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

Ramer-Douglas-Peucker Algorithm for Line Simplification

This PHP code implements the Ramer-Douglas-Peucker (RDP) algorithm, which is used to simplify a polyline (a set of connected points) by removing unnecessary points while maintaining the overall shape.

Function perpendicular_distance

• Calculates the perpendicular distance between a point (pt) and a line defined by two points (line).
• It first normalizes the delta x and y of the line and calculates the magnitude.
• Then, it calculates the dot product of the normalized delta and a vector from the point to the start of the line.
• Finally, it scales the normalized delta by the dot product and subtracts it from the vector to get the perpendicular distance.

Function ramer_douglas_peucker

• Implements the RDP algorithm.
• It takes a list of points and an epsilon value as input.
• Finds the point with the maximum perpendicular distance from the line connecting the start and end points.
• If the maximum distance is greater than the epsilon, it recursively simplifies the two sublines formed by the maximum distance point.
• If the maximum distance is less than or equal to the epsilon, it returns a line with just the start and end points.

Example Usage

The code includes an example usage with a polyline and an epsilon value of 1.0. It simplifies the polyline using the RDP algorithm and prints the simplified result.

Explanation of the Example

The input polyline has 9 points. The algorithm first finds the point with the maximum perpendicular distance from the line connecting the start and end points. This point is the 4th point, with a distance of 1.02.

Since the distance is greater than the epsilon, the algorithm recursively simplifies the two sublines formed by the 4th point.

The first subline from the start to the 4th point is simplified to [0,0], [1,0.1], [2,-0.1], [3,5].

The second subline from the 4th point to the end is simplified to [3,5], [4,6], [5,7], [6,8.1], [7,9], [8,9], [9,9].

Finally, the algorithm returns the simplified result as a line with 8 points: [0,0], [1,0.1], [2,-0.1], [3,5], [4,6], [5,7], [6,8.1], [9,9].

This simplified line preserves the overall shape of the original polyline while removing unnecessary points.

function perpendicular_distance(array $pt, array $line) {
  // Calculate the normalized delta x and y of the line.
  $dx = $line[1][0] - $line[0][0];
  $dy = $line[1][1] - $line[0][1];
  $mag = sqrt($dx * $dx + $dy * $dy);
  if ($mag > 0) {
    $dx /= $mag;
    $dy /= $mag;
  }

  // Calculate dot product, projecting onto normalized direction.
  $pvx = $pt[0] - $line[0][0];
  $pvy = $pt[1] - $line[0][1];
  $pvdot = $dx * $pvx + $dy * $pvy;

  // Scale line direction vector and subtract from pv.
  $dsx = $pvdot * $dx;
  $dsy = $pvdot * $dy;
  $ax = $pvx - $dsx;
  $ay = $pvy - $dsy;

  return sqrt($ax * $ax + $ay * $ay);
}

function ramer_douglas_peucker(array $points, $epsilon) {
  if (count($points) < 2) {
    throw new InvalidArgumentException('Not enough points to simplify');
  }

  // Find the point with the maximum distance from the line between start/end.
  $dmax = 0;
  $index = 0;
  $end = count($points) - 1;
  $start_end_line = [$points[0], $points[$end]];
  for ($i = 1; $i < $end; $i++) {
    $dist = perpendicular_distance($points[$i], $start_end_line);
    if ($dist > $dmax) {
      $index = $i;
      $dmax = $dist;
    }
  }

  // If max distance is larger than epsilon, recursively simplify.
  if ($dmax > $epsilon) {
    $new_start = ramer_douglas_peucker(array_slice($points, 0, $index + 1), $epsilon);
    $new_end = ramer_douglas_peucker(array_slice($points, $index), $epsilon);
    array_pop($new_start);
    return array_merge($new_start, $new_end);
  }

  // Max distance is below epsilon, so return a line from with just the
  // start and end points.
  return [ $points[0], $points[$end]];
}

$polyline = [
  [0,0],
  [1,0.1],
  [2,-0.1],
  [3,5],
  [4,6],
  [5,7],
  [6,8.1],
  [7,9],
  [8,9],
  [9,9],
];

$result = ramer_douglas_peucker($polyline, 1.0);
print "Result:\n";
foreach ($result as $point) {
  print $point[0] . ',' . $point[1] . "\n";
}