Self-intersecting Quadrilateral

      A quadrilateral is a polygon having four sides, four angles, and four vertices. A polygon means that the figure is a closed shape, meaning the last line segment connects back to the first one, effectively enclosing an area.

      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!) It could also be irregularly shaped with no name.

      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).

      This self-intersecting quadrilateral is still four-sided and closed, so it is no less a quadrilateral than the original.

      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)

End