Getting any point along an NSBezier path

Question

For a program I'm writing, I need to be able to trace a virtual line (that is not straight) that an object must travel along. I was thinking to use NSBezierPath to draw the line, but I cannot find a way to get any point along the line, which I must do so I can move the object along it.

Can anyone suggest a way to find a point along an NSBezierPath? If thats not possible, can anyone suggest a method to do the above?

Answer 1

EDIT: The below code is still accurate, but there are much faster ways to calculate it. See Introduction to Fast Bezier and Even Faster Bezier.

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There are two ways to approach this. If you just need to move something along the line, use a CAKeyframeAnimation. This is pretty straightforward and you never need to calculate the points.

If on the other hand you actually need to know the point for some reason, you have to calculate the Bézier yourself. For an example, you can pull the sample code for Chapter 18 from iOS 5 Programming Pushing the Limits. (It is written for iOS, but it applies equally to Mac.) Look in CurvyTextView.m.

Given control points P0_ through P3_, and an offset between 0 and 1 (see below), pointForOffset: will give you the point along the path:

static double Bezier(double t, double P0, double P1, double P2,
                     double P3) {
  return 
                   pow(1-t, 3) *     P0
     + 3 *         pow(1-t, 2) * t * P1
     + 3 * (1-t) * pow(t,   2) *     P2
     +             pow(t,   3) *     P3;
}

- (CGPoint)pointForOffset:(double)t {
  double x = Bezier(t, P0_.x, P1_.x, P2_.x, P3_.x);
  double y = Bezier(t, P0_.y, P1_.y, P2_.y, P3_.y);
  return CGPointMake(x, y);
}

NOTE: This code violates one of my cardinal rules of always using accessors rather than accessing ivars directly. It's because in it's called many thousands of times, and eliminating the method call has a significant performance impact.

"Offset" is not a trivial thing to work out. It does not proceed linearly along the curve. If you need evenly spaced points along the curve, you'll need to calculate the correct offset for each point. This is done with this routine:

// Simplistic routine to find the offset along Bezier that is
// aDistance away from aPoint. anOffset is the offset used to
// generate aPoint, and saves us the trouble of recalculating it
// This routine just walks forward until it finds a point at least
// aDistance away. Good optimizations here would reduce the number
// of guesses, but this is tricky since if we go too far out, the
// curve might loop back on leading to incorrect results. Tuning
// kStep is good start.
- (double)offsetAtDistance:(double)aDistance 
                 fromPoint:(CGPoint)aPoint
                    offset:(double)anOffset {
  const double kStep = 0.001; // 0.0001 - 0.001 work well
  double newDistance = 0;
  double newOffset = anOffset + kStep;
  while (newDistance <= aDistance && newOffset < 1.0) {
    newOffset += kStep;
    newDistance = Distance(aPoint, 
                           [self pointForOffset:newOffset]);
  }
  return newOffset;
}

I leave Distance() as an exercise for the reader, but it's in the example code of course.

The referenced code also provides BezierPrime() and angleForOffset: if you need those. Chapter 18 of iOS:PTL covers this in more detail as part of a discussion on how to draw text along an arbitrary path.