How to determine if a point is in a 2D triangle?

Determining if a Point Lies Within a 2D Triangle

Explore various algorithms to accurately check if a given 2D point is inside, outside, or on the edge of a triangle.

Determining whether a point lies within a 2D triangle is a fundamental problem in computational geometry, graphics, and game development. This article explores several common and effective algorithms to solve this problem, providing both conceptual understanding and practical code examples. We'll cover methods like the Barycentric Coordinate System, the Same-Side Technique, and the Cross Product Method, each with its own advantages and use cases.

Understanding the Problem: Point-in-Triangle

Given three vertices of a triangle, P1, P2, P3, and a test point P, we need to ascertain if P is located strictly inside the triangle, strictly outside, or exactly on one of its edges or vertices. The definition of 'inside' can sometimes vary (e.g., including edges or not), but typically, we aim for a clear distinction.

graph TD
    A[Start] --> B{Given: P1, P2, P3 (Triangle Vertices) and P (Test Point)}
    B --> C{Choose an Algorithm}
    C --> D["Barycentric Coordinates"]
    C --> E["Same-Side Technique"]
    C --> F["Cross Product Method"]
    D --> G{Calculate Barycentric Coordinates (u, v, w)}
    E --> H{Check if P is on the 'same side' of all edges}
    F --> I{Calculate Cross Products for all edges}
    G --> J{Are u, v, w all between 0 and 1?}
    H --> J
    I --> J
    J --> K{Result: Point is Inside}
    J --> L{Result: Point is Outside/On Edge}
    K --> M[End]
    L --> M

Flowchart of Point-in-Triangle Determination Process

Method 1: Barycentric Coordinates

The Barycentric Coordinate System provides a powerful way to represent any point within a triangle as a weighted average of its vertices. For a point P to be inside a triangle defined by vertices P1, P2, P3, its barycentric coordinates (u, v, w) must all be non-negative and sum to 1. Specifically, if 0 <= u <= 1, 0 <= v <= 1, 0 <= w <= 1, and u + v + w = 1, then P is inside or on the edge of the triangle. If any coordinate is exactly 0 or 1, the point lies on an edge or vertex. If any coordinate is negative, the point is outside.

def point_in_triangle_barycentric(p, p1, p2, p3):
    # Unpack coordinates
    x, y = p
    x1, y1 = p1
    x2, y2 = p2
    x3, y3 = p3

    # Calculate area of the main triangle (P1P2P3)
    # This is 2 * signed area, or determinant
    denom = (y2 - y3) * (x1 - x3) + (x3 - x2) * (y1 - y3)
    if denom == 0: # Degenerate triangle
        return False

    # Calculate barycentric coordinates
    u = ((y2 - y3) * (x - x3) + (x3 - x2) * (y - y3)) / denom
    v = ((y3 - y1) * (x - x3) + (x1 - x3) * (y - y3)) / denom
    w = 1 - u - v

    # Check if point is inside or on edge
    return 0 <= u <= 1 and 0 <= v <= 1 and 0 <= w <= 1

# Example usage:
p1 = (0, 0)
p2 = (5, 0)
p3 = (2.5, 5)

point_inside = (2, 2)
point_outside = (6, 1)
point_on_edge = (2.5, 0)

print(f"Point {point_inside} inside? {point_in_triangle_barycentric(point_inside, p1, p2, p3)}")
print(f"Point {point_outside} inside? {point_in_triangle_barycentric(point_outside, p1, p2, p3)}")
print(f"Point {point_on_edge} inside? {point_in_triangle_barycentric(point_on_edge, p1, p2, p3)}")

Python implementation of the Barycentric Coordinates method.

Method 2: Same-Side Technique

The Same-Side Technique relies on the observation that a point P is inside a triangle if and only if it lies on the same side of all three lines formed by the triangle's edges. To determine which 'side' a point is on, we can use the sign of the cross product (or a 2D equivalent) or a simple 2D orientation test. If the point P is consistently on the 'left' or 'right' side of each directed edge (P1P2, P2P3, P3P1), then it's inside.

function sign(p1, p2, p3) {
    return (p1.x - p3.x) * (p2.y - p3.y) - (p2.x - p3.x) * (p1.y - p3.y);
}

function pointInTriangleSameSide(pt, v1, v2, v3) {
    let d1, d2, d3;
    let has_neg, has_pos;

    d1 = sign(pt, v1, v2);
    d2 = sign(pt, v2, v3);
    d3 = sign(pt, v3, v1);

    has_neg = (d1 < 0) || (d2 < 0) || (d3 < 0);
    has_pos = (d1 > 0) || (d2 > 0) || (d3 > 0);

