How to check if a polygon is regular or convex

Question

Is there a function in R that checks whether a polygon is regular or not and some that checks whether it is convex or not?

If either answer is no, any idea? As I thought:

#Por exemplo um quadrilátero, a primeira coluna é x e a segunda coluna é y.
quadrilatero = matrix(c(0,2,-7,1,4,3, 4,5), ncol = 2)

#o input (poligono) tem duas colunas obrigatoriamente 
verifica_regular = function(poligono){
  distancias = apply(poligono, 1, dist)
  if(distancias == distancias[1]) return(T)
  else return(F)
}
verifica_regular(quadrilatero)

But the function dist within the apply calculates the difference between a value and the value in the next column.

To check the convexity I have no idea.

Answer 1

To check that the polygon is regular, you need the distance of all points, including the "last" point of your polygon to the "first" - in the code examples below, if you do not "close" the polygon, you will have an incorrect result in the first case.

And in that case you don’t need the function apply. dist already gives you the distance between all points, so what you need is to take the diagonal of the bottom (or top) of the matrix that it returns to you.

And as @Jjoao noted, the condition of equal sides is necessary, but not enough for the polygon to be regular. All their angles need to be identical as well. One way to verify this, given that the sides are equal, is to verify that the center of the polygon is equidistant from all points. The updated code is below.

verifica_lados_iguais <- function(poligono) {
    no_vertices <- nrow(poligono)
    poligono_fechado <- rbind(poligono, poligono[1])
    distancias <- diag(as.matrix(dist(poligono_fechado))[1:no_vertices, 2:(no_vertices + 1)])
    lados_iguais <- all(distancias == distancias[1])
}

verifica_equidistancia_centro <- function(poligono) {
    c <- apply(poligono, 2, sum) / nrow(poligono)
    dist_centro <- apply(poligono, 1, function(p) sqrt((c[1] - p[1])^2 + (c[2] - p[2])^2))
    equidistanteCentro <- all((abs(dist_centro - dist_centro[1]) < 0.00001))
}

verifica_regular <- function(poligono) {
    regular <- verifica_lados_iguais(poligono) &&
        verifica_equidistancia_centro(poligono)
}

print(quadrilatero <- matrix(c(0, 0, 0, 5, 5, 5, 2, 1), ncol = 2, byrow = TRUE))
print(verifica_regular(quadrilatero))
print(quadrado <- matrix(c(0, 0, 0, 5, 5, 5, 5, 0), ncol = 2, byrow = TRUE))
print(verifica_regular(quadrado))
print(losango <- matrix(c(0, 0, 2, 1, 4, 0, 2, -1), ncol = 2, byrow = TRUE))
print(verifica_regular(losango))