Package 'geometry'

Title: Mesh Generation and Surface Tessellation
Description: Makes the 'Qhull' library <http://www.qhull.org> available in R, in a similar manner as in Octave and MATLAB. Qhull computes convex hulls, Delaunay triangulations, halfspace intersections about a point, Voronoi diagrams, furthest-site Delaunay triangulations, and furthest-site Voronoi diagrams. It runs in 2D, 3D, 4D, and higher dimensions. It implements the Quickhull algorithm for computing the convex hull. Qhull does not support constrained Delaunay triangulations, or mesh generation of non-convex objects, but the package does include some R functions that allow for this.
Authors: Jean-Romain Roussel [cph, ctb] (wrote tsearch function with QuadTrees), C. B. Barber [cph], Kai Habel [cph, aut], Raoul Grasman [cph, aut], Robert B. Gramacy [cph, aut], Pavlo Mozharovskyi [cph, aut], David C. Sterratt [cph, aut, cre]
Maintainer: David C. Sterratt <david.c.sterratt@ed.ac.uk>
License: GPL (>= 3)
Version: 0.4.7
Built: 2024-02-10 09:02:10 UTC
Source: CRAN

Help Index


Conversion of Barycentric to Cartesian coordinates

Description

Given the barycentric coordinates of one or more points with respect to a simplex, compute the Cartesian coordinates of these points.

Usage

bary2cart(X, Beta)

Arguments

X

Reference simplex in NN dimensions represented by a N+1N+1-by-NN matrix

Beta

MM points in barycentric coordinates with respect to the simplex X represented by a MM-by-N+1N+1 matrix

Value

MM-by-NN matrix in which each row is the Cartesian coordinates of corresponding row of Beta

Author(s)

David Sterratt

See Also

cart2bary

Examples

## Define simplex in 2D (i.e. a triangle)
X <- rbind(c(0, 0),
           c(0, 1),
           c(1, 0))
## Cartesian cooridinates of points
beta <- rbind(c(0, 0.5, 0.5),
              c(0.1, 0.8, 0.1))
## Plot triangle and points
trimesh(rbind(1:3), X)
text(X[,1], X[,2], 1:3) # Label vertices
P <- bary2cart(X, beta)
points(P)

Conversion of Cartesian to Barycentric coordinates.

Description

Given the Cartesian coordinates of one or more points, compute the barycentric coordinates of these points with respect to a simplex.

Usage

cart2bary(X, P)

Arguments

X

Reference simplex in NN dimensions represented by a N+1N+1-by-NN matrix

P

MM-by-NN matrix in which each row is the Cartesian coordinates of a point.

Details

Given a reference simplex in NN dimensions represented by a N+1N+1-by-NN matrix an arbitrary point PP in Cartesian coordinates, represented by a 1-by-NN row vector, can be written as

P=βXP = \beta X

where β\beta is an N+1N+1 vector of the barycentric coordinates. A criterion on β\beta is that

iβi=1\sum_i\beta_i = 1

Now partition the simplex into its first NN rows XNX_N and its N+1N+1th row XN+1X_{N+1}. Partition the barycentric coordinates into the first NN columns βN\beta_N and the N+1N+1th column βN+1\beta_{N+1}. This allows us to write

PN+1XN+1=βNXN+βN+1XN+1XN+1P_{N+1} - X_{N+1} = \beta_N X_N + \beta_{N+1} X_{N+1} - X_{N+1}

which can be written

PN+1XN+1=βN(XN1NXN+1)P_{N+1} - X_{N+1} = \beta_N(X_N - 1_N X_{N+1})

where 1N1_N is an NN-by-1 matrix of ones. We can then solve for βN\beta_N:

βN=(PN+1XN+1)(XN1NXN+1)1\beta_N = (P_{N+1} - X_{N+1})(X_N - 1_N X_{N+1})^{-1}

and compute

βN+1=1i=1Nβi\beta_{N+1} = 1 - \sum_{i=1}^N\beta_i

This can be generalised for multiple values of PP, one per row.

Value

MM-by-N+1N+1 matrix in which each row is the barycentric coordinates of corresponding row of P. If the simplex is degenerate a warning is issued and the function returns NULL.

Note

Based on the Octave function by David Bateman.

Author(s)

David Sterratt

See Also

bary2cart

Examples

## Define simplex in 2D (i.e. a triangle)
X <- rbind(c(0, 0),
           c(0, 1),
           c(1, 0))
## Cartesian coordinates of points
P <- rbind(c(0.5, 0.5),
           c(0.1, 0.8))
## Plot triangle and points
trimesh(rbind(1:3), X)
text(X[,1], X[,2], 1:3) # Label vertices
points(P)
cart2bary(X, P)

Transform Cartesian coordinates to polar or cylindrical coordinates.

Description

The inputs x, y (, and z) must be the same shape, or scalar. If called with a single matrix argument then each row of C represents the Cartesian coordinate (x, y (, z)).

Usage

cart2pol(x, y = NULL, z = NULL)

Arguments

x

x-coordinates or matrix with three columns

y

y-coordinates (optional, if x) is a matrix

z

z-coordinates (optional, if x) is a matrix

Value

A matrix P where each row represents one polar/(cylindrical) coordinate (theta, r, (, z)).

Author(s)

Kai Habel

David Sterratt

See Also

pol2cart, cart2sph, sph2cart


Transform Cartesian to spherical coordinates

Description

If called with a single matrix argument then each row of c represents the Cartesian coordinate (x, y, z).