    // If all signs are the same (all positive or all negative), the point is inside.
    // If any sign is zero, the point is on an edge.
    return !(has_neg && has_pos);
}

// Example usage:
const v1 = { x: 0, y: 0 };
const v2 = { x: 5, y: 0 };
const v3 = { x: 2.5, y: 5 };

const pointInside = { x: 2, y: 2 };
const pointOutside = { x: 6, y: 1 };
const pointOnEdge = { x: 2.5, y: 0 };

console.log(`Point (${pointInside.x}, ${pointInside.y}) inside? ${pointInTriangleSameSide(pointInside, v1, v2, v3)}`);
console.log(`Point (${pointOutside.x}, ${pointOutside.y}) inside? ${pointInTriangleSameSide(pointOutside, v1, v2, v3)}`);
console.log(`Point (${pointOnEdge.x}, ${pointOnEdge.y}) inside? ${pointInTriangleSameSide(pointOnEdge, v1, v2, v3)}`);

JavaScript implementation of the Same-Side Technique.

Method 3: Cross Product Method

Similar to the Same-Side Technique, the Cross Product Method leverages vector math. For a point P to be inside a triangle P1P2P3, it must lie on the same side of each edge when traversing the triangle in a consistent direction (e.g., counter-clockwise). This can be checked by computing the Z-component of the 2D cross product for vectors formed by the edges and the test point. If all cross products have the same sign (or are zero), the point is inside or on an edge.

#include <iostream>

struct Point {
    double x, y;
};

// Function to compute the 2D cross product (z-component)
// (P2 - P1) x (P - P1)
double cross_product(Point p1, Point p2, Point p) {
    return (p2.x - p1.x) * (p.y - p1.y) - (p2.y - p1.y) * (p.x - p1.x);
}

bool point_in_triangle_cross_product(Point p, Point p1, Point p2, Point p3) {
    // Compute cross products for each edge
    double cp1 = cross_product(p1, p2, p);
    double cp2 = cross_product(p2, p3, p);
    double cp3 = cross_product(p3, p1, p);

    // Check if all cross products have the same sign (or are zero)
    // This means the point is on the same side of all edges
    bool s1 = cp1 >= 0;
    bool s2 = cp2 >= 0;
    bool s3 = cp3 >= 0;

    return (s1 == s2) && (s2 == s3);
}

int main() {
    Point p1 = {0, 0};
    Point p2 = {5, 0};
    Point p3 = {2.5, 5};

    Point point_inside = {2, 2};
    Point point_outside = {6, 1};
    Point point_on_edge = {2.5, 0};

    std::cout << "Point (" << point_inside.x << ", " << point_inside.y << ") inside? " 
              << (point_in_triangle_cross_product(point_inside, p1, p2, p3) ? "true" : "false") << std::endl;
    std::cout << "Point (" << point_outside.x << ", " << point_outside.y << ") inside? " 
              << (point_in_triangle_cross_product(point_outside, p1, p2, p3) ? "true" : "false") << std::endl;
    std::cout << "Point (" << point_on_edge.x << ", " << point_on_edge.y << ") inside? " 
              << (point_in_triangle_cross_product(point_on_edge, p1, p2, p3) ? "true" : "false") << std::endl;

    return 0;
}

C++ implementation of the Cross Product Method.

Choosing the Right Method

Each method has its strengths:

• Barycentric Coordinates: Excellent for interpolation, provides more information than just inside/outside. Can be slightly more computationally intensive due to divisions.
• Same-Side / Cross Product: Generally faster as they primarily involve multiplications and additions. Ideal when you only need a boolean (inside/outside) result. The Cross Product method is essentially a specific implementation of the Same-Side logic.

For most applications, the Same-Side or Cross Product methods are sufficient and often preferred for their simplicity and performance. If you need to interpolate values across the triangle, Barycentric Coordinates are the way to go.

1. Define Triangle Vertices

Establish the coordinates of the three vertices of your 2D triangle (e.g., P1(x1, y1), P2(x2, y2), P3(x3, y3)).

2. Define Test Point

Specify the coordinates of the point you wish to test (e.g., P(x, y)).

3. Select Algorithm

Choose an algorithm based on your needs: Barycentric for interpolation, Same-Side/Cross Product for simple inside/outside checks.

4. Implement and Test

Implement the chosen algorithm in your preferred programming language and test it with various points (inside, outside, and on edges) to ensure correctness.

5. Handle Edge Cases

Consider how your application should behave if the point lies exactly on an edge or vertex, and adjust the comparison operators (<, <=, >, >=) accordingly.