Package {alphahull}

Generalization of the Convex Hull of a Sample of Points in the Plane

Description

Computation of the \alpha-shape and \alpha-convex hull of a given sample of points in the plane. The concepts of \alpha-shape and \alpha-convex hull generalize the definition of the convex hull of a finite set of points. The programming is based on the duality between the Voronoi diagram and Delaunay triangulation. The package also includes a function that returns the Delaunay mesh of a given sample of points and its dual Voronoi diagram in one single object.

Details

Author(s)

Beatriz Pateiro-Lopez, Alberto Rodriguez-Casal.

Maintainer: Beatriz Pateiro-Lopez <beatriz.pateiro@usc.es>

alpha-convex hull calculation

Description

This function calculates the \alpha-convex hull of a given sample of points in the plane for \alpha>0.

Usage

ahull(x, y = NULL, alpha)

Arguments

Details

An attempt is made to interpret the arguments x and y in a way suitable for computing the \alpha-convex hull. Any reasonable way of defining the coordinates is acceptable, see xy.coords.

The \alpha-convex hull is defined for any finite number of points. However, since the algorithm is based on the Delaunay triangulation, at least three non-collinear points are required.

If y is NULL and x is an object of class "delvor", then the \alpha-convex hull is computed with no need to invoke again the function delvor (it reduces the computational cost).

The complement of the \alpha-convex hull can be written as the union of O(n) open balls and halfplanes, see complement. The boundary of the \alpha-convex hull is formed by arcs of open balls of radius \alpha (besides possible isolated sample points). The arcs are determined by the intersections of some of the balls that define the complement of the \alpha-convex hull. The extremes of an arc are given by c+rA_\theta v and c+rA_{-\theta}v where c and r represent the center and radius of the arc, repectively, and A_\theta v represents the clockwise rotation of angle \theta of the unitary vector v. Joining the end points of adjacent arcs we can define polygons that help us to determine the area of the estimator , see areaahull.

Value

A list with the following components:

References

Edelsbrunner, H., Kirkpatrick, D.G. and Seidel, R. (1983). On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4), pp.551-559.

Rodriguez-Casal, R. (2007). Set estimation under convexity type assumptions. Annales de l'I.H.P.- Probabilites & Statistiques, 43, pp.763-774.

Pateiro-Lopez, B. (2008). Set estimation under convexity type restrictions. Phd. Thesis. Universidad de Santiago de Compostela. ISBN 978-84-9887-084-8.

See Also

plot.ahull.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(100), nc = 2)
# Value of alpha
alpha <- 0.2
# Alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
plot(ahull.obj)

# Uniform sample of size n=300 in the annulus B(c,0.5)\B(c,0.25), 
# with c=(0.5,0.5). 
n <- 300
theta<-runif(n,0,2*pi)
r<-sqrt(runif(n,0.25^2,0.5^2))
x<-cbind(0.5+r*cos(theta),0.5+r*sin(theta))
# Value of alpha
alpha <- 0.1
# Alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
# The arcs defining the boundary of the alpha-convex hull are ordered
plot(x)
for (i in 1:dim(ahull.obj$arcs)[1]){
arc(ahull.obj$arcs[i,1:2],ahull.obj$arcs[i,3],ahull.obj$arcs[i,4:5],
ahull.obj$arcs[i,6],col=2)
Sys.sleep(0.5)
}

# Random sample  from a uniform distribution on a Koch snowflake 
# with initial side length 1 and 3 iterations
x <- rkoch(2000, side = 1, niter = 3)
# Value of alpha
alpha <- 0.05
# Alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
plot(ahull.obj)

## End(Not run)

alpha-convex hull calculation of tracking data

Description

This function approximates the \alpha-convex hull of tracking data and returns a list of geom_path objects of the boundary.

Usage

ahull_track(x, y = NULL, alpha, nps = 10000, sc = 100)

Arguments

Details

An attempt is made to interpret the arguments x and y in a way suitable for computing the \alpha-convex hull. Any reasonable way of defining the coordinates is acceptable, see xy.coords.