Usage

cart2sph(x, y = NULL, z = NULL)

Arguments

x

x-coordinates or matrix with three columns

y

y-coordinates (optional, if x) is a matrix

z

z-coordinates (optional, if x) is a matrix

Value

Matrix with columns:

theta

the angle relative to the positive x-axis

phi

the angle relative to the xy-plane

r

the distance to the origin (0, 0, 0)

Author(s)

Kai Habel

David Sterratt

See Also

sph2cart, cart2pol, pol2cart


Compute smallest convex hull that encloses a set of points

Description

Returns information about the smallest convex complex of a set of input points in NN-dimensional space (the convex hull of the points). By default, indices to points forming the facets of the hull are returned; optionally normals to the facets and the generalised surface area and volume can be returned. This function interfaces the Qhull library.

Usage

convhulln(
  p,
  options = "Tv",
  output.options = NULL,
  return.non.triangulated.facets = FALSE
)

Arguments

p

An MM-by-NN matrix. The rows of p represent MM points in NN-dimensional space.

options

String containing extra options for the underlying Qhull command; see details below and Qhull documentation at ../doc/qhull/html/qconvex.html#synopsis.

output.options

String containing Qhull options to generate extra output. Currently n (normals) and FA (generalised areas and volumes) are supported; see ‘Value’ for details. If output.options is TRUE, select all supported options.

return.non.triangulated.facets

logical defining whether the output facets should be triangulated; FALSE by default.

Value

By default (return.non.triangulated.facets is FALSE), return an MM-by-NN matrix in which each row contains the indices of the points in p forming an N1N-1-dimensional facet. e.g In 3 dimensions, there are 3 indices in each row describing the vertices of 2-dimensional triangles.

If return.non.triangulated.facets is TRUE then the number of columns equals the maximum number of vertices in a facet, and each row defines a polygon corresponding to a facet of the convex hull with its vertices followed by NAs until the end of the row.

If the output.options or options argument contains FA or n, return a list with class convhulln comprising the named elements:

p

The points passed to convnhulln

hull

The convex hull, represented as a matrix indexing p, as described above

area

If FA is specified, the generalised area of the hull. This is the surface area of a 3D hull or the length of the perimeter of a 2D hull. See ../doc/qhull/html/qh-optf.html#FA.

vol

If FA is specified, the generalised volume of the hull. This is volume of a 3D hull or the area of a 2D hull. See ../doc/qhull/html/qh-optf.html#FA.

normals

If n is specified, this is a matrix hyperplane normals with offsets. See ../doc/qhull/html/qh-opto.html#n.

Note

This function was originally a port of the Octave convhulln function written by Kai Habel.

See further notes in delaunayn.

Author(s)

Raoul Grasman, Robert B. Gramacy, Pavlo Mozharovskyi and David Sterratt david.c.sterratt@ed.ac.uk

References

Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., “The Quickhull algorithm for convex hulls,” ACM Trans. on Mathematical Software, Dec 1996.

http://www.qhull.org

See Also

intersectn, delaunayn, surf.tri, convex.hull

Examples

## Points in a sphere
ps <- matrix(rnorm(3000), ncol=3)
ps <- sqrt(3)*ps/drop(sqrt((ps^2) %*% rep(1, 3)))
ts.surf <- t(convhulln(ps))  # see the qhull documentations for the options
## Not run: 
rgl::triangles3d(ps[ts.surf,1],ps[ts.surf,2],ps[ts.surf,3],col="blue",alpha=.2)
for(i in 1:(8*360)) rgl::view3d(i/8)

## End(Not run)

## Square
pq <- rbox(0, C=0.5, D=2)
# Return indices only
convhulln(pq)
# Return convhulln object with normals, generalised area and volume
ch <- convhulln(pq, output.options=TRUE)
plot(ch)

## Cube
pc <- rbox(0, C=0.5, D=3)
# Return indices of triangles on surface
convhulln(pc)
# Return indices of squares on surface
convhulln(pc, return.non.triangulated.facets=TRUE)

Delaunay triangulation in N dimensions

Description

The Delaunay triangulation is a tessellation of the convex hull of the points such that no NN-sphere defined by the NN- triangles contains any other points from the set.

Usage

delaunayn(p, options = NULL, output.options = NULL, full = FALSE)

Arguments

p

An MM-by-NN matrix whose rows represent MM points in NN-dimensional space.

options

String containing extra control options for the underlying Qhull command; see the Qhull documentation (../doc/qhull/html/qdelaun.html) for the available options.

The Qbb option is always passed to Qhull. The remaining default options are Qcc Qc Qt Qz for N<4N<4 and Qcc Qc Qt Qx for N>=4N>=4. If neither of the QJ or Qt options are supplied, the Qt option is passed to Qhull. The Qt option ensures all Delaunay regions are simplical (e.g., triangles in 2D). See ../doc/qhull/html/qdelaun.html for more details. Contrary to the Qhull documentation, no degenerate (zero area) regions are returned with the Qt option since the R function removes them from the triangulation.

If options is specified, the default options are overridden. It is recommended to use output.options for options controlling the outputs.

output.options

String containing Qhull options to control output. Currently Fn (neighbours) and Fa (areas) are supported. Causes an object of return value for details. If output.options is TRUE, select all supported options.

full

Deprecated and will be removed in a future release. Adds options Fa and Fn.

Value

If output.options is NULL (the default), return the Delaunay triangulation as a matrix with MM rows and N+1N+1 columns in which each row contains a set of indices to the input points p. Thus each row describes a simplex of dimension NN, e.g. a triangle in 2D or a tetrahedron in 3D.

