# deldir: Delaunay triangulation and Dirichlet tessellation In deldir: Delaunay Triangulation and Dirichlet (Voronoi) 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

 ```1 2``` ```deldir(x, y, dpl=NULL, rw=NULL, eps=1e-09, sort=TRUE, plot=FALSE, round=TRUE,digits=6, z=NULL, zdum=NULL, suppressMsge=FALSE, ...) ```

## Arguments

 `x,y` These arguments specify the coordinates of the point set being triangulated or tessellated. These can be given by two separate arguments x and y which are vectors or by a single argument x which is either a data frame or a generic list, possibly one of class `ppp`. (See package `spatstat`.) If `x` is a data frame then the `x` coordinates of the points to be triangulated or tessellated 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. If there is no column named “y” but there are columns named “x” and “z” then the `y` coordinates are taken to be the first “other” column. If there no columns named either “x” or “y”, then the `x` coordinates are taken to be the first column not named “z” and the `y` coordinates are taken to be the second column not named “z”. If there is a column named “z” and if the argument `z` (see below) 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 or tessellated. If `x` is not of class `ppp`, if it has a component `z` and if argument `z` is `NULL`, then the `z` argument is set equal to this component `z`. If `x` is of class “ppp”, if the argument `z` is `NULL`, if `x` is “marked” (see package `spatstat`) and if the marks of `x` are a vector or a factor (as opposed to a data frame) then the `z` argument is set equal to these marks. In this case `x` should not have a component `z`, and at any rate such a component would be ignored. `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 numeric “multiplicative 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. `plot` 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; `round` Logical scalar. Should the data stored in the returned value be rounded to `digits` decimal digits? This is essentially for cosmetic purposes. This argument is a “new addtion” to `deldir()`, as of version 0.1-26. Previously rounding was done willy-nilly. The change was undertaken when Kodi Arfer pointed out that the rounding might have unwanted effects upon “downstream” operations. `digits` The number of decimal places to which all numeric values in the returned list should be rounded. Defaults to 6. Ignored if `round` (see above) is set to `FALSE`. `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. The so-called weights in fact need not be numeric.) If `z` 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. If `z` is left `NULL` and if `x` is of class “ppp” and is “marked” (see package `spatstat`) and if in addition the marks are atomic (i.e. not a data frame) then `z` is taken to be the marks of `x`. `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. Currently (package version \version) no such message is produced, so this argument has no effect. `...` 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 `plot=TRUE`.

## Details

This package had its origins (way back in the mists of time!) as an Splus library section named “delaunay”. That library section in turn was a re-write of a stand-alone Fortran program written in 1987/88 while the author was with the Division of Mathematics and Statistics, CSIRO, Sydney, Australia. This program was an implementation of the second (iterative) Lee-Schacter algorithm. The stand-alone Fortran program was re-written as an Splus function (which called upon dynamically loaded Fortran code) while the author was 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 Professor of Statistics at Curtin University). 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, was deprecated in favour of the Statlib package “delaunay”. Don also collaborated in the preparation of that package. It is not clear to me whether the “delaunay” package, or indeed Statlib (or indeed Splus!) still exist.

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

## Value

A list (of class `deldir`), invisible if `plot=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 10 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, in the columns named `ind1` and `ind2`, are the indices of the two points, in the set being triangulated, which are separated by that edge. The seventh and eighth entries, in the columns named `bp1` and `bp2` are logical values. The entry in column `bp1` 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 entry in column `bp2` and the second endpoint of the edge. The nineth and tenth entries, in columns named `thirdv1` and `thirdv2` are the indices of the respective third vertices of the Delaunay triangle whose circumcentres constitute the corresponding endpoints of the edge under consideration. (The other two vertices of the triangle in question are indexed by the entries of columns `ind1` and `ind2`.) The entries of columns `thirdv1` and `thirdv2` may (also) take the values \$-1, -2, -3\$, and \$-4\$. This will be the case if the circumcentre in question lies outside of the rectangular window `rw`. In these circumstances the corresponding endpoint of the tile edge is the intersection of the line joining the two circumcentres with the boundary of `rw`, and the numeric value of the entry of column “thirdv1” (respectively “thirdv2”) indicates which side. The numbering follows the convention for numbering the sides of a plot region in `R`: 1 for the bottom side, 2 for the left hand side, 3 for the top side and 4 for the right hand side. Note that the entry in column `thirdv1` will be negative if and only if the corresponding entry in column `bp1` is `TRUE`. Similarly for columns `thirdv2` and `bp2`. `summary` a data frame with 9, 10 or 11 columns and `n.data + n.dum` rows (see below). The rows correspond to the points in the set being triangulated. Note that the row names are the indices of the points in the orginal sequence of points being triangulated/tessellated. Usually these will be the sequence 1, 2, ..., npd ("n plus dummy"). However if there were duplicated points then the row name corresponding to a point is the first of the indices of the set of duplicated points in which the given point appears. The columns are: `x` (the x-coordinate of the point) `y` (the y-coordinate of the point) `pt.type` (a character vector with entries “data” and “dummy”; present only if `n.dum > 0`) `z` (the auxiliary values or “weights”; present only 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) `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 (as data points or as dummy points) to `deldir()` before duplicate points (if any) were removed. These indices 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 `plot=TRUE` a plot of the triangulation and/or tessellation is produced or added to an existing plot.

