```
#' Variable covariant linear mapping
#'
#' This function encodes a linear mapping that sends a vector in the vector
#' space containing a point cloud $M^J$ to the vector space of variables over
#' $J$, whose coordinates are the covariant coordinates of the cloud, relative
#' to a given point $P$, with respect to the axis generated by the vector.
#' @param cloud An m-by-n matrix of coordinates for m points in n-dimensional
#' space.
#' @param point An n-coordinate point in the same space as the cloud.
#' @param dir_vec An n-dimensional vector giving the direction of the axis
#' through the cloud.
#' @export
vac <-
function(cloud, point, dir_vec) {
# If point is unspecified, use barycenter
if (missing(point)) point <- barycenter(cloud)
# Transform cloud relative to point
transform_cloud <- cloud - rep(point, each = nrow(cloud))
# Calculate coordinates
transform_cloud %*% matrix(dir_vec, ncol = 1)
}
```

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