Increase nps if the trajectory is not contained in the computed estimator.

Value

A list of geom_path objects defining the boundary of the \alpha-convex

References

Cholaquidis, A., Fraiman, R., Lugosi, G. and Pateiro-Lopez, B. (2014) Set estimation from reflected Brownian motion. arXiv:1411.0433.

Wikelski, M., and Kays, R. (2014). Movebank: archive, analysis and sharing of animal movement data. World Wide Web electronic publication.

Examples

## Not run: 
# Data from Movebank
# Study Name: Missouri Bison Tracking Project
# Principal Investigator: Stephen Blake
library(move)
# Login to Movebank (requires a Movebank account)
curl <- movebankLogin(username = "xxx", password = "zzz")
# Download study data (requires agreed to the license terms in the web interface)
study <- "Missouri Bison Tracking Project"
animalName <-"AHP2059"
track <- getMovebankData(study = study, animalName = animalName, login = curl)
dat <- track@data
# Plot animal locations
p <- ggplot2::ggplot(
  dat,
  ggplot2::aes(x = location_long, y = location_lat)
) +
  ggplot2::geom_path(
    colour = "red",
    linewidth = 0.3
  ) +
  ggplot2::coord_quickmap()
p
# Alpha-convex hull of the animal trajectory
ah_out <- ahull_track(x = dat[, c("location_long", "location_lat")], alpha = 0.005)
# Plot the trajectory and the alpha-convex hull
p + ah_out

## End(Not run)

Angles of the extremes of an arc

Description

Given a vector v and an angle \theta, anglesArc returns the angles that A_\theta v and A_{-\theta} v form with the axis OX, where A_\theta v represents the clockwise rotation of angle \theta of the vector v.

Usage

anglesArc(v, theta)

Arguments

Details

The angle that forms the vector v with the axis OX takes its value in [0,2\pi).

Value

Examples

## Not run: 
# Let v=c(0,1) and theta=pi/4
# Consider the arc such that v is the internal angle bisector that 
# divides the angle 2*theta into two equal angles
# The angles that the arc forms with the OX axis are pi/4 and 3*pi/4
v <- c(0,1)
theta <- pi/4
anglesArc(v,theta)

## End(Not run)

Add an arc to a plot

Description

This function adds the arc of B(c,r) between the angles that A_\theta v and A_{-\theta} v form with the axis OX, where A_\theta v represents the clockwise rotation of angle \theta of the vector v.

Usage

arc(c, r, v, theta, ...)

Arguments

See Also

plot.ahull.

Examples

## Not run: 
# Plot of the circumference of radius 1
theta <- seq(0, 2*pi, length = 100)
r <- 1
plot(r*cos(theta), r*sin(theta), type = "l")
# Add in red the arc between pi/4 and 3*pi/4
arc(c(0,0), 1, c(0,1), pi/4, col = 2, lwd = 2)

## End(Not run)

Area of the alpha-convex hull

Description

This function calculates the area of the \alpha-convex hull of a sample of points.

Usage

areaahull(x, timeout = 5)

Arguments

Value

See Also

ahull.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(500), nc = 2)
# Value of alpha
alpha <- 1
# alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
# Area of the alpha-convex hull
areaahull(ahull.obj)

## End(Not run)

Area of the alpha-convex hull

Description

This function calculates the area of the \alpha-convex hull of a sample of points.

Usage

areaahulleval(x)

Arguments

Value

See Also

ahull.

alpha-shape calculation

Description

This function calculates the \alpha-shape of a given sample for \alpha>0.

Usage

ashape(x, y = NULL, alpha)

Arguments

Details

An attempt is made to interpret the arguments x and y in a way suitable for computing the \alpha-shape, see xy.coords.

The \alpha-shape is defined for any finite number of points. However, since the algorithm is based on the Delaunay triangulation, at least three non-collinear points are required.

If y is NULL and x is an object of class "delvor", then the \alpha-shape is computed without invoking again the function delvor (it reduces the computational cost).

