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!)

     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 quadrilaterallibrary(ggplot2)# Define coordinates for original quadrilateraloriginal_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 sidecolor_map <- c("AB" = "red", "BC" = "blue", "CD" = "green", "DA" = "purple")# Function to plot the quadrilateralplot_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 visibilityx_range <- c(-1, 7)  y_range <- c(-3, 7)  # Plot the original quadrilateralp1 <- plot_quad(original_quad, "Original Quadrilateral", x_range, y_range)# Plot the self-intersecting quadrilateralp2 <- plot_quad(stretched_quad, "Self-Intersecting Quadrilateral", x_range, y_range)# Display both plotslibrary(gridExtra)grid.arrange(p1, p2, ncol = 2)

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