If the output.options argument is TRUE or is a string containing Fn or Fa, return a list with class delaunayn comprising the named elements:

tri

The Delaunay triangulation described above

areas

If TRUE or if Fa is specified, an MM-dimensional vector containing the generalised area of each simplex (e.g. in 2D the areas of triangles; in 3D the volumes of tetrahedra). See ../doc/qhull/html/qh-optf.html#Fa.

neighbours

If TRUE or if Fn is specified, a list of neighbours of each simplex. Note that a negative number corresponds to "facet" (="edge" in 2D or "face" in 3D) that has no neighbour, as will be the case for some simplices on the boundary of the triangulation. See ../doc/qhull/html/qh-optf.html#Fn

Note

This function interfaces the Qhull library and is a port from Octave (https://octave.org/) to R. Qhull computes convex hulls, Delaunay triangulations, halfspace intersections about a point, Voronoi diagrams, furthest-site Delaunay triangulations, and furthest-site Voronoi diagrams. It runs in 2D, 3D, 4D, and higher dimensions. It implements the Quickhull algorithm for computing the convex hull. Qhull handles round-off errors from floating point arithmetic. It computes volumes, surface areas, and approximations to the convex hull. See the Qhull documentation included in this distribution (the doc directory ../doc/qhull/index.html).

Qhull does not support constrained Delaunay triangulations, triangulation of non-convex surfaces, mesh generation of non-convex objects, or medium-sized inputs in 9D and higher. A rudimentary algorithm for mesh generation in non-convex regions using Delaunay triangulation is implemented in distmesh2d (currently only 2D).

Author(s)

Raoul Grasman and Robert B. Gramacy; based on the corresponding Octave sources of Kai Habel.

References

Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., “The Quickhull algorithm for convex hulls,” ACM Trans. on Mathematical Software, Dec 1996.

http://www.qhull.org

See Also

tri.mesh, convhulln, surf.tri, distmesh2d

Examples

# example delaunayn
d <- c(-1,1)
pc <- as.matrix(rbind(expand.grid(d,d,d),0))
tc <- delaunayn(pc)

# example tetramesh
## Not run: 
rgl::view3d(60)
rgl::light3d(120,60)
tetramesh(tc,pc, alpha=0.9)

## End(Not run)

tc1 <- delaunayn(pc, output.options="Fa")
## sum of generalised areas is total volume of cube
sum(tc1$areas)

A simple mesh generator for non-convex regions

Description

An unstructured simplex requires a choice of mesh points (vertex nodes) and a triangulation. This is a simple and short algorithm that improves the quality of a mesh by relocating the mesh points according to a relaxation scheme of forces in a truss structure. The topology of the truss is reset using Delaunay triangulation. A (sufficiently smooth) user supplied signed distance function (fd) indicates if a given node is inside or outside the region. Points outside the region are projected back to the boundary.

Usage

distmesh2d(
  fd,
  fh,
  h0,
  bbox,
  p = NULL,
  pfix = array(0, dim = c(0, 2)),
  ...,
  dptol = 0.001,
  ttol = 0.1,
  Fscale = 1.2,
  deltat = 0.2,
  geps = 0.001 * h0,
  deps = sqrt(.Machine$double.eps) * h0,
  maxiter = 1000,
  plot = TRUE
)

Arguments

fd

Vectorized signed distance function, for example mesh.dcircle or mesh.diff, accepting an n-by-2 matrix, where n is arbitrary, as the first argument.

fh

Vectorized function, for example mesh.hunif, that returns desired edge length as a function of position. Accepts an n-by-2 matrix, where n is arbitrary, as its first argument.

h0

Initial distance between mesh nodes. (Ignored of p is supplied)

bbox

Bounding box cbind(c(xmin,xmax), c(ymin,ymax))

p

An n-by-2 matrix. The rows of p represent locations of starting mesh nodes.

pfix

nfix-by-2 matrix with fixed node positions.

...

parameters to be passed to fd and/or fh

dptol

Algorithm stops when all node movements are smaller than dptol

ttol

Controls how far the points can move (relatively) before a retriangulation with delaunayn.

Fscale

“Internal pressure” in the edges.

deltat

Size of the time step in Euler's method.

geps

Tolerance in the geometry evaluations.

deps

Stepsize Δx\Delta x in numerical derivative computation for distance function.

maxiter

Maximum iterations.

plot

logical. If TRUE (default), the mesh is plotted as it is generated.

Details

This is an implementation of original Matlab software of Per-Olof Persson.

Excerpt (modified) from the reference below:

‘The algorithm is based on a mechanical analogy between a triangular mesh and a 2D truss structure. In the physical model, the edges of the Delaunay triangles of a set of points correspond to bars of a truss. Each bar has a force-displacement relationship f(,0)f(\ell, \ell_{0}) depending on its current length \ell and its unextended length 0\ell_{0}.’

‘External forces on the structure come at the boundaries, on which external forces have normal orientations. These external forces are just large enough to prevent nodes from moving outside the boundary. The position of the nodes are the unknowns, and are found by solving for a static force equilibrium. The hope is that (when fh = function(p) return(rep(1,nrow(p)))), the lengths of all the bars at equilibrium will be nearly equal, giving a well-shaped triangular mesh.’

See the references below for all details. Also, see the comments in the source file.

Value

n-by-2 matrix with node positions.