The function ashape returns (among other values) the matrix edges. The structure of edges is that of matrix mesh returned by the function delvor. Note that the \alpha-shape is a subgraph of the Delaunay triangulation and, therefore, edges is a submatrix of mesh.

Value

A list with the following components:

References

Edelsbrunner, H., Kirkpatrick, D.G. and Seidel, R. (1983). On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4), pp.551-559.

See Also

plot.ashape, delvor.

Examples

## Not run: 
# Uniform sample of size n=300 in the annulus B(c,0.5)\B(c,0.25), 
# with c=(0.5,0.5). 
n <- 300
theta<-runif(n,0,2*pi)
r<-sqrt(runif(n,0.25^2,0.5^2))
x<-cbind(0.5+r*cos(theta),0.5+r*sin(theta))
# Value of alpha
alpha <- 0.1
# alpha-shape 
ashape.obj <- ashape(x, alpha = alpha)
# If we change the value of alpha there is no need to compute
# again the Delaunay triangulation and Voronoi Diagram
alpha <- 0.4
ashape.obj.new <- ashape(ashape.obj$delvor.obj, alpha = alpha)

# Random sample  from a uniform distribution on a Koch snowflake 
# with initial side length 1 and 3 iterations
x <- rkoch(2000, side = 1, niter = 3)
# Value of alpha
alpha <- 0.05
# alpha-shape 
ashape.obj <- ashape(x, alpha = alpha)

## End(Not run)

Complement of the alpha-convex hull

Description

This function calculates the complement of the \alpha-convex hull of a given sample for \alpha>0.

Usage

complement(x, y = NULL, alpha)

Arguments

Details

An attempt is made to interpret the arguments x and y in a way suitable for computing the \alpha-shape. Any reasonable way of defining the coordinates is acceptable, see xy.coords.

If y is NULL and x is an object of class "delvor", then the complement of the \alpha-convex hull is computed with no need to invoke again the function delvor (it reduces the computational cost).

The complement of the \alpha-convex hull is calculated as a union of open balls and halfplanes that do not contain any point of the sample. See Edelsbrunnner et al. (1983) for a basic description of the algorithm. The construction of the complement is based on the Delaunay triangulation and Voronoi diagram of the sample, provided by the function delvor. The function complement returns a matrix compl. For each row i, compl[i,] contains the information relative to an open ball or halfplane of the complement. The first three columns are assigned to the characterization of the ball or halfplane i. The information relative to the edge of the Delaunay triangulation that generates the ball or halfplane i is contained in compl[i,4:16]. Thus, if the row i refers to an open ball, compl[i,1:3] contains the center and radius of the ball. Furthermore, compl[i,17:18] and compl[i,19] refer to the unitary vector v and the angle \theta that characterize the arc that joins the two sample points that define the ball i. If the row i refers to a halfplane, compl[i,1:3] determines its equation. For the halfplane y>a+bx, compl[i,1:3]=(a,b,-1). In the same way, for the halfplane y<a+bx, compl[i,1:3]=(a,b,-2), for the halfplane x>a, compl[i,1:3]=(a,0,-3) and for the halfplane x<a, compl[i,1:3]=(a,0,-4).

Value

References

Edelsbrunner, H., Kirkpatrick, D.G. and Seidel, R. (1983) On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4), pp.551-559.

See Also

delvor, ahull.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(100), nc = 2)
# Value of alpha
alpha <- 0.2
# Complement of the alpha-convex hull
compl <- complement(x, alpha = alpha)

## End(Not run)

Delaunay triangulation and Voronoi diagram

Description

This function returns a list containing the Delaunay triangulation and Voronoi diagram of a given planar sample.

Usage

delvor(x, y = NULL)

Arguments

Details

The coordinates are obtained using xy.coords. At least three non-collinear points are required.

The Delaunay triangulation is computed with delaunayn from the geometry package, which uses Qhull. The corresponding Voronoi diagram is reconstructed from the triangle incidences and circumcentres. For an interior Delaunay edge, its dual Voronoi edge joins the circumcentres of the two adjacent triangles.

