Recursive Algorithms in Computational Geometry

Recursion Algorithms: Divide and Conquer in Computational Geometry

Introduction

Recursion algorithms play a vital role in problem-solving, especially in computational geometry. These algorithms follow the divide and conquer approach, which involves breaking a complex problem into smaller and simpler subproblems until they become easier to solve. In this blog post, we will explore various recursive algorithms used in computational geometry and understand how they work.

Point in a Polygon

The "Point in a Polygon" problem is a classical problem in computational geometry, where we are given a polygon and a point and we need to determine whether the point lies inside the polygon or not. To solve this problem recursively, we can follow these steps:

1. Divide the polygon into smaller sub-polygons or triangles.
2. Check if the point lies within any of the sub-polygons or triangles. If it does, recursively apply the same process to that sub-polygon or triangle.
3. Repeat this process until we find a sub-polygon or triangle that either contains the point or does not intersect it.

Here's an example implementation in Python:

def point_in_polygon(polygon, point):
    if len(polygon) < 3:
        return False

    for i in range(len(polygon)):
        p1 = polygon[i]
        p2 = polygon[(i + 1) % len(polygon)]

        if (p1[1] > point[1]) != (p2[1] > point[1]) and point[0] < (p2[0] - p1[0]) * (point[1] - p1[1]) / (p2[1] - p1[1]) + p1[0]:
            return True
    
    return False

In this code snippet, we check if the given point is on the left or right side of each edge of the polygon. If the number of intersections with the edges is even, the point lies outside the polygon. Otherwise, it lies inside the polygon.

Convex Hull

Another important problem in computational geometry is finding the convex hull of a set of points. The convex hull is the smallest convex polygon that contains all the given points. To solve this problem recursively, we can follow these steps:

1. Choose a point from the given set of points. This point will always be part of the convex hull.
2. Divide the set of points into two subsets based on their position relative to the line formed by the chosen point and the current point under consideration.
3. Recursively apply the same process to each subset of points until we obtain the convex hull.

Here's a recursive implementation in C++:

struct Point {
    int x, y;
};

bool compare_points(Point a, Point b) {
    if (a.x == b.x) {
        return a.y < b.y;
    }
    return a.x < b.x;
}

int orientation(Point p, Point q, Point r) {
    int val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y);

    if (val == 0) {
        return 0;
    }
    return (val > 0) ? 1 : 2;
}

void find_convex_hull(Point points[], int n, vector<Point>& hull) {
    if (n < 3) {
        return;
    }

    int leftmost_index = 0;
    for (int i = 1; i < n; i++) {
        if (points[i].x < points[leftmost_index].x) {
            leftmost_index = i;
        }
    }

    int current_point = leftmost_index;
    int next_point;
    do {
        hull.push_back(points[current_point]);
        next_point = (current_point + 1) % n;

        for (int i = 0; i < n; i++) {
            if (orientation(points[current_point], points[i], points[next_point]) == 2) {
                next_point = i;
            }
        }

        current_point = next_point;
    } while (current_point != leftmost_index);
}

This code snippet demonstrates the use of the Graham's scan algorithm to find the convex hull. The algorithm starts with the leftmost point and repeatedly chooses the point with the smallest counterclockwise angle until it reaches the starting point again.

Conclusion

In this blog post, we delved into the world of recursion algorithms in the field of computational geometry. We explored the "Point in a Polygon" problem and the convex hull problem, providing detailed explanations and code examples for a better understanding. By implementing these recursive algorithms correctly, programmers can efficiently solve complex problems in computational geometry.

Happy coding!