Wishlist

Author(s)

Raoul Grasman

References

http://persson.berkeley.edu/distmesh/

P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004

See Also

tri.mesh, delaunayn, mesh.dcircle, mesh.drectangle, mesh.diff, mesh.union, mesh.intersect

Examples

# examples distmesh2d
fd <- function(p, ...) sqrt((p^2)%*%c(1,1)) - 1
     # also predefined as `mesh.dcircle'
fh <- function(p,...)  rep(1,nrow(p))
bbox <- matrix(c(-1,1,-1,1),2,2)
p <- distmesh2d(fd,fh,0.2,bbox, maxiter=100)
    # this may take a while:
    # press Esc to get result of current iteration

# example with non-convex region
fd <- function(p, ...) mesh.diff(p , mesh.drectangle, mesh.dcircle, radius=.3)
     # fd defines difference of square and circle

p <- distmesh2d(fd,fh,0.05,bbox,radius=0.3,maxiter=4)
p <- distmesh2d(fd,fh,0.05,bbox,radius=0.3, maxiter=10)
     # continue on previous mesh

A simple mesh generator for non-convex regions in n-D space

Description

An unstructured simplex requires a choice of mesh points (vertex nodes) and a triangulation. This is a simple and short algorithm that improves the quality of a mesh by relocating the mesh points according to a relaxation scheme of forces in a truss structure. The topology of the truss is reset using Delaunay triangulation. A (sufficiently smooth) user supplied signed distance function (fd) indicates if a given node is inside or outside the region. Points outside the region are projected back to the boundary.

Usage

distmeshnd(
  fdist,
  fh,
  h,
  box,
  pfix = array(dim = c(0, ncol(box))),
  ...,
  ptol = 0.001,
  ttol = 0.1,
  deltat = 0.1,
  geps = 0.1 * h,
  deps = sqrt(.Machine$double.eps) * h
)

Arguments

fdist

Vectorized signed distance function, for example mesh.dsphere, accepting an m-by-n matrix, where m is arbitrary, as the first argument.

fh

Vectorized function, for example mesh.hunif, that returns desired edge length as a function of position. Accepts an m-by-n matrix, where n is arbitrary, as its first argument.

h

Initial distance between mesh nodes.

box

2-by-n matrix that specifies the bounding box. (See distmesh2d for an example.)

pfix

nfix-by-2 matrix with fixed node positions.

...

parameters that are passed to fdist and fh

ptol

Algorithm stops when all node movements are smaller than dptol

ttol

Controls how far the points can move (relatively) before a retriangulation with delaunayn.

deltat

Size of the time step in Euler's method.

geps

Tolerance in the geometry evaluations.

deps

Stepsize Δx\Delta x in numerical derivative computation for distance function.

Details

This is an implementation of original Matlab software of Per-Olof Persson.

Excerpt (modified) from the reference below:

‘The algorithm is based on a mechanical analogy between a triangular mesh and a n-D truss structure. In the physical model, the edges of the Delaunay triangles of a set of points correspond to bars of a truss. Each bar has a force-displacement relationship f(,0)f(\ell, \ell_{0}) depending on its current length \ell and its unextended length 0\ell_{0}.’

‘External forces on the structure come at the boundaries, on which external forces have normal orientations. These external forces are just large enough to prevent nodes from moving outside the boundary. The position of the nodes are the unknowns, and are found by solving for a static force equilibrium. The hope is that (when fh = function(p) return(rep(1,nrow(p)))), the lengths of all the bars at equilibrium will be nearly equal, giving a well-shaped triangular mesh.’

See the references below for all details. Also, see the comments in the source file of distmesh2d.

Value

m-by-n matrix with node positions.

Wishlist

Author(s)

Raoul Grasman; translated from original Matlab sources of Per-Olof Persson.

References

http://persson.berkeley.edu/distmesh/

P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004

See Also

distmesh2d, tri.mesh, delaunayn, mesh.dsphere, mesh.hunif,
mesh.diff, mesh.union, mesh.intersect

Examples

## Not run: 
# examples distmeshnd
require(rgl)

fd = function(p, ...) sqrt((p^2)%*%c(1,1,1)) - 1
     # also predefined as `mesh.dsphere'
fh = function(p,...)  rep(1,nrow(p))
     # also predefined as `mesh.hunif'
bbox = matrix(c(-1,1),2,3)
p = distmeshnd(fd,fh,0.2,bbox, maxiter=100)
    # this may take a while:
    # press Esc to get result of current iteration

## End(Not run)

Compute the dot product of two vectors

Description

If x and y are matrices, calculate the dot-product along the first non-singleton dimension. If the optional argument d is given, calculate the dot-product along this dimension.

Usage

dot(x, y, d = NULL)

Arguments

x

Matrix of vectors

y

Matrix of vectors

d

Dimension along which to calculate the dot product

Value

Vector with length of dth dimension

Author(s)

David Sterratt


Retrieve or set a list of array element values

Description

entry.value retrieves or sets the values in an array a at the positions indicated by the rows of a matrix idx.

Usage

entry.value(a, idx)

Arguments

a

An array.

idx

Numerical matrix with the same number of columns as the number of dimensions of a. Each row indices a cell in a of which the value is to be retrieved or set.

value

An array of length nrow(idx).

Value

entry.value(a,idx) returns a vector of values at the indicated cells. entry.value(a,idx) <- val changes the indicated cells of a to val.

Author(s)

Raoul Grasman

Examples

a = array(1:(4^4),c(4,4,4,4))
entry.value(a,cbind(1:4,1:4,1:4,1:4))
entry.value(a,cbind(1:4,1:4,1:4,1:4)) <- 0

entry.value(a, as.matrix(expand.grid(1:4,1:4,1:4,1:4)))
     # same as `c(a[1:4,1:4,1:4,1:4])' which is same as `c(a)'

Compute external- or ‘cross’- product of 3D vectors.

