Create linear map to move simplex with p vertices to p-1 dimensional space

This approach comes for https://en.wikipedia.org/wiki/Simplex#Cartesian_coordinates_for_regular_n-dimensional_simplex_in_Rn and centers the projected points at 0.

simplex_project_mat(p)

Arguments

p	number of vertices

Value

A matrix that transforms each vertices (column) to (p-1) dimensional space

Examples

A4 <- simplex_project_mat(4)
# expected for dim 4 (for examples on wikipedia)
A4_expected <- matrix(c(1,0,0,
                        -1/3, sqrt(8)/3,0,
                        -1/3,-sqrt(2)/3, sqrt(2/3),
                        -1/3, -sqrt(2)/3, -sqrt(2/3)),
                      ncol = 4)
all.equal(A4, A4_expected) # minor numerical inconsistencies
#> [1] TRUE

library(dplyr)

if (interactive()){
  # visualizing this 4d projection
  s4in3 <- simplex_project_mat(4) %>% t %>% data.frame %>%
    rbind(., data.frame(X1 = 0, X2 = 0, X3 = 0)) %>%
    mutate(color = factor(c(rep(1,4), 2)))#center


  library(plotly)
  plot_ly(s4in3,
          type = "scatter3d",
          mode = "markers", x = ~X1, y = ~X2, z = ~X3,
          color = ~color)
}
# visualizing a 3d projection
s3in2 <- simplex_project_mat(3) %>% t %>% data.frame %>%
 rbind(., data.frame(X1 = 0, X2 = 0)) %>%
 mutate(color = factor(c(rep(1,3), 2))) #center

library(ggplot2)
ggplot(s3in2) +
  geom_point(aes(x = X1, y = X2, color = color))