Step by Step solution about How to resolve the algorithm Ramer-Douglas-Peucker line simplification step by step in the C programming language

This C code implements the Douglas-Peucker (DP) algorithm to simplify a given array of points, reducing the number of points while preserving the overall shape of the original curve. Below is a breakdown of the code:

1. Data Structures:

   • point_t: A C struct representing a point with x and y coordinates.

2. Functions:

   • perpendicular_distance: Calculates the perpendicular distance from a point p to a line segment defined by points p1 and p2.
   • douglas_peucker: Implements the Douglas-Peucker algorithm, taking an array of points, the number of points, a distance tolerance epsilon, and destination buffer dest to store the simplified points.

3. Algorithm Description:

   • Main Loop:
     • The algorithm iterates over all points between points[1] and points[n-1], except the endpoints.
     • For each point, it calculates the perpendicular distance to the line segment connecting points[0] and points[n-1].
     • It tracks the point with the maximum distance from this line segment and stores its index in index.
   • Recursive Calls:
     • If the maximum distance exceeds the tolerance epsilon, the algorithm recursively calls itself on two subsets of points:
       • points[0] to points[index+1]
       • points[index] to points[n-1]
     • The number of output points in each recursive call is returned and accumulated in the total count.
   • Base Case:
     • If the maximum distance is less than or equal to epsilon, the algorithm returns the two endpoints as the simplified curve.

4. print_points: A utility function to print the points in an array.

5. main Function:

   • Defines an array of sample points (points) and its length (len).
   • Invokes douglas_peucker to simplify the points with a tolerance of 1.0 and stores the result in out.
   • Prints the simplified points using print_points.

In summary, this code reduces the number of points in an input array while maintaining the original curve's essential shape. This technique is useful in various applications, such as data simplification, image compression, and computer graphics.

#include <assert.h>
#include <math.h>
#include <stdio.h>

typedef struct point_tag {
    double x;
    double y;
} point_t;

// Returns the distance from point p to the line between p1 and p2
double perpendicular_distance(point_t p, point_t p1, point_t p2) {
    double dx = p2.x - p1.x;
    double dy = p2.y - p1.y;
    double d = sqrt(dx * dx + dy * dy);
    return fabs(p.x * dy - p.y * dx + p2.x * p1.y - p2.y * p1.x)/d;
}

// Simplify an array of points using the Ramer–Douglas–Peucker algorithm.
// Returns the number of output points.
size_t douglas_peucker(const point_t* points, size_t n, double epsilon,
                       point_t* dest, size_t destlen) {
    assert(n >= 2);
    assert(epsilon >= 0);
    double max_dist = 0;
    size_t index = 0;
    for (size_t i = 1; i + 1 < n; ++i) {
        double dist = perpendicular_distance(points[i], points[0], points[n - 1]);
        if (dist > max_dist) {
            max_dist = dist;
            index = i;
        }
    }
    if (max_dist > epsilon) {
        size_t n1 = douglas_peucker(points, index + 1, epsilon, dest, destlen);
        if (destlen >= n1 - 1) {
            destlen -= n1 - 1;
            dest += n1 - 1;
        } else {
            destlen = 0;
        }
        size_t n2 = douglas_peucker(points + index, n - index, epsilon, dest, destlen);
        return n1 + n2 - 1;
    }
    if (destlen >= 2) {
        dest[0] = points[0];
        dest[1] = points[n - 1];
    }
    return 2;
}

void print_points(const point_t* points, size_t n) {
    for (size_t i = 0; i < n; ++i) {
        if (i > 0)
            printf(" ");
        printf("(%g, %g)", points[i].x, points[i].y);
    }
    printf("\n");
}

int main() {
    point_t points[] = {
        {0,0}, {1,0.1}, {2,-0.1}, {3,5}, {4,6},
        {5,7}, {6,8.1}, {7,9}, {8,9}, {9,9}
    };
    const size_t len = sizeof(points)/sizeof(points[0]);
    point_t out[len];
    size_t n = douglas_peucker(points, len, 1.0, out, len);
    print_points(out, n);
    return 0;
}