Minimum Bounding Polygon (Convex Hull; Smallest Enclosing A Set of Points) — With C Code Sample
©2009 Darel Rex Finley.  This complete article, unmodified, may be freely distributed for educational purposes.

This C code sample takes a set of 2D points, like this:

and obtains the smallest enclosing polygon; that is, the shortest polygon that encloses all of the points, like so:

The same technique also works to find the smallest (shortest) polygon around a set of polygons — just pass all the corners of the polygons into the function as one big set of points.

This code has not been thoroughly tested, so if you encounter any problems with it, please do let me know.  This function should have no trouble with multiple points having the same coordinates — it will still return a simple polygon with no repeated corners.  If all of your points have the same coordinates, the resultant polygon will have a single corner.

//  public-domain code by Darel Rex Finley, January 2009

#define  CIRCLE_RADIANS  6.283185307179586476925286766559

//  Determines the radian angle of the specified point (as it relates to the origin).
//  Warning:  Do not pass zero in both parameters, as this will cause division-by-zero.

double angleOf(double x, double y) {

  double  dist=sqrt(x*x+y*y) ;

  if (y>=0.) return acos( x/dist)                  ;
  else       return acos(-x/dist)+.5*CIRCLE_RADIANS; }

//  Pass in a set of 2D points in x,y,points.  Returns a polygon in polyX,polyY,polyCorners.
//  To be safe, polyX and polyY should have enough space to store all the points passed in x,y,points.

void findSmallestPolygon(double *x, double *y, long points, double *polyX, double *polyY, long *polyCorners) {

  double  newX=x[0], newY=y[0], xDif, yDif, oldAngle=.5*CIRCLE_RADIANS, newAngle, angleDif, minAngleDif ;
  long    i ;

  //  Find a starting point.
  for (i=0; i<points; i++) if (y[i]>newY || y[i]==newY && x[i]<newX) {
    newX=x[i]; newY=y[i]; }

  //  Polygon-construction loop.
  while (!(*polyCorners) || newX!=polyX[0] || newY!=polyY[0]) {
    polyY[*polyCorners]=newY; minAngleDif=CIRCLE_RADIANS;
    for (i=0; i<points; i++) {
      if (xDif || yDif) {
        newAngle=angleOf(xDif,yDif);     angleDif =oldAngle-newAngle;
        while (angleDif< 0.            ) angleDif+=CIRCLE_RADIANS;
        while (angleDif>=CIRCLE_RADIANS) angleDif-=CIRCLE_RADIANS;
        if (angleDif<minAngleDif) {
          minAngleDif=angleDif; newX=x[i]; newY=y[i]; }}}
    (*polyCorners)++; oldAngle+=.5*CIRCLE_RADIANS-minAngleDif; }}

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