mvmesh-geom {mvmesh}

Miscellaneous computational geometry and utility functions

Description

EdgeSubdivision calculates an equal area/volume subdivision of a simplex. AffineTransform defines a new mesh by translating all points x to x' = A Rotate2D and Rotate3D calculate rotation matrices for use by AffineTransform.

Icosahedron returns the vertices of an icosahedron with vertices on the unit sphere

Other functions are internal functions, use at your own risk.

Usage

EdgeSubdivision( n, k )
EdgeSubdivisionMulti( V, SVI, k, normalize = FALSE, p = 2)
ConvertBase( m, b, n)
NumVertices( n, k, single = TRUE)
PointCoord( S, color )
SimplexCoord( S, color )
SVIFromColor( S, T )

MatchRow(v, table, first = 1, last = nrow(table))
AffineTransform( mesh, A, shift ) 
Rotate2D( theta )
Rotate3D( theta )
Icosahedron( )

V2Hrep( S ) 
H2Vrep( H ) 
SatisfyHrep( x, Hsingle )
HrepCones( S )
IntersectMultipleSimplicesV( S1, S2 )
IntersectMultipleSimplicesH( H1, H2, skip.redundant=FALSE )
Intersect2SimplicesH( H1, H2, tessellate=FALSE, skip.redundant=FALSE )
Lift2UnitSimplex(S)

Arguments

v

a vector of length n

table

matrix of size m3 x n

first

row to start search

last

row to end search

mesh

object of class "mvmesh"

A

n x n matrix

shift

shift vector of length n

theta

rotation angle; in 2D, this is a single angle; in 3D is it a vector of length 3, with theta[i] giving rotation around i-th axis

k

number of subdivisions

n

dimension of simplex

V

matrix of vertices; each row is a point in R^n

normalize

TRUE to normalize vertices to lie on the unit sphere in the l^p norm

p

power in the l^p norm

S, S1, S2

matrix of size (vps x n) specifying the vertices of a single simplex; S[j,] is the j-th vertex of S

SVI

Simplex Vertex Indices, see mvmesh

m

positive integer to be converted to base 'b'

b

positive integer, the base used to expess 'x'

single

If TRUE, return only one value; if FALSE, return table of values

color

color matrix, internal matrix used by EdgeSubdivision to subdivide a simplex

T

array giving a list of color matrices

H, H1, H2

array of simplices in the H-representation, H[,,k] is the H-representation for the k-th simplex

x

matrix with columns giving the points

Hsingle

matrix giving the H-representation of a single simplex

tessellate

TRUE to tessellate the resulting intersection

skip.redundant

TRUE to skip the call to rcdd::redundant

Details

AffineTransform computes a new mesh from a previous one, with each vertex v being replaced by A Rotate3D computes a 3D rotation matrix.

Icosahedron returns the vertices of the icosahedron with vertices on the unit sphere

H2Vrep converts from the half-space (H) representation to the vertex (V) representation of a simplex. V2Hrep converts from the V-representation to the H-representation. It is assumed that all the resulting value are of the same dimension. If this is not the case, an error will occur. To work with such cases, call the function separately for each simplex and save the result in different size objects. The one place where this can occur with mvmesh objects is with a PolarSphere or PolarBall: at the places where polar coordinates are nonunique, vertices will repeat and the H-representation will have fewer constraints than other simplices.

IntersectMultipleSimplicesV computes the pairwise intersection of two lists of simplices given in the V-representation. IntersectMultipleSimplicesH computes the pairwise intersection of two lists of simplices given in the H-representation. Intersect2SimplicesH computes the intersection of two simplices, both specified in the H-representation.

Lift2UnitSimplex reverses the projection from the unit simplex in n-space to the first (n-1) coordinates. That is, it 'lifts' each (n-1) dimensional simplex in R^(n-1) to the unit simplex in R^n by appending an n-th coordinate, with x[n] <- 1-sum(x[1:(n-1)]).

Value

MatchRow returns an integer vector, showing which rows of table match v. If there are no matches, it returns integer(0).

AffineTransform returns an object of class "mvmesh". Rotate2D returns a 2 x 2 rotation matrix, Rotate3D returns a 3 x 3 rotation matrix.

EdgeSubdivision computes an edgewise subdivision of a simplex using the method of Edelsbrunner and Grayson. The algorithm of Concalves, et. al. was implemented in R. It is a coordinate free method. ConvertBase is an internal routine used by the subdivision algorithm. NumVertices is a utility routine to recursively calculate the number of vertices in an edgewise subdivision.

EdgeSubdivMulti is roughly a vectorized version of EdgeSubdivison. It takes a list of simplices, and performs a k-subdivision of each simplex for function UnitSphere and related functions. Since some simplices may share edges, the same vertex can be occur multiple times, so this function goes through the resulting vertices and eliminates repeats. This function is not meant to be called by an end user; it is not guaranteed to be general.

ConvertBase is an internal function that converts a positve integer 'x' to an 'n' digit base 'b' representation. NumVertices is an internal function that computes the number of simplices in an edgewise subdivision (without doing the subdivision). PointCoord is an internal function that computes a single vertex of a simplex. SimplexCoord is an internal function that computes the coordinates of a simplex 'S' given color matrix 'color'. SVIFromColor is an internal function that computes the SVI from a starting simplex 'S' and color array 'T'.

Note that rays and lines are not allowed in V2Hrep; use rcdd funtion makeH directly to use them.

EdgeSubdivision returns a color matrix, a coordinate free representaion of the subdivision. One generally uses UnitSimplex or UnitBall to get a vertex representation of the subdivision.

EdgeSubdivMulti returns a list of class 'mvmesh'

References

Edelsbrunner and Grayson, Discrete Comput. Geom., Vol 24, 707-719 (2000).

Goncalves, Palhares, Takahashi, and Mesquita, Algorithm 860: SimpleS – an extension of Freudenthal's simplex subdivision, ACM Trans. Math. Softw., 32, 609-621 (2006).

Examples

Icosahedron( )

T <- EdgeSubdivision( n=2, k=2 )
T

ConvertBase( 10, 2, 6 )  # note order of digits

NumVertices( n=4, k=8, single=FALSE )

S <- rbind( diag(rep(1,2)), c(0,0) ) # solid simplex in 2D
PointCoord( S, T[,,1] )

SimplexCoord( S, T[,,1] )

SVIFromColor( S, T )

S1 <- rbind( c(0,0,0), diag( rep(1,3) ) )
S2 <- rbind( c(1,1,1), diag( rep(1,3) ) )
S3 <- rbind( c(1,1,1), c(0,1,0), c(1,0,0), c(1,1,0) )
S <- array( c(S1,S2,S3), dim=c(4,3,3) )

( H1 <- V2Hrep( S ) )
( S4 <- H2Vrep( H1 ) )

( H2 <- HrepCones( UnitSphere(n=2,k=1)$S )[,,2] )  # cone between 0 <= y <= x, x >= 0
x <- matrix( rnorm(100), ncol=2 )
( i <- SatisfyHrep( x, H2 ) )
x[i,]

(table <- matrix( c(1:12,1:3 ), ncol=3, byrow=TRUE ))
MatchRow( 1:3, table )

## Not run: 
plot( Icosahedron( ), col="green" )

mesh <- SolidSimplex( n=3, k=2 )
plot(mesh, col="blue")
mesh2 <- AffineTransform( mesh, A=Rotate3D( rep(pi/2,3) ), shift=c(1,1,1) ) 
plot(mesh2, new.plot=FALSE, col="red" )



## End(Not run)