This vignette contains part II of a set of examples on how to use `clugenr` in 3D. Examples require the following setup code:

    library(clugenr)    # The clugenr library
    options(rgl.useNULL = TRUE)  # Create RGL plots in systems without displays (CI)
    library(rgl)
    setupKnitr(autoprint = TRUE) # Render RGL plots directly on generated page

    # Load helper functions for plotting examples
    source("plot_examples_3d.R", local = knitr::knit_global())

    # Keep examples reproducible in newer R versions
    RNGversion("3.6.0")

The 3D examples were plotted with the `plot_examples_3d()` function available here.

## Lateral dispersion and placement of point projections on the line

    seed <- 456

### Normal projection placement (default): `proj_dist_fn = "norm"`

    e064 <- clugen(3, 4, 300, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 0.0, seed = seed)
    e065 <- clugen(3, 4, 300, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 1.0, seed = seed)
    e066 <- clugen(3, 4, 300, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 3.0, seed = seed)

    plot_examples_3d(list(e = e064, t = "e064: lateral_disp = 0"),
                     list(e = e065, t = "e065: lateral_disp = 1"),
                     list(e = e066, t = "e066: lateral_disp = 3"))

### Uniform projection placement: `proj_dist_fn = "unif"`

    e067 <- clugen(3, 4, 300, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 0.0, seed = seed,
                   proj_dist_fn = "unif")
    e068 <- clugen(3, 4, 300, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 1.0, seed = seed,
                   proj_dist_fn = "unif")
    e069 <- clugen(3, 4, 300, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 3.0, seed = seed,
                   proj_dist_fn = "unif")

    plot_examples_3d(list(e = e067, t = "e067: lateral_disp = 0"),
                     list(e = e068, t = "e068: lateral_disp = 1"),
                     list(e = e069, t = "e069: lateral_disp = 3"))

### Custom projection placement using the Beta distribution

    # Custom proj_dist_fn: point projections placed using the Beta distribution
    proj_beta <- function(len, n) len * rbeta(n, 0.1, 0.1) - len / 2

    e070 <- clugen(3, 4, 400, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 0.0, seed = seed,
                   proj_dist_fn = proj_beta)
    e071 <- clugen(3, 4, 400, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 1.0, seed = seed,
                   proj_dist_fn = proj_beta)
    e072 <- clugen(3, 4, 400, c(1, 0, 0), pi / 2, c(20, 20, 20), 13, 2, 3.0, seed = seed,
                   proj_dist_fn = proj_beta)

    plot_examples_3d(list(e = e070, t = "e070: lateral_disp = 0"),
                     list(e = e071, t = "e071: lateral_disp = 1"),
                     list(e = e072, t = "e072: lateral_disp = 3"))

## Controlling final point positions from their projections on the cluster-supporting line

    seed <- 12321

    # Custom proj_dist_fn: point projections placed using the Beta distribution
    proj_beta <- function(len, n) len * rbeta(n, 0.1, 0.1) - len / 2

### Points on hyperplane orthogonal to cluster-supporting line (default): `point_dist_fn = "n-1"`

    e073 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed)
    e074 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   proj_dist_fn = "unif")
    e075 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   proj_dist_fn = proj_beta)

    plot_examples_3d(list(e = e073, t = "e073: proj_dist_fn = 'norm' (default)"),
                     list(e = e074, t = "e074: proj_dist_fn = 'unif'"),
                     list(e = e075, t = "e075: custom proj_dist_fn (Beta dist.)"))

### Points around projection on cluster-supporting line: `point_dist_fn = "n"`

    e076 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   point_dist_fn = "n")
    e077 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   point_dist_fn = "n", proj_dist_fn = "unif")
    e078 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   point_dist_fn = "n", proj_dist_fn = proj_beta)

    plot_examples_3d(list(e = e076, t = "e076: proj_dist_fn = 'norm' (default)"),
                     list(e = e077, t = "e077: proj_dist_fn = 'unif'"),
                     list(e = e078, t = "e078: custom proj_dist_fn (Beta dist.)"))

### Custom point placement using the exponential distribution

    # Custom point_dist_fn: final points placed using the Exponential distribution
    clupoints_n_1_exp <- function(projs, lat_std, len, clu_dir, clu_ctr) {
      dist_exp <- function(npts, lstd) lstd * rexp(npts, rate = 2 / lstd)
      clupoints_n_1_template(projs, lat_std, clu_dir, dist_exp)
    }

    e079 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   point_dist_fn = clupoints_n_1_exp)
    e080 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   point_dist_fn = clupoints_n_1_exp, proj_dist_fn = "unif")
    e081 <- clugen(3, 5, 400, c(1, 0, 0), pi / 3, c(20, 20, 20), 22, 3, 2, seed = seed,
                   point_dist_fn = clupoints_n_1_exp, proj_dist_fn = proj_beta)

    plot_examples_3d(list(e = e079, t = "e079: proj_dist_fn = 'norm' (default)"),
                     list(e = e080, t = "e080: proj_dist_fn = 'unif'"),
                     list(e = e081, t = "e081: custom proj_dist_fn (Beta dist.)"))

### Manipulating cluster sizes

    seed <- 87

    # Custom clucenters_fn (all): yields fixed positions for the clusters
    centers_fixed <- function(nclu, csep, coff) {
      matrix(c(-csep[1], -csep[2], -csep[3], csep[1], -csep[2], -csep[3],
               -csep[1], csep[2], csep[3], csep[1], csep[2], csep[3]),
             nrow = nclu, byrow = TRUE)
    }

    # Custom clusizes_fn (e083): cluster sizes determined via the uniform distribution,
    # no correction for total points
    clusizes_unif <- function(nclu, npts, ae) sample(2 * npts / nclu, nclu, replace = TRUE)

    # Custom clusizes_fn (e084): clusters all have the same size, no correction for
    # total points
    clusizes_equal <- function(nclu, npts, ae) npts %/% nclu * rep.int(1, nclu)

    e082 <- clugen(3, 4, 400, c(1, 1, 1), pi, c(20, 20, 20), 0, 0, 5, seed = seed,
                   point_dist_fn = "n",
                   clucenters_fn = centers_fixed)
    e083 <- clugen(3, 4, 400, c(1, 1, 1), pi, c(20, 20, 20), 0, 0, 5, seed = seed,
                   clusizes_fn = clusizes_unif, point_dist_fn = "n",
                   clucenters_fn = centers_fixed)
    e084 <- clugen(3, 4, 400, c(1, 1, 1), pi, c(20, 20, 20), 0, 0, 5, seed = seed,
                   clusizes_fn = clusizes_equal, point_dist_fn = "n",
                   clucenters_fn = centers_fixed)

    plot_examples_3d(list(e = e082, t = "e082: normal dist. (default)"),
                     list(e = e083, t = "e083: unif. dist. (custom)"),
                     list(e = e084, t = "e084: equal size (custom)"))