Description

Computes the external product

(x2y3x3y2,  x3y1x1y3,  x1y2x2y1)\left(x_2 y_3 - x_3 y_2,\; x_3 y_1 - x_1 y_3,\; x_1 y_2 - x_2 y_1 \right)

of the 3D vectors in x and y.

Usage

extprod3d(x, y, drop = TRUE)

Arguments

x

n-by-3 matrix. Each row is one x-vector

y

n-by-3 matrix. Each row is one y-vector

drop

logical. If TRUE and if the inputs are one row matrices or vectors, then delete the dimensions of the array returned.

Value

If n is greater than 1 or drop is FALSE, n-by-3 matrix; if n is 1 and drop is TRUE, a vector of length 3.

Author(s)

Raoul Grasman

See Also

drop


Find point in intersection of convex hulls

Description

Find point that lies somewhere in interesction of two convex hulls. If such a point does not exist, return NA. The feasible point is found using a linear program similar to the one suggested at ../doc/qhull/html/qhalf.html#notes

Usage

feasible.point(ch1, ch2, tol = 0)

Arguments

ch1

First convex hull with normals

ch2

Second convex hull with normals

tol

The point must be at least this far within the facets of both convex hulls


Compute halfspace intersection about a point

Description

Compute halfspace intersection about a point

Usage

halfspacen(p, fp, options = "Tv")

Arguments

p

An MM-by-N+1N+1 matrix. Each row of p represents a halfspace by a NN-dimensional normal to a hyperplane and the offset of the hyperplane.

fp

A “feasible” point that is within the space contained within all the halfspaces.

options

String containing extra options, separated by spaces, for the underlying Qhull command; see Qhull documentation at ../doc/qhull/html/qhalf.html.

Value

A NN-column matrix containing the intersection points of the hyperplanes ../doc/qhull/html/qhalf.html.

Note

halfspacen was introduced in geometry 0.4.0, and is still under development. It is worth checking results for unexpected behaviour.

Author(s)

David Sterratt

References

Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., “The Quickhull algorithm for convex hulls,” ACM Trans. on Mathematical Software, Dec 1996.

http://www.qhull.org

See Also

convhulln

Examples

p <- rbox(0, C=0.5)  # Generate points on a unit cube centered around the origin
ch <- convhulln(p, "n") # Generate convex hull, including normals to facets, with "n" option
# Intersections of half planes
# These points should be the same as the orginal points
pn <- halfspacen(ch$normals, c(0, 0, 0))

Test if points lie in convex hull

Description

Tests if a set of points lies within a convex hull, returning a boolean vector in which each element is TRUE if the corresponding point lies within the hull and FALSE if it lies outwith the hull or on one of its facets.

Usage

inhulln(ch, p)

Arguments

ch

Convex hull produced using convhulln

p

An MM-by-NN matrix of points to test. The rows of p represent MM points in NN-dimensional space.

Value

A boolean vector with MM elements

Note

inhulln was introduced in geometry 0.4.0, and is still under development. It is worth checking results for unexpected behaviour.

Author(s)

David Sterratt

See Also

convhulln, point.in.polygon in sp

Examples

p <- cbind(c(-1, -1, 1), c(-1, 1, -1))
ch <- convhulln(p)
## First point should be in the hull; last two outside
inhulln(ch, rbind(c(-0.5, -0.5),
                  c( 1  ,  1),
                  c(10  ,  0)))

## Test hypercube
p <- rbox(D=4, B=1)
ch <- convhulln(p)
tp <-  cbind(seq(-1.9, 1.9, by=0.2), 0, 0, 0)
pin <- inhulln(ch, tp)
## Points on x-axis should be in box only betw,een -1 and 1
pin == (tp[,1] < 1 & tp[,1] > -1)

Compute convex hull of intersection of two sets of points

Description

Compute convex hull of intersection of two sets of points

Usage

intersectn(
  ps1,
  ps2,
  tol = 0,
  return.chs = TRUE,
  options = "Tv",
  fp = NULL,
  autoscale = FALSE
)

Arguments

ps1

First set of points

ps2

Second set of points

tol

Tolerance used to determine if a feasible point lies within the convex hulls of both points and to round off the points generated by the halfspace intersection, which sometimes produces points very close together.

return.chs

If TRUE (default) return the convex hulls of the first and second sets of points, as well as the convex hull of the intersection.

options

Options passed to halfspacen. By default this is Tv.

fp

Coordinates of feasible point, i.e. a point known to lie in the hulls of ps1 and ps2. The feasible point is required for halfspacen to find the intersection. intersectn tries to find the feasible point automatically using the linear program in feasible.point, but currently the linear program fails on some examples where there is an obvious solution. This option overrides the automatic search for a feasible point

autoscale

Experimental in v0.4.2 Automatically scale the points to lie in a sensible numeric range. May help to correct some numerical issues.

Value

List containing named elements: ch1, the convex hull of the first set of points, with volumes, areas and normals (see convhulln; ch2, the convex hull of the first set of points, with volumes, areas and normals; ps, the intersection points of convex hulls ch1 and ch2; and ch, the convex hull of the intersection points, with volumes, areas and normals.

Note

intersectn was introduced in geometry 0.4.0, and is still under development. It is worth checking results for unexpected behaviour.

