Using CAMediaTimingFunction to calculate value at time (t)

There's some simple math behind calculating the positions, you can read about it in every paper discussing Bézier curves, even on wikipedia. Anyway, I can relate to everybody who's in trouble to actually implement it in code, so I wrote this sample UIView as it's probably the easiest way to get you started.

#import "MBBezierView.h"

CGFloat bezierInterpolation(CGFloat t, CGFloat a, CGFloat b, CGFloat c, CGFloat d) {
    CGFloat t2 = t * t;
    CGFloat t3 = t2 * t;
    return a + (-a * 3 + t * (3 * a - a * t)) * t
    + (3 * b + t * (-6 * b + b * 3 * t)) * t
    + (c * 3 - c * 3 * t) * t2
    + d * t3;
}

@implementation MBBezierView

- (void)drawRect:(CGRect)rect {
    CGPoint p1, p2, p3, p4;
    p1 = CGPointMake(30, rect.size.height * 0.33);
    p2 = CGPointMake(CGRectGetMidX(rect), CGRectGetMinY(rect));
    p3 = CGPointMake(CGRectGetMidX(rect), CGRectGetMaxY(rect));
    p4 = CGPointMake(-30 + CGRectGetMaxX(rect), rect.size.height * 0.66);

    [[UIColor blackColor] set];
    [[UIBezierPath bezierPathWithRect:rect] fill];

    [[UIColor redColor] setStroke];

    UIBezierPath *bezierPath = [[[UIBezierPath alloc] init] autorelease];   
    [bezierPath moveToPoint:p1];
    [bezierPath addCurveToPoint:p4 controlPoint1:p2 controlPoint2:p3];
    [bezierPath stroke];

    [[UIColor brownColor] setStroke];
    for (CGFloat t = 0.0; t <= 1.00001; t += 0.05) {
        CGPoint point = CGPointMake(bezierInterpolation(t, p1.x, p2.x, p3.x, p4.x), bezierInterpolation(t, p1.y, p2.y, p3.y, p4.y));
        UIBezierPath *pointPath = [UIBezierPath bezierPathWithArcCenter:point radius:5 startAngle:0 endAngle:2*M_PI clockwise:YES];
        [pointPath stroke];
    }   
}

@end

This is what I get:

I found this:

Cocoa with Love - Parametric acceleration curves in Core Animation

But I think it can be made a little simpler and more readable by using blocks. So we can define a category on CAKeyframeAnimation that looks something like this:

CAKeyframeAnimation+Parametric.h:

// this should be a function that takes a time value between 
//  0.0 and 1.0 (where 0.0 is the beginning of the animation
//  and 1.0 is the end) and returns a scale factor where 0.0
//  would produce the starting value and 1.0 would produce the
//  ending value
typedef double (^KeyframeParametricBlock)(double);

@interface CAKeyframeAnimation (Parametric)

+ (id)animationWithKeyPath:(NSString *)path 
      function:(KeyframeParametricBlock)block
      fromValue:(double)fromValue
      toValue:(double)toValue;

CAKeyframeAnimation+Parametric.m:

@implementation CAKeyframeAnimation (Parametric)

+ (id)animationWithKeyPath:(NSString *)path 
      function:(KeyframeParametricBlock)block
      fromValue:(double)fromValue
      toValue:(double)toValue {
  // get a keyframe animation to set up
  CAKeyframeAnimation *animation = 
    [CAKeyframeAnimation animationWithKeyPath:path];
  // break the time into steps
  //  (the more steps, the smoother the animation)
  NSUInteger steps = 100;
  NSMutableArray *values = [NSMutableArray arrayWithCapacity:steps];
  double time = 0.0;
  double timeStep = 1.0 / (double)(steps - 1);
  for(NSUInteger i = 0; i < steps; i++) {
    double value = fromValue + (block(time) * (toValue - fromValue));
    [values addObject:[NSNumber numberWithDouble:value]];
    time += timeStep;
  }
  // we want linear animation between keyframes, with equal time steps
  animation.calculationMode = kCAAnimationLinear;
  // set keyframes and we're done
  [animation setValues:values];
  return(animation);
}

@end

Now usage will look something like this:

// define a parametric function
KeyframeParametricBlock function = ^double(double time) {
  return(1.0 - pow((1.0 - time), 2.0));
};

if (layer) {
  [CATransaction begin];
    [CATransaction 
      setValue:[NSNumber numberWithFloat:2.5]
      forKey:kCATransactionAnimationDuration];

    // make an animation
    CAAnimation *drop = [CAKeyframeAnimation 
      animationWithKeyPath:@"position.y"
      function:function fromValue:30.0 toValue:450.0];
    // use it
    [layer addAnimation:drop forKey:@"position"];

  [CATransaction commit];
}

I know it might not be quite as simple as what you wanted, but it's a start.