For a Delaunay edge on the convex hull, the dual Voronoi edge is semi-infinite. The function dummycoor supplies fictitious coordinates along the outward-pointing ray for plotting; these are not additional Voronoi vertices. Its direction is determined using the sample centroid.

The matrix mesh contains one row for each Delaunay edge: ind1, ind2 are its sample-point indices; x1, y1, x2, y2 are its endpoint coordinates; mx1, my1, mx2, my2 are the coordinates of the corresponding Voronoi edge endpoints; and bp1, bp2 identify fictitious endpoints of unbounded Voronoi rays.

Value

A list of class "delvor" with the following components:

References

Barber, C. B., Dobkin, D. P. and Huhdanpaa, H. (1996). The Quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software, 22(4), pp. 469–483.

See Also

plot.delvor.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(20), ncol = 2)
# Delaunay triangulation and Voronoi diagram calculation
delvor.obj <- delvor(x)

## End(Not run)

Fictitious endpoint of a semi-infinite Voronoi edge

Description

Computes a fictitious endpoint in the outward direction of a semi-infinite Voronoi edge, for plotting purposes.

Usage

dummycoor(tri.obj, l1, l2, m, away)

Arguments

Details

A Delaunay edge on the convex hull corresponds to a semi-infinite Voronoi edge whose finite endpoint is the circumcentre m. The direction is perpendicular to the Delaunay edge joining l1 and l2. The outward orientation is selected using the centroid of the sample coordinates in tri.obj.

The fictitious endpoint is placed along the ray at a distance depending on away and the projection of the Delaunay edge midpoint. This ensures that the plotting endpoint extends beyond that projection. The returned point is not an actual Voronoi vertex.

Value

A numeric vector of length two giving the fictitious endpoint of the semi-infinite Voronoi ray.

See Also

delvor.

Examples

xy <- rbind(c(0, 0), c(1, 0), c(0, 1))
dummycoor(xy, c(0, 0), c(1, 0), c(0.5, 0.5), away = 2)

Devroye-Wise estimator

Description

This function calculates the Devroye-Wise estimator of a given sample of points in the plane for \epsilon>0.

Usage

dw(x, y = NULL, eps)

Arguments

Details

An attempt is made to interpret the arguments x and y in a way suitable for computing the Devroye-Wise estimator. Any reasonable way of defining the coordinates is acceptable, see xy.coords.

Value

Given a sample of points in the plane, the estimator is defined as union of balls of radius \epsilon with centers in the sample points. For each arc in the boundary of the Devroye-Wise estimator, the columns of the output matrix store the center c and radius r of the arc, the unitary vector v, the angle \theta that define the arc and the indices of the end points.

References

Devroye, L. and Wise, G. (1980) Detection of abnormal behaviour via nonparametric estimation of the support. SIAM J. Appl. Math. 3, pp. 480-488.

Examples

## Not run: 
# Uniform sample of size n = 200 in the annulus B(c, 0.5)\B(c, 0.25), 
# with c = (0.5, 0.5).
n <- 200
theta <- runif(n, 0, 2*pi)
r <- sqrt(runif(n, 0.25^2, 0.5^2))
x <- cbind(0.5 + r*cos(theta), 0.5 + r*sin(theta))
eps <- 0.05
dw.obj <- dw(x, eps = eps)
plot(x)
for(i in 1:dim(dw.obj)[1]){arc(dw.obj[i, 1:2], eps, dw.obj[i, 4:5], dw.obj[i, 6])}

## End(Not run)

RBM-sausage calculation of tracking data

Description

This function approximates the RBM-sausage of tracking data and returns a list of geom_path objects of the boundary.

Usage

dw_track(x, y = NULL, eps, nps = 20000, sc = 100)

Arguments

Details

An attempt is made to interpret the arguments x and y in a way suitable for computing the RBM-sausage. Any reasonable way of defining the coordinates is acceptable, see xy.coords. Given a trajectory in the plane, the estimator is defined as the set of points whose distance to the trajectory is less than or equal to \epsilon (this estimator is analogous to the one of Devroye and Wise (1980) for i.i.d. data). Increase nps if the trajectory is not contained in the computed estimator.

