How to check if a polygon is convex

To see if it is a convex polygon, you have to calculate the angles at each of the vertices of the polygon. If all angles have the same signal (positive or negative, depending on orientation), then the polygon is convex.

Instead of finding the angles, you just need to find the cross product of the segments on both sides of the angles. If the segments at point B are AB and BC, the cross product has a |AB| * |BC| * Sin(teta) value where theta is the angle between the two segments. Since the lengths are always positive, the result is positive if the angle is positive and negative if the angle is negative.

The code below shows an implementation of the algorithm in R.

# Codigo (e descrição) adaptado de
# http://csharphelper.com/blog/2014/07/determine-whether-a-polygon-is-convex-in-c/
verifica_convexo <- function(polygon) {
    crossProductLength <- function(ax, ay, bx, by, cx, cy) {
        BAx = ax - bx
        BAy = ay - by
        BCx = cx - bx
        BCy = cy - by

        (BAx * BCy - BAy * BCx)
    }

    no_vertices <- nrow(polygon)

    testFor <- function(a) {
        b <- (a + 1) %% no_vertices
        c <- (b + 1) %% no_vertices
        sign(crossProductLength(polygon[a + 1,1], polygon[a + 1,2],
                                polygon[b + 1,1], polygon[b + 1,2],
                                polygon[c + 1,1], polygon[c + 1,2]))
    }

    signs <- sapply(0:(no_vertices - 1), testFor)
    convexo <- all(signs == signs[1])
}

quadrado <- matrix(c(0, 0, 0, 5, 5, 5, 5, 0), ncol = 2, byrow = TRUE)
print(quadrado)
print(verifica_convexo(quadrado))
seta <- matrix(c(0, 0, 1, 4, 2, 0, 1, 1), ncol = 2, byrow = TRUE)
print(seta)
print(verifica_convexo(seta))