implementing easing function

Question

I'm trying to port and implement an easing function I found

EDIT

: I pasted in the wrong easing function, Sorry! Here is the correct one:

Math.easeOutQuart = function (t, b, c, d) {
    t /= d;
    t--;
    return -c * (t*t*t*t - 1) + b;
};

---

The language i'm using is not Flash or Actionscript. Here is my code:

ease:{outquart:{function(t as float,b as float,c as float,d as float) as float
        t=t/d
        t=t-1
        return -c * (t*t*t*t - 1) + b
    end function}}

I'm calling the function in a loop with:

EDIT2 - the calling function.

m.move is set to 1 or -1 for direction to move, or -5 +5 to move by 5 lengths. setspritemoves is called as often as possible, currently it is as fast as the system can call, but I could trigger the call on a millisecond timer.

setspritemoves:function()
                if m.move=1 then
                m.duration=1
                    if m.ishd then
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]+m.move*324
                        next i
                    else
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]+m.move*224
                        next i
                    end if                          
                else if m.move=5 then
                    m.duration=5
                    if m.ishd then
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]+m.move*324
                        next i
                    else
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]+m.move*224
                        next i
                    end if      
                else if m.move=-1 then
                m.duration=1
                    if m.ishd then
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]-m.move*324
                        next i
                    else
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]-m.move*224
                        next i
                    end if      
                else if m.move=-5 then
                    m.duration=5
                    if m.ishd then
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]-m.move*324
                        next i
                    else
                        for i=0 to m.spriteposx.count()-1
                            m.moveto[i]=m.spriteposx[i]-m.move*224
                        next i
                    end if
                end if
                end function

m.moveto[i] is the destination x coordinate, m.time is an integer I increment, m.duration is what I assume to be the amount of time I want the change to take to complete, m.spriteposx is the current position of the object I'm moving. [i] is the current sprite.

What should the increment value be for time what should the duration be, if I want to move 345 pixels in 1/2 second?

In all my experiments, I either overshoot by a huge factor, or only move a few pixels.

currently m.time is incremented by 1 every iteration, and m.duration is 100. I"ve tried all kinds of values and none seems to work consistently.

Answer 1

Why haven't you copied the logic across 1-1? The tween is a simple algorithm, it simply maps co-ordinates from b to b+c in a quartic fashion, i.e. b + c*t^4 where t gets values in the interval [0,1]. You can see by substitution that when t=0 the value is the initial value, b, and as t->1 the position is the required b+c.

That's the reason for the line t \= d, so d is an arbitrary duration and t, the time passed since the beginning of the animation gets a value in the aforementioned range. But you've done t=t-1 and taken negatives, etc. Why?

For example, moving 345px in 0.5s, you would have an initial position, b and c=345 assuming px is the units of measure. d=0.5 and you split the animation into intervals of a length of your choosing (depending on the power of the machine that will run the animation. Mobile devices aren't as powerful as desktops, so you choose a reasonable framerate under the circumstances). Let's say we choose 24 fps, so we split the interval into 0.5*24 = 12 frames, and call the function once every 1/24^th of a second, each time with t taking values of 1/24, 2/24, etc. If it's more comfortable to work not in seconds but in frames, then d=12 and t takes values 1,2,...,12. The calculations are the same either way.

Here's a nice example (click the box to run the demo), feel free to fiddle with the values:

http://jsfiddle.net/nKhxw/

Answer 2

Bezier functions

Borrowed from http://blog.greweb.fr/2012/02/bezier-curve-based-easing-functions-from-concept-to-implementation/

/**
* KeySpline - use bezier curve for transition easing function
* is inspired from Firefox's nsSMILKeySpline.cpp
* Usage:
* var spline = new KeySpline(0.25, 0.1, 0.25, 1.0)
* spline.get(x) => returns the easing value | x must be in [0, 1] range
*/
function KeySpline (mX1, mY1, mX2, mY2) {

  this.get = function(aX) {
    if (mX1 == mY1 && mX2 == mY2) return aX; // linear
    return CalcBezier(GetTForX(aX), mY1, mY2);
  }

  function A(aA1, aA2) { return 1.0 - 3.0 * aA2 + 3.0 * aA1; }
  function B(aA1, aA2) { return 3.0 * aA2 - 6.0 * aA1; }
  function C(aA1)      { return 3.0 * aA1; }

  // Returns x(t) given t, x1, and x2, or y(t) given t, y1, and y2.
  function CalcBezier(aT, aA1, aA2) {
    return ((A(aA1, aA2)*aT + B(aA1, aA2))*aT + C(aA1))*aT;
  }

  // Returns dx/dt given t, x1, and x2, or dy/dt given t, y1, and y2.
  function GetSlope(aT, aA1, aA2) {
    return 3.0 * A(aA1, aA2)*aT*aT + 2.0 * B(aA1, aA2) * aT + C(aA1);
  }

  function GetTForX(aX) {
    // Newton raphson iteration
    var aGuessT = aX;
    for (var i = 0; i < 4; ++i) {
      var currentSlope = GetSlope(aGuessT, mX1, mX2);
      if (currentSlope == 0.0) return aGuessT;
      var currentX = CalcBezier(aGuessT, mX1, mX2) - aX;
      aGuessT -= currentX / currentSlope;
    }
    return aGuessT;
  }
}

Aliases for common curves:

{
    "ease":        [0.25, 0.1, 0.25, 1.0], 
    "linear":      [0.00, 0.0, 1.00, 1.0],
    "ease-in":     [0.42, 0.0, 1.00, 1.0],
    "ease-out":    [0.00, 0.0, 0.58, 1.0],
    "ease-in-out": [0.42, 0.0, 0.58, 1.0]
}

Should be easy to make your own curves...