Author(s)

David Sterratt

See Also

convhulln, halfspacen, inhulln, feasible.point

Examples

# Two overlapping boxes
ps1 <- rbox(0, C=0.5)
ps2 <- rbox(0, C=0.5) + 0.5
out <- intersectn(ps1, ps2)
message("Volume of 1st convex hull: ", out$ch1$vol)
message("Volume of 2nd convex hull: ", out$ch2$vol)
message("Volume of intersection convex hull: ", out$ch$vol)

Row-wise matrix functions

Description

Compute maximum or minimum of each row, or sort each row of a matrix, or a set of (equal length) vectors.

Usage

matmax(...)

Arguments

...

A numeric matrix or a set of numeric vectors (that are column-wise bind together into a matrix with cbind).

Value

matmin and matmax return a vector of length nrow(cbind(...)). matsort returns a matrix of dimension dim(cbind(...)) with in each row of cbind(...) sorted. matsort(x) is a lot faster than, e.g., t(apply(x,1,sort)), if x is tall (i.e., nrow(x)>>ncol(x) and ncol(x)<30. If ncol(x)>30 then matsort simply calls 't(apply(x,1,sort))'. matorder returns a permutation which rearranges its first argument into ascending order, breaking ties by further arguments.

Author(s)

Raoul Grasman

Examples

example(Unique)

Circle distance function

Description

Signed distance from points p to boundary of circle to allow easy definition of regions in distmesh2d.

Usage

mesh.dcircle(p, radius = 1, ...)

Arguments

p

A matrix with 2 columns (3 in mesh.dsphere), each row representing a point in the plane.

radius

radius of circle

...

additional arguments (not used)

Value

A vector of length nrow(p) containing the signed distances to the circle

Author(s)

Raoul Grasman; translated from original Matlab sources of Per-Olof Persson.

References

http://persson.berkeley.edu/distmesh/

P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004

See Also

distmesh2d, mesh.drectangle, mesh.diff, mesh.intersect, mesh.union

Examples

example(distmesh2d)

Difference, union and intersection operation on two regions

Description

Compute the signed distances from points p to a region defined by the difference, union or intersection of regions specified by the functions regionA and regionB. regionA and regionB must accept a matrix p with 2 columns as their first argument, and must return a vector of length nrow(p) containing the signed distances of the supplied points in p to their respective regions.

Usage

mesh.diff(p, regionA, regionB, ...)

Arguments

p

A matrix with 2 columns (3 in mesh.dsphere), each row representing a point in the plane.

regionA

vectorized function describing region A in the union / intersection / difference

regionB

vectorized function describing region B in the union / intersection / difference

...

additional arguments passed to regionA and regionB

Value

A vector of length nrow(p) containing the signed distances to the boundary of the region.

Author(s)

Raoul Grasman; translated from original Matlab sources of Per-Olof Persson.

See Also

distmesh2d, mesh.dcircle, mesh.drectangle mesh.dsphere


Rectangle distance function

Description

Signed distance from points p to boundary of rectangle to allow easy definition of regions in distmesh2d.

Usage

mesh.drectangle(p, x1 = -1/2, y1 = -1/2, x2 = 1/2, y2 = 1/2, ...)

Arguments

p

A matrix with 2 columns, each row representing a point in the plane.

x1

lower left corner of rectangle

y1

lower left corner of rectangle

x2

upper right corner of rectangle

y2

upper right corner of rectangle

...

additional arguments (not used)

Value

a vector of length nrow(p) containing the signed distances

Author(s)

Raoul Grasman; translated from original Matlab sources of Per-Olof Persson.

References

http://persson.berkeley.edu/distmesh/

P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004

See Also

distmesh2d, mesh.drectangle, mesh.diff, mesh.intersect, mesh.union

Examples

example(distmesh2d)

Sphere distance function

Description

Signed distance from points p to boundary of sphere to allow easy definition of regions in distmeshnd.

Usage

mesh.dsphere(p, radius = 1, ...)

Arguments

p

A matrix with 2 columns (3 in mesh.dsphere), each row representing a point in the plane.

radius

radius of sphere

...

additional arguments (not used)

Value

A vector of length nrow(p) containing the signed distances to the sphere

Author(s)

Raoul Grasman; translated from original Matlab sources of Per-Olof Persson.

References

http://persson.berkeley.edu/distmesh/

P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004

See Also

distmeshnd

Examples

example(distmeshnd)

Uniform desired edge length

Description

Uniform desired edge length function of position to allow easy definition of regions when passed as the fh argument of distmesh2d or distmeshnd.

Usage

mesh.hunif(p, ...)

Arguments

p

A n-by-m matrix, each row representing a point in an m-dimensional space.

...

additional arguments (not used)

Value

Vector of ones of length n.

Author(s)

Raoul Grasman; translated from original Matlab sources of Per-Olof Persson.

See Also

distmesh2d and distmeshnd.


Transform polar or cylindrical coordinates to Cartesian coordinates.

Description

The inputs theta, r, (and z) must be the same shape, or scalar. If called with a single matrix argument then each row of P represents the polar/(cylindrical) coordinate (theta, r (, z)).

Usage

pol2cart(theta, r = NULL, z = NULL)

Arguments

theta

describes the angle relative to the positive x-axis.

r

is the distance to the z-axis (0, 0, z).

z

(optional) is the z-coordinate

Value

a matrix C where each row represents one Cartesian coordinate (x, y (, z)).

Author(s)

Kai Habel

David Sterratt

See Also

cart2pol, sph2cart, cart2sph


Determines area of a polygon by triangle method.

