Algorithm and implementation of finding convex hull in C language
The convex hull problem is an important problem in computational geometry, which describes the smallest convex polygon in a point set. In this article, we will explore the algorithm and its implementation using C language to solve the convex hull problem.
Algorithm and implementation of finding convex hull in C language
The key to the convex hull algorithm is how to determine whether a point is on the convex hull. For a given set of points, we can choose one point as the starting point and connect other points to it in a certain order. If the connecting lines of a point are within the boundary of the convex hull, then the point is on the convex hull. Based on this idea, we can design the following algorithm to solve the convex hull problem.
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Find the leftmost point P0 in the point set as the starting point.
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Sort the other points in the point set according to their polar angle with P0.
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Connect the sorted points in order to form a convex polygon.
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Traverse the connecting line and determine whether each point is within the boundary of the convex hull.
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If all points are within the boundaries of the convex hull, the algorithm ends; otherwise, the farthest point is deleted from the convex hull and returns to step 4.
The following is a sample code implemented in C language:
#include
// Define a point structure
typedef struct {
int x;
int y;
} Point;
// Calculate the square of the distance between two points
int distance(Point p1, Point p2) {
int dx = p1.x - p2.x;
int dy = p1.y - p2.y;
return dx * dx + dy * dy;
}
// Determine whether point p is within the boundary of the convex hull
int inConvexHull(Point p, Point hull[], int size) {
// Traverse the boundaries of the convex hull
for (int i = 0; i < size - 1; i++) {
Point p1 = hull[i];
Point p2 = hull[i + 1];
// Calculate the distance from the boundary of the convex hull
int d1 = (p1.x - p.x) * (p2.y - p.y) - (p1.y - p.y) * (p2.x - p.x);
int d2 = distance(p1, p2);
// If the distance is less than 0, the point is outside the boundary of the convex hull
if (d1 < 0) {
return 0;
}
// If the distance is equal to 0, it means that the point is on the boundary of the convex hull
if ( d1 == 0 & & d2 >= distance ( p1 , p )) {
return 0;
}
}
return 1;
}
// Algorithm for finding convex hull
void convexHull(Point points[], int n) {
// Find the leftmost point P0
int leftmost = 0;
for (int i = 1; i < n; i++) {
if (points[i].x < points[leftmost].x) {
leftmost = i;
}
}
// Sort other points by polar angle
// The specific implementation of the sorting algorithm is omitted here.
// Connect the points to form a convex polygon
int count = 0;
Point hull[n];
hull[count++] = points[leftmost];
int next;
do {
next = (leftmost + 1) % n;
for (int i = 0; i < n; i++) {
if (i != leftmost && i != next && inConvexHull(points[i], hull, count)) {
next = i;
}
}
hull[count++] = points[next];
leftmost = next;
} while (leftmost != 0);
// Output the points of the convex hull
for (int i = 0; i < count; i++) {
printf(\d, %d) \ hull[i].x, hull[i].y);
}
printf(\n}
int main() {
// Suppose we have the following point set
Point points[] = { {0, 3}, {1, 1}, {2, 2}, {4, 4},
{0, 0}, {1, 2}, {3, 1}, {3, 3}};
int n = sizeof(points) / sizeof(points[0]);
// Call the convex hull algorithm
convexHull(points, n);
return 0;
}
Through the above algorithm and implementation, we can find the convex hull of a given point set. The time complexity of this algorithm is O(n^2), where n is the size of the point set. The key to the algorithm is to determine whether a point is within the boundary of the convex hull. This determination can be effectively realized through distance calculation and comparison.
To sum up, the algorithm and implementation of finding the convex hull in C language are based on the connection and position judgment of points. By selecting the starting point, sorting by polar angle, connecting the points and judging whether the point is within the boundary of the convex hull, we can obtain the convex hull of the point set. This algorithm has wide applications in computational geometry and graphics processing. I hope the explanation in this article will be helpful to readers.
Part of the code is transferred from: https://www.ktiao.com/c/2023-08/254131.html