Answer 3

Thank you, Jonny!

Here is how to implement Bezier easing functions: C or Objective-C for iOS

// APPLE ORIGINAL TIMINGS:
//    linear        (0.00, 0.00), (0.00, 0.00), (1.00, 1.00), (1.00, 1.00)
//    easeIn        (0.00, 0.00), (0.42, 0.00), (1.00, 1.00), (1.00, 1.00)
//    easeOut       (0.00, 0.00), (0.00, 0.00), (0.58, 1.00), (1.00, 1.00)
//    easeInEaseOut (0.00, 0.00), (0.42, 0.00), (0.58, 1.00), (1.00, 1.00)
//    default       (0.00, 0.00), (0.25, 0.10), (0.25, 1.00), (1.00, 1.00)

+(double)defaultEase_Linear:(double)t
{
    return t;
}

// Замедление в начале
+(double)defaultEase_In:(double)t
{
    return [AnimationMath easeBezier_t:t

                              point0_x:0
                              point0_y:0

                              point1_x:0.42
                              point1_y:0

                              point2_x:1
                              point2_y:1

                              point3_x:1
                              point3_y:1];
}

// Замедление в конце
+(double)defaultEase_Out:(double)t
{
    return [AnimationMath easeBezier_t:t

                              point0_x:0
                              point0_y:0

                              point1_x:0
                              point1_y:0

                              point2_x:0.58
                              point2_y:1

                              point3_x:1
                              point3_y:1];
}

+(double)defaultEase_InOut:(double)t
{
    return [AnimationMath easeBezier_t:t

                              point0_x:0
                              point0_y:0

                              point1_x:0.42
                              point1_y:0

                              point2_x:0.58
                              point2_y:1

                              point3_x:1
                              point3_y:1];
}

+(double)defaultEase_default:(double)t
{
    return [AnimationMath easeBezier_t:t

                              point0_x:0
                              point0_y:0

                              point1_x:0.25
                              point1_y:0.1

                              point2_x:0.25
                              point2_y:1.0

                              point3_x:1
                              point3_y:1];
}


// For *better understanding* there is p1 and p2, because it is a Bezier curve from 0,0 to 1,0. So, you can remove p1 and p2 from this method, it is just for better understanding what's going on here

double ease_bezier_A(double aA1, double aA2) { return 1.0 - 3.0 * aA2 + 3.0 * aA1; }
double ease_bezier_B(double aA1, double aA2) { return 3.0 * aA2 - 6.0 * aA1; }
double ease_bezier_C(double aA1)      { return 3.0 * aA1; }

// Returns x(t) given t, x1, and x2, or y(t) given t, y1, and y2.
double ease_bezier_calc(double aT, double aA1, double aA2) {
    return ((ease_bezier_A(aA1, aA2)*aT + ease_bezier_B(aA1, aA2))*aT + ease_bezier_C(aA1))*aT;
}

// Returns dx/dt given t, x1, and x2, or dy/dt given t, y1, and y2.
double ease_bezier_get_slope(double aT, double aA1, double aA2) {
    return 3.0 * ease_bezier_A(aA1, aA2)*aT*aT + 2.0 * ease_bezier_B(aA1, aA2) * aT + ease_bezier_C(aA1);
}

double ease_bezier_get_t_for_x(double aX, double mX1, double mX2) {
    // Newton raphson iteration
    double aGuessT = aX;
    for (int i = 0; i < 4; ++i) {
        double currentSlope = ease_bezier_get_slope(aGuessT, mX1, mX2);
        if (currentSlope == 0.0) return aGuessT;
        double currentX = ease_bezier_calc(aGuessT, mX1, mX2) - aX;
        aGuessT -= currentX / currentSlope;
    }
    return aGuessT;
}



// Objective-C
// For ***better understanding*** there is p1 and p2, because it is a Bezier curve from 0,0 to 1,0. So, you can remove p1 and p2 from this method, it is just for better understanding what's going on here
// p1_x always = 0
// p1_y always = 0
// p2_x always = 1.0
// p2_y always = 1.0
+(double)easeBezier_t:(double)t
             point0_x:(double)point0_x point0_y:(double)point0_y
             point1_x:(double)point1_x point1_y:(double)point1_y
             point2_x:(double)point2_x point2_y:(double)point2_y
             point3_x:(double)point3_x point3_y:(double)point3_y
{
    if (point0_x != 0 || point0_y != 0 || point3_x != 1 || point3_y != 1) {
        [NSException raise:@"Error! Your bezier is wrong!!!" format:@""];
    }

    double v = ease_bezier_calc(ease_bezier_get_t_for_x(t, point1_x, point2_x), point1_y, point2_y);

    return v;
}