Description

Determines area of a polygon by triangle method. The variables x and y define the vertex pairs, and must therefore have the same shape. They can be either vectors or arrays. If they are arrays then the columns of x and y are treated separately and an area returned for each.

Usage

polyarea(x, y, d = 1)

Arguments

x

X coordinates of vertices.

y

Y coordinates of vertices.

d

Dimension of array to work along.

Details

If the optional dim argument is given, then polyarea works along this dimension of the arrays x and y.

Value

Area(s) of polygon(s).

Author(s)

David Sterratt based on the octave sources by David M. Doolin

Examples

x <- c(1, 1, 3, 3, 1)
y <- c(1, 3, 3, 1, 1)
polyarea(x, y)
polyarea(cbind(x, x), cbind(y, y)) ##  c(4, 4)
polyarea(cbind(x, x), cbind(y, y), 1) ##  c(4, 4)
polyarea(rbind(x, x), rbind(y, y), 2) ##  c(4, 4)

Generate various point distributions

Description

Default is corners of a hypercube.

Usage

rbox(n = 3000, D = 3, B = 0.5, C = NA)

Arguments

n

number of random points in hypercube

D

number of dimensions of hypercube

B

bounding box coordinate - faces will be -B and B from origin

C

add a unit hypercube to the output - faces will be -C and C from origin

Value

Matrix of points

Author(s)

David Sterratt


Transform spherical coordinates to Cartesian coordinates

Description

The inputs theta, phi, and r must be the same shape, or scalar. If called with a single matrix argument then each row of S represents the spherical coordinate (theta, phi, r).

Usage

sph2cart(theta, phi = NULL, r = NULL)

Arguments

theta

describes the angle relative to the positive x-axis.

phi

is the angle relative to the xy-plane.

r

is the distance to the origin (0, 0, 0).

If only a single return argument is requested then return a matrix C where each row represents one Cartesian coordinate (x, y, z).

Author(s)

Kai Habel

David Sterratt

See Also

cart2sph, pol2cart, cart2pol


Find surface triangles from tetrahedral mesh

Description

Find surface triangles from tetrahedral mesh typically obtained with delaunayn.

Usage

surf.tri(p, t)

Arguments

p

An n-by-3 matrix. The rows of p represent n points in dim-dimensional space.

t

Matrix with 4 columns, interpreted as output of delaunayn.

Details

surf.tri and convhulln serve a similar purpose in 3D, but surf.tri also works for non-convex meshes obtained e.g. with distmeshnd. It also does not produce currently unavoidable diagnostic output on the console as convhulln does at the Rterm console–i.e., surf.tri is silent.

Value

An m-by-3 index matrix of which each row defines a triangle. The indices refer to the rows in p.

Note

surf.tri was based on Matlab code for mesh of Per-Olof Persson (http://persson.berkeley.edu/distmesh/).

Author(s)

Raoul Grasman

See Also

tri.mesh, convhulln, surf.tri, distmesh2d

Examples

## Not run: 
# more extensive example of surf.tri

# url's of publically available data:
data1.url = "http://neuroimage.usc.edu/USCPhantom/mesh_data.bin"
data2.url = "http://neuroimage.usc.edu/USCPhantom/CT_PCS_trans.bin"

meshdata = R.matlab::readMat(url(data1.url))
elec = R.matlab::readMat(url(data2.url))$eeg.ct2pcs/1000
brain = meshdata$mesh.brain[,c(1,3,2)]
scalp = meshdata$mesh.scalp[,c(1,3,2)]
skull = meshdata$mesh.skull[,c(1,3,2)]
tbr = t(surf.tri(brain, delaunayn(brain)))
tsk = t(surf.tri(skull, delaunayn(skull)))
tsc = t(surf.tri(scalp, delaunayn(scalp)))
rgl::triangles3d(brain[tbr,1], brain[tbr,2], brain[tbr,3],col="gray")
rgl::triangles3d(skull[tsk,1], skull[tsk,2], skull[tsk,3],col="white", alpha=0.3)
rgl::triangles3d(scalp[tsc,1], scalp[tsc,2], scalp[tsc,3],col="#a53900", alpha=0.6)
rgl::view3d(-40,30,.4,zoom=.03)
lx = c(-.025,.025); ly = -c(.02,.02);
rgl::spheres3d(elec[,1],elec[,3],elec[,2],radius=.0025,col='gray')
rgl::spheres3d( lx, ly,.11,radius=.015,col="white")
rgl::spheres3d( lx, ly,.116,radius=.015*.7,col="brown")
rgl::spheres3d( lx, ly,.124,radius=.015*.25,col="black")

## End(Not run)

Render tetrahedron mesh (3D)

Description

tetramesh(T, X, col) uses the rgl package to display the tetrahedrons defined in the m-by-4 matrix T as mesh. Each row of T specifies a tetrahedron by giving the 4 indices of its points in X.

Usage

tetramesh(T, X, col = grDevices::heat.colors(nrow(T)), clear = TRUE, ...)

Arguments

T

T is a m-by-3 matrix in trimesh and m-by-4 in tetramesh. A row of T contains indices into X of the vertices of a triangle/tetrahedron. T is usually the output of delaunayn.

X

X is an n-by-2/n-by-3 matrix. The rows of X represent n points in 2D/3D space.

col

The tetrahedron colour. See rgl documentation for details.

clear

Should the current rendering device be cleared?

...

Parameters to the rendering device. See the rgl package.