Value

A list of geom_path objects defining the boundary of the estimator

References

Cholaquidis, A., Fraiman, R., Lugosi, G. and Pateiro-Lopez, B. (2014) Set estimation from reflected Brownian motion. arXiv:1411.0433.

Devroye, L. and Wise, G. (1980) Detection of abnormal behaviour via nonparametric estimation of the support. SIAM J. Appl. Math. 3, pp. 480-488.

Wikelski, M., and Kays, R. (2014). Movebank: archive, analysis and sharing of animal movement data. World Wide Web electronic publication.

Examples

## Not run: 
# Data from Movebank
# Study Name: Missouri Bison Tracking Project
# Principal Investigator: Stephen Blake
library(move)
# Login to Movebank (requires a Movebank account)
curl <- movebankLogin(username = "xxx", password = "zzz")
# Download study data (requires agreed to the license terms in the web interface)
study <- "Missouri Bison Tracking Project"
animalName <-"AHP2059"
track <- getMovebankData(study = study, animalName = animalName, login = curl)
dat <- track@data
# Plot animal locations
p <- ggplot2::ggplot(
  dat,
  ggplot2::aes(x = location_long, y = location_lat)
) +
  ggplot2::geom_path(
    colour = "red",
    linewidth = 0.3
  ) +
  ggplot2::coord_quickmap()
p
# Devroye-Wise of the animal trajectory
dw_out <- dw_track(x = dat[, c("location_long", "location_lat")], eps = 0.001)
# Plot the trajectory and the Devroye-Wise
p + dw_out

## End(Not run)

Determines for one or more points whether they belong to the alpha-convex hull

Description

This function determines for one or more points p whether they belong to the \alpha-convex hull of a sample.

Usage

inahull(ahull.obj, p)

Arguments

Details

The complement of the \alpha-convex hull of a sample is calculated by complement. The function inahull checks whether each point in p belongs to any of the open balls or halfplanes that define the complement.

Value

See Also

ahull, complement.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(100), nc = 2)
# Value of alpha
alpha <- 0.2
# alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
# Check if the point (0.5, 0.5) belongs to the alpha-convex hull
inahull(ahull.obj, p = c(0.5, 0.5))
# Check if the points (0.5, 0.5) and (2, 2) belong to the alpha-convex hull
inahull(ahull.obj, p = rbind(c(0.5, 0.5), c(2, 2)))

## End(Not run)

Intersection of two circumferences

Description

This function calculates the intersection of two circumferences, given their centers and radius c1,r1 and c2,r2, respectively.

Usage

inter(c11, c12, r1, c21, c22, r2)

Arguments

Details

The function inter is internally called by the function ahull.

Value

A list with the following components:

Construct a Kock snowflake curve

Description

This function uses recursion to construct a Kock snowflake curve.

Usage

koch(side = 3, niter = 5)

Arguments

Details

The Koch snowflake is a fractal curve described by the Swedish mathematician Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process.

Value

References

von Koch, H. (1904). Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire. Arkiv for Matematik, 1, pp.681-704.

See Also

rkoch.

Examples

## Not run: 
# The first four iterations of a Koch snowflake 
# with side length of the initial equilateral triangle equal to 3.
vertices <- koch(side = 2, niter = 4)
plot(vertices[, 1], vertices[, 2], type = "l", asp = TRUE, 
main = "Koch snowflake", xlab = "", ylab = "", col = 4)
polygon(vertices[, 1], vertices[, 2] , col = 4)

## End(Not run)

Length of the boundary of the alpha-convex hull

Description

This function calculates the length of the boundary of the \alpha-convex hull of a given sample.

Usage

lengthahull(ahull.arcs)

Arguments

Details

The function lengthahull is internally called by the function ahull.

Value

See Also

ahull.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(100), nc = 2)
# Value of alpha
alpha <- 0.2
# alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
# Length of the alpha-convex hull
ahull.obj$length

## End(Not run)

Plot the alpha-convex hull

Description

This function returns a plot of the \alpha-convex hull. If desired, it also adds the Delaunay triangulation, Voronoi diagram and \alpha-shape of the sample.

