simplex.createTransform: Transform points on an (n-1)-simplex to n-dimensional space

In hitandrun: "Hit and Run" and "Shake and Bake" for Sampling Uniformly from Convex Shapes

Transform points on an (n-1)-simplex to n-dimensional space

Description

This function generates a matrix to transform points in an (n-1) dimensional homogeneous coordinate representation of the (n-1) simplex to n-dimensional Cartesian coordinates.

The inverse transform can also be generated, and conversion can be to homogeneous coordinates instead of Cartesion ones.

Usage

simplex.createTransform(n, inverse=FALSE, keepHomogeneous=inverse)

Arguments

n	Dimension of the space
inverse	TRUE to convert from n-space coordinates to (n-1)-simplex coordinates
keepHomogeneous	TRUE to convert to homogeneous coordinates rather than Cartesian

Details

Multiply a coordinate vector in homogeneous coordinates by pre-multiplying by the generated matrix (see examples).

Value

A transformation matrix.

Author(s)

Gert van Valkenhoef

See Also

simplex.createConstraints har

Examples

E <- 1E-12 # Allowed numerical error

# The origin in (n-1)-dimensional space should be the centroid of the simplex
# when transformed to n-dimensional space
transform <- simplex.createTransform(3)
x <- transform %*% c(0, 0, 1)
x
stopifnot(abs(x - c(1/3, 1/3, 1/3)) < E)

# The same should hold for the inverse transformation
invTransform <- simplex.createTransform(3, inverse=TRUE)
y <- invTransform %*% c(1/3, 1/3, 1/3, 1)
y
stopifnot(abs(y - c(0, 0, 1)) < E)

# Of course, an arbitrary weight vector should transform back to itself
transform <- simplex.createTransform(3, keepHomogeneous=TRUE)
x <- c(0.2, 0.5, 0.3, 1.0)
y <- transform %*% invTransform %*% x
y
stopifnot(abs(y - x) < E)

# And we can apply the tranform to a matrix:
a <- cbind(x, x, x)
b <- transform %*% invTransform %*% a
b
stopifnot(abs(b - a) < E)

hitandrun documentation built on May 28, 2022, 1:09 a.m.