We usually think of quadrilaterals as squares, rectangles, parallelograms, trapezoids, rhombuses, or kites. (I was impressed that my four year-old granddaughter knew the last one, although she called it a diamond!)

      However, a polygon may intersect itself. A five-sided star is one example, where the sides are connected to alternating vertices.

      A quarilateral may also intersect itself. In the following diagram, the original quadrilateral has points A (0,0), B (4,0), C (3,3), D (1,4). The self-intersecting quadrilateral is formed from the original quadrilateral by shifting point D from (1,4) to (2, -2), so side CD crosses AB).

      Here is some R code:

# Self-intersecting quadrilateral
library(ggplot2)

# Define coordinates for original quadrilateral
original_quad <- data.frame(
  x1 = c(0, 4, 3, 1),
  y1 = c(0, 0, 3, 4),
  x2 = c(4, 3, 1, 0),
  y2 = c(0, 3, 4, 0),
  group = c("AB", "BC", "CD", "DA"),
  labels = c("A", "B", "C", "D")
)

# Define coordinates for self-intersecting quadrilateral
# Shift point D from (1,4) to (2, -2), so side CD crosses AB)

stretched_quad <- data.frame(
  x1 = c(0, 4, 3, 2),  
  y1 = c(0, 0, 3, -2), 
  x2 = c(4, 3, 2, 0),
  y2 = c(0, 3, -2, 0),
  group = c("AB", "BC", "CD", "DA"),
  labels = c("A", "B", "C", "D")
)

# Define colors for each side
color_map <- c("AB" = "red", "BC" = "blue", "CD" = "green", "DA" = "purple")

# Function to plot the quadrilateral
plot_quad <- function(data, title, x_lim, y_lim) {
  ggplot(data) +
    geom_segment(aes(x = x1, y = y1, xend = x2, yend = y2, color = group), size = 1.5) +  # Draw each side
    geom_point(aes(x = x1, y = y1), size = 3, color = "black") +  # Show points
    geom_text(aes(x = x1, y = y1, label = labels), vjust = -1, hjust = -0.5, size = 6, fontface = "bold") +  # Label A, B, C, D
    scale_color_manual(values = color_map) +
    coord_fixed() +
    xlim(x_lim[1], x_lim[2]) +
    ylim(y_lim[1], y_lim[2]) +
    theme_minimal() +
    ggtitle(title)
}

# Expanded limits for full visibility
x_range <- c(-1, 7)  
y_range <- c(-3, 7)  

# Plot the original quadrilateral
p1 <- plot_quad(original_quad, "Original Quadrilateral", x_range, y_range)

# Plot the self-intersecting quadrilateral
p2 <- plot_quad(stretched_quad, "Self-Intersecting Quadrilateral", x_range, y_range)

# Display both plots
library(gridExtra)
grid.arrange(p1, p2, ncol = 2)