## Notes on Memory Allocation

It is difficult-to-impossible to determine a priori how much memory needs to be allocated for storing the edges of the Delaunay triangles and Dirichlet tiles, and for storing the “adjacency list” used by the Lee-Schacter algorithm. In the code, an attempt is made to allocate sufficient storage. If, during the course of running the algorithm, the amount of storage turns out to be inadquate, the algorithm is halted, the storage is incremented, and the algorithm is restarted (with an informative message). This message may be suppressed by wrapping the call to `deldir()` in `suppressMessages()`.

## Notes on error messages

In previous versions of this package, error traps were set in the underlying Fortran code for 17 different errors. These were identified by an error number which was passed back up the call stack and finally printed out by `deldir()` which then invisibly returned a `NULL` value. A glossary of the meanings of the values of was provided in a file to be found in a file located in the `inst` directory (“folder” if you are a Windoze weenie).

This was a pretty shaganappi system. Consequently and as of version 1.2-1 conversion to “proper” error trapping was implemented. Such error trapping is effected via the `rexit()` subroutine which is now available in `R`. (See “Writing R Extensions”, section 6.2.1.)

Note that when an error is detected `deldir()` now exits with a genuine error, rather than returning `NULL`. The glossary of the meanings of “error numbers” is now irrelevant and has been removed from the `inst` directory.

An error trap that merits particular mention was introduced in version `0.1-16` of `deldir`. This error trap relates to “triangle problems”. It was drawn to my attention by Adam Dadvar (on 18 December, 2018) that in some data sets collinearity problems may cause the “triangle finding” procedure, used by the algorithm to successively add new points to a tessellation, to go into an infinite loop. A symptom of the collinearity is that the vertices of a putative triangle appear not to be in anticlockwise order irrespective of whether they are presented in the order `i, j, k` or `k, j, i`. The result of this anomaly is that the procedure keeps alternating between moving to “triangle” `i, j, k` and moving to “triangle” `k, j, i`, forever.

The error trap in question is set in `trifnd`, the triangle finding subroutine. It detects such occurrences of “clockwise in either orientation” vertices. The trap causes the `deldir()` function to throw an error rather than disappearing into a black hole.

When an error of the “triangle problems” nature occurs, a possible remedy is to increase the value of the `eps` argument of `deldir()`. (See the Examples.) There may conceiveably be other problems that lead to infinite loops and so I put in another error trap to detect whether the procedure has inspected more triangles than actually exist, and if so to throw an error.

Note that the strategy of increasing the value of `eps` is probably the appropriate response in most (if not all) of the cases where errors of this nature arise. Similarly this strategy is probably the appropriate response to the errors

• Vertices of 'triangle' are collinear and vertex 2 is not between 1 and 3. Error in circen.

• Vertices of triangle are collinear. Error in dirseg.

• Vertices of triangle are collinear. Error in dirout.

However it is impossible to be sure. The intricacy and numerical delicacy of triangulations is too great for anyone to be able to foresee all the possibilities that could arise.

If there is any doubt as the appropriateness of the “increase `eps`” strategy, the user is advised to do his or her best to explore the data set, graphically or by other means, and thereby determine what is actually going on and why problems are occurring.

## Warnings

1. 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.)

2. 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`.

3. 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

## References

Lee, D. T. and Schacter, B. J. (1980) Two algorithms for constructing a Delaunay triangulation, International Journal of Computer and Information Sciences 9 (3), pp. 219 – 242.

Ahuja, N. and Schacter, B. J. (1983). Pattern Models. New York: Wiley.

`plot.deldir()`, `tile.list()`, `triang.list()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31``` ```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) # Let deldir() choose the rectangular window. dxy1 <- deldir(x,y) # User chooses the rectangular window. dxy2 <- deldir(x,y,rw=c(0,10,0,10)) # Put dummy points at the corners of the rectangular # window, i.e. at (0,0), (10,0), (10,10), and (0,10) dxy3 <- deldir(x,y,dpl=list(ndx=2,ndy=2),rw=c(0,10,0,10)) # Plot the triangulation created (but not the tesselation). ## Not run: dxy2 <- deldir(x,y,rw=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 <- c(1.63,0.79,2.84,1.56,0.22,1.07) zdum <- rep(42,4) dxy4 <- deldir(x,y,dpl=list(ndx=2,ndy=2),rw=c(0,10,0,10),z=z,zdum=zdum) # Example of collinearity error. ## Not run: dniP <- deldir(niProperties) # Throws an error ## End(Not run) dniP <- deldir(niProperties,eps=1e-8) # No error. ```