An algorithm to find bounding box of closed bezier curves?

Question

I\'m looking for an algorithm to find bounding box (max/min points) of a closed quadratic bezier curve in Cartesian axis:

input: C (a closed bezier curve)
out         


        

Answer 1

Timo-s first variant adapted to Objective-C

CGPoint CubicBezierPointAt(CGPoint p1, CGPoint p2, CGPoint p3, CGPoint p4, CGFloat t) {

   CGFloat x = CubicBezier(p1.x, p2.x, p3.x, p4.x, t);
   CGFloat y = CubicBezier(p1.y, p2.y, p3.y, p4.y, t);

   return CGPointMake(x, y);
}

// array containing TopLeft and BottomRight points for curve`s enclosing bounds
NSArray* CubicBezierExtremums(CGPoint p1, CGPoint p2, CGPoint p3, CGPoint p4) {

   CGFloat a, b, c, t, t1, t2, b2ac, sqrtb2ac;
   NSMutableArray *tValues = [NSMutableArray new];

   for (int i = 0; i < 2; i++) {
      if (i == 0) {
         a = 3 * (-p1.x + 3 * p2.x - 3 * p3.x + p4.x);
         b = 6 * (p1.x - 2 * p2.x +  p3.x);
         c = 3 * (p2.x - p1.x);
      }
      else {
         a = 3 * (-p1.y + 3 * p2.y - 3 * p3.y + p4.y);
         b = 6 * (p1.y - 2 * p2.y +  p3.y);
         c = 3 * (p2.y - p1.y);
      }

      if(ABS(a) < CGFLOAT_MIN) {// Numerical robustness
         if (ABS(b) < CGFLOAT_MIN) {// Numerical robustness
            continue;
         }

         t = -c / b;

         if (t > 0 && t < 1) {
            [tValues addObject:[NSNumber numberWithDouble:t]];
         }
         continue;
      }

      b2ac = pow(b, 2) - 4 * c * a;

      if (b2ac < 0) {
         continue;
      }

      sqrtb2ac = sqrt(b2ac);

      t1 = (-b + sqrtb2ac) / (2 * a);

      if (t1 > 0.0 && t1 < 1.0) {
         [tValues addObject:[NSNumber numberWithDouble:t1]];
      }

      t2 = (-b - sqrtb2ac) / (2 * a);

      if (t2 > 0.0 && t2 < 1.0) {
         [tValues addObject:[NSNumber numberWithDouble:t2]];
      }
   }

   int j = (int)tValues.count;

   CGFloat x = 0;
   CGFloat y = 0;
   NSMutableArray *xValues = [NSMutableArray new];
   NSMutableArray *yValues = [NSMutableArray new];

   while (j--) {
      t = [[tValues objectAtIndex:j] doubleValue];
      x = CubicBezier(p1.x, p2.x, p3.x, p4.x, t);
      y = CubicBezier(p1.y, p2.y, p3.y, p4.y, t);
      [xValues addObject:[NSNumber numberWithDouble:x]];
      [yValues addObject:[NSNumber numberWithDouble:y]];
   }

   [xValues addObject:[NSNumber numberWithDouble:p1.x]];
   [xValues addObject:[NSNumber numberWithDouble:p4.x]];
   [yValues addObject:[NSNumber numberWithDouble:p1.y]];
   [yValues addObject:[NSNumber numberWithDouble:p4.y]];

   //find minX, minY, maxX, maxY
   CGFloat minX = [[xValues valueForKeyPath:@"@min.self"] doubleValue];
   CGFloat minY = [[yValues valueForKeyPath:@"@min.self"] doubleValue];
   CGFloat maxX = [[xValues valueForKeyPath:@"@max.self"] doubleValue];
   CGFloat maxY = [[yValues valueForKeyPath:@"@max.self"] doubleValue];

   CGPoint origin = CGPointMake(minX, minY);
   CGPoint bottomRight = CGPointMake(maxX, maxY);

   NSArray *toReturn = [NSArray arrayWithObjects:
                        [NSValue valueWithCGPoint:origin],
                        [NSValue valueWithCGPoint:bottomRight],
                        nil];

   return toReturn;
}