is not supplied then the auxiliary values or weights associated with the dummy points are all taken to be missing values (i.e. NA).

Author(s)

Rolf Turner r.turner@auckland.ac.nz https://www.stat.auckland.ac.nz/~rolf

References

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.

Examples

# Puts dummy points at the corners of the rectangular
# window, i.e. at (0,0), (10,0), (10,10), and (0,10)
# An "alerting note" is printed.
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).
## Not run: 
tv   <- deldir(x,y,list(ndx=2,ndy=2),c(0,10,0,10),plot=TRUE,wl='tr')

## End(Not run)

# 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)

[Package deldir version 0.1-9 Index]