Author(s)

Raoul Grasman

See Also

trimesh, rgl, delaunayn, convhulln, surf.tri

Examples

## Not run: 
# example delaunayn
d = c(-1,1)
pc = as.matrix(rbind(expand.grid(d,d,d),0))
tc = delaunayn(pc)

# example tetramesh
clr = rep(1,3) %o% (1:nrow(tc)+1)
rgl::view3d(60,fov=20)
rgl::light3d(270,60)
tetramesh(tc,pc,alpha=0.7,col=clr)

## End(Not run)

Convert convhulln object to RGL mesh

Description

Convert convhulln object to RGL mesh

Usage

to.mesh3d(x, ...)

Arguments

x

convhulln object

...

Arguments to qmesh3d or tmesh3d

Value

mesh3d object, which can be displayed in RGL with dot3d, wire3d or shade3d

See Also

as.mesh3d


Display triangles mesh (2D)

Description

trimesh(T, p) displays the triangles defined in the m-by-3 matrix T and points p as a mesh. Each row of T specifies a triangle by giving the 3 indices of its points in X.

Usage

trimesh(T, p, p2, add = FALSE, axis = FALSE, boxed = FALSE, ...)

Arguments

T

T is a m-by-3 matrix. A row of T contains indices into X of the vertices of a triangle. T is usually the output of delaunayn.

p

A vector or a matrix.

p2

if p is not a matrix p and p2 are bind to a matrix with cbind.

add

Add to existing plot in current active device?

axis

Draw axes?

boxed

Plot box?

...

Parameters to the rendering device. See the rgl package.

Author(s)

Raoul Grasman

See Also

tetramesh, rgl, delaunayn, convhulln, surf.tri

Examples

#example trimesh
p = cbind(x=rnorm(30), y=rnorm(30))
tt = delaunayn(p)
trimesh(tt,p)

Search for the enclosing Delaunay convex hull

Description

For t <- delaunay(cbind(x, y)), where (x, y) is a 2D set of points, tsearch(x, y, t, xi, yi) finds the index in t containing the points (xi, yi). For points outside the convex hull the index is NA.

Usage

tsearch(x, y, t, xi, yi, bary = FALSE, method = "quadtree")

Arguments

x

X-coordinates of triangulation points

y

Y-coordinates of triangulation points

t

Triangulation, e.g. produced by t <- delaunayn(cbind(x, y))

xi

X-coordinates of points to test

yi

Y-coordinates of points to test

bary

If TRUE return barycentric coordinates as well as index of triangle.

method

One of "quadtree" or "orig". The Quadtree algorithm is much faster and new from version 0.4.0. The orig option uses the tsearch algorithm adapted from Octave code. Its use is deprecated and it may be removed from a future version of the package.

Value

If bary is FALSE, the index in t containing the points (xi, yi). For points outside the convex hull the index is NA. If bary is TRUE, a list containing:

list("idx")

the index in t containing the points (xi, yi)

list("p")

a 3-column matrix containing the barycentric coordinates with respect to the enclosing triangle of each point (xi, yi).

Note

The original Octave function is Copyright (C) 2007-2012 David Bateman

Author(s)

Jean-Romain Roussel (Quadtree algorithm), David Sterratt (Octave-based implementation)

See Also

tsearchn, delaunayn


Search for the enclosing Delaunay convex hull

Description

For t = delaunayn(x), where x is a set of points in NN dimensions, tsearchn(x, t, xi) finds the index in t containing the points xi. For points outside the convex hull, idx is NA. tsearchn also returns the barycentric coordinates p of the enclosing triangles.

Usage

tsearchn(x, t, xi, ...)

Arguments

x

An NN-column matrix, in which each row represents a point in NN-dimensional space.

t

A matrix with N+1N+1 columns. A row of t contains indices into x of the vertices of an NN-dimensional simplex. t is usually the output of delaunayn.

xi

An MM-by-NN matrix. The rows of xi represent MM points in NN-dimensional space whose positions in the mesh are being sought.

...

Additional arguments

Details

If x is NA and the t is a delaunayn object produced by delaunayn with the full option, then use the Qhull library to perform the search. Please note that this is experimental in geometry version 0.4.0 and is only partly tested for 3D hulls, and does not yet work for hulls of 4 dimensions and above.

Value

A list containing:

idx

An MM-long vector containing the indices of the row of t in which each point in xi is found.

p

An MM-by-N+1N+1 matrix containing the barycentric coordinates with respect to the enclosing simplex of each point in xi.

Note

Based on the Octave function Copyright (C) 2007-2012 David Bateman.

Author(s)

David Sterratt

See Also

tsearch, delaunayn


Extract Unique Rows

Description

‘Unique’ returns a vector, data frame or array like 'x' but with duplicate elements removed.

Usage

Unique(X, rows.are.sets = FALSE)

Arguments

X

Numerical matrix.

rows.are.sets

If ‘TRUE’, rows are treated as sets - i.e., to define uniqueness, the order of the rows does not matter.

Value

Matrix of the same number of columns as x, with the unique rows in x sorted according to the columns of x. If rows.are.sets = TRUE the rows are also sorted.

Note

Unique’ is (under circumstances) much quicker than the more generic base function ‘unique’.

Author(s)

Raoul Grasman

Examples

# `Unique' is faster than `unique'
x = matrix(sample(1:(4*8),4*8),ncol=4)
y = x[sample(1:nrow(x),3000,TRUE), ]
gc(); system.time(unique(y))
gc(); system.time(Unique(y))

#
z = Unique(y)
x[matorder(x),]
z[matorder(z),]