Point inside a rectangle - On which side of the diagonal is it?

Point Inside a Rectangle: Determining Diagonal Side

Learn how to mathematically determine which side of a rectangle's diagonal a given point lies on, using vector cross products and geometric principles.

When working with geometric problems, it's often necessary to determine the relative position of a point to a line. For a rectangle, a common scenario involves its diagonals. This article will guide you through the process of figuring out whether a point lies to one side or the other of a specific diagonal, using vector math and cross products. This technique is fundamental in computer graphics, collision detection, and various computational geometry tasks.

Understanding the Problem

Consider a rectangle defined by four vertices. We'll focus on one of its diagonals. A point can either be on the diagonal, or on one of the two sides it divides the rectangle into. Our goal is to establish a robust method to distinguish between these two sides. We'll assume a 2D coordinate system for simplicity.

flowchart TD
    A[Define Rectangle Vertices] --> B{Choose a Diagonal}
    B --> C[Identify Diagonal Endpoints P1, P2]
    C --> D[Define Point Q(x,y)]
    D --> E{Calculate Vector Cross Product}
    E --> F{Interpret Result}
    F --> G1[Point is on one side]
    F --> G2[Point is on the other side]
    F --> G3[Point is on the diagonal]
    G1 & G2 & G3 --> H[End]

Flowchart for determining a point's position relative to a diagonal.

Mathematical Approach: Vector Cross Product

The most effective way to determine which side of a line a point lies on in 2D is by using the concept of the vector cross product (or more accurately, its 2D equivalent, the scalar 'z' component of a 3D cross product). Given two vectors ( \vec{A} ) and ( \vec{B} ), their 2D cross product is calculated as ( A_x B_y - A_y B_x ). The sign of this result tells us the orientation of ( \vec{B} ) relative to ( \vec{A} ).

Let the diagonal be defined by points ( P_1(x_1, y_1) ) and ( P_2(x_2, y_2) ). Let the point in question be ( Q(x_q, y_q) ).

1. Form Vector 1 (Diagonal Vector): ( \vec{V_1} = P_2 - P_1 = (x_2 - x_1, y_2 - y_1) )
2. Form Vector 2 (Point Vector): ( \vec{V_2} = Q - P_1 = (x_q - x_1, y_q - y_1) )

Now, calculate the 2D cross product of ( \vec{V_1} ) and ( \vec{V_2} ): ( \text{Cross Product} = (x_2 - x_1)(y_q - y_1) - (y_2 - y_1)(x_q - x_1) )

Interpretation of the result:

• If the result is positive, the point ( Q ) is to one side (e.g., 'left' if ( \vec{V_1} ) points 'forward').
• If the result is negative, the point ( Q ) is to the other side (e.g., 'right').
• If the result is zero, the point ( Q ) lies exactly on the line defined by the diagonal.

def cross_product_2d(v1_x, v1_y, v2_x, v2_y):
    return v1_x * v2_y - v1_y * v2_x

def point_side_of_diagonal(p1, p2, q):
    # p1, p2 are diagonal endpoints (x, y) tuples
    # q is the point in question (x, y) tuple

    # Vector 1: P1 to P2 (the diagonal vector)
    v1_x = p2[0] - p1[0]
    v1_y = p2[1] - p1[1]

    # Vector 2: P1 to Q (vector from diagonal start to point)
    v2_x = q[0] - p1[0]
    v2_y = q[1] - p1[1]

    cp = cross_product_2d(v1_x, v1_y, v2_x, v2_y)

    if cp > 0:
        return "Left Side" # Or 'Side A'
    elif cp < 0:
        return "Right Side" # Or 'Side B'
    else:
        return "On Diagonal"

# Example Usage:
# Rectangle vertices: (0,0), (5,0), (5,3), (0,3)
# Diagonal from (0,0) to (5,3)
p1_diag = (0, 0)
p2_diag = (5, 3)

point_q1 = (2, 1) # Should be on one side
point_q2 = (4, 2) # Should be on the other side
point_q3 = (2.5, 1.5) # Should be on the diagonal

print(f"Point {point_q1} is: {point_side_of_diagonal(p1_diag, p2_diag, point_q1)}")
print(f"Point {point_q2} is: {point_side_of_diagonal(p1_diag, p2_diag, point_q2)}")
print(f"Point {point_q3} is: {point_side_of_diagonal(p1_diag, p2_diag, point_q3)}")

# Another diagonal: (0,3) to (5,0)
p1_diag_alt = (0, 3)
p2_diag_alt = (5, 0)

point_q4 = (1, 1) # Test with the other diagonal
print(f"Point {point_q4} relative to {p1_diag_alt}-{p2_diag_alt}: {point_side_of_diagonal(p1_diag_alt, p2_diag_alt, point_q4)}")

Python implementation for determining a point's side relative to a diagonal.

Handling Edge Cases and Considerations

While the cross product method is robust, consider these points:

• Floating Point Precision: When comparing the cross product result to zero, use a small epsilon value (e.g., abs(cp) < 1e-9) instead of direct cp == 0 to account for floating-point inaccuracies.
• Point within Rectangle: This method only tells you which side of the infinite line defined by the diagonal the point lies on. It does not inherently check if the point is within the rectangle's bounds. For that, you would need additional checks, such as checking if the point is within the min/max X and Y coordinates of the rectangle, or using a point-in-polygon test.
• Diagonal Choice: A rectangle has two diagonals. Ensure you are using the correct diagonal for your specific problem. The method works identically for both, but the 'sides' will be relative to the chosen diagonal.