Usage

## S3 method for class 'ahull'
plot(x, add = FALSE, do.shape = FALSE, 
	wlines = c("none", "both", "del", "vor"), wpoints = TRUE, 
	number = FALSE, col = NULL, xlim = NULL, 
	ylim = NULL, lwd = NULL, ...)

Arguments

See Also

ahull, ashape.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(100), nc = 2)
# Value of alpha
alpha <- 0.2
# alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
# Plot including the alpha-convex hull in pink, alpha-shape in blue, 
# sample points in black, voronoi diagram in green 
# and Delaunay triangulation in red 
plot(ahull.obj, do.shape = TRUE, wlines = "both", col = c(6, 4, 1, 2, 3))

# Random sample  from a uniform distribution on a Koch snowflake 
# with initial side length 1 and 3 iterations
x <- rkoch(2000, side = 1, niter = 3)
# Value of alpha
alpha <- 0.05
# Alpha-convex hull
ahull.obj <- ahull(x, alpha = alpha)
plot(ahull.obj)

## End(Not run)

Plot the alpha-shape

Description

This function returns a plot of the \alpha-shape. If desired, it also adds the Delaunay triangulation and Voronoi diagram of the sample.

Usage

## S3 method for class 'ashape'
plot(x, add = FALSE, wlines = c("none", "both", "del", "vor"),
	wpoints = TRUE, number = FALSE, col = NULL, 
	xlim = NULL, ylim = NULL, lwd = NULL, ...)

Arguments

See Also

ashape.

Examples

## Not run: 
# Uniform sample of size n=300 in the annulus B(c, 0.5)\B(c, 0.25) 
# with c=(0.5, 0.5). 
n <- 300
theta<-runif(n,0,2*pi)
r<-sqrt(runif(n,0.25^2,0.5^2))
x<-cbind(0.5+r*cos(theta),0.5+r*sin(theta))
# Value of alpha
alpha <- 0.1
# alpha-shape 
ashape.obj <- ashape(x, alpha = alpha)
# Plot alpha-shape in blue, sample points in black, 
# and Delaunay triangulation in red
plot(ashape.obj, wlines= "del", col = c(4, 1, 2))

# Random sample  from a uniform distribution on a Koch snowflake 
# with initial side length 1 and 3 iterations
x <- rkoch(2000, side = 1, niter = 3)
# Value of alpha
alpha <- 0.05
# alpha-shape 
ashape.obj <- ashape(x, alpha = alpha)
# Plot alpha-shape in blue
plot(ashape.obj, col = c(4, 1))

## End(Not run)

Plot the Voronoi diagram and Delaunay traingulation

Description

This function returns a plot of the Voronoi diagram and Delaunay traingulation.

Usage

## S3 method for class 'delvor'
plot(x, add = FALSE, wlines = c("both", "del", "vor"),
	wpoints = TRUE, number = FALSE, col = NULL, 
	xlim = NULL, ylim = NULL, ...)

Arguments

See Also

delvor.

Examples

## Not run: 
# Random sample in the unit square
x <- matrix(runif(100), nc = 2)
# Delaunay triangulation and Voronoi diagram
delvor.obj <- delvor(x)
# Plot Voronoi diagram and Delaunay triangulation 
plot(delvor.obj)

## End(Not run)

Random generation on a Koch snowflake curve

Description

This function generates ramdom points from a uniform distribution on a Koch snowflake.

Usage

rkoch(n, side = 3, niter = 5)

Arguments

Value

A 2-column matrix with the coordinates of generated points.

See Also

koch.

Examples

## Not run: 
unifkoch <- rkoch(2000, side = 1, niter = 3)
plot(unifkoch, asp = TRUE)

## End(Not run)

Clockwise rotation

Description

This function calculates the clockwise rotation of angle \theta of a given vector v in the plane.

Usage

rotation(v, theta)

Arguments

Value

Examples

## Not run: 
# Rotation of angle pi/4 of the vector (0,1)
rotation(v = c(0, 1), theta = pi/4)

## End(Not run)