Interpolate

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For the complete details on all available functions please read the doc comments. If you have any questions feel free to contact me directly at fernando@cpudreams.com.
 
For the complete details on all available functions please read the doc comments. If you have any questions feel free to contact me directly at fernando@cpudreams.com.

Revision as of 20:09, 2 January 2010

Author: Fernando Zapata (fernando@cpudreams.com)

Contents

Description

Interpolation utility functions for easing, Bezier splines, and Catmull-Rom splines. Provides consistent calling conventions across these three interpolation types. Provides low level access via individual easing functions, for example EaseInOutCirc(), Bezier(), and CatmullRom(). Provides high level access using sequence generators, NewEase(), NewBezier(), and NewCatmullRom(). Functionality is available at different levels of abstraction, making the functions easy to use and making your own high level abstractions easy to build.

Usage

You can use the low level functions similar to how you might use Unity's built-in Mathf.Lerp().

<javascript> var start = 0.0; var distance = 3.0; var duration = 2.0; private var elapsedTime = 0.0;

function Update() {

   transform.position.y = Interpolate.EaseOutSine(start, distance,
                                                  elapsedTime, duration);
   elapsedTime += Time.deltaTime;

} </javascript>

Instead of hard coding the easing function you can use the Ease(EaseType) function to look up a concrete easing function.

<javascript> // set using Unity's property inspector to any easing type (ex. EaseInCirc) var easeType : EaseType; var ease : Function;

function Awake() {

 ease = Interpolate.Ease(easeType); // get easing function by type
  // ease can now be used:
 // transform.position.y = ease(start, distance, elapsedTime, duration);

} </javascript>

You can also use higher level sequence generator functions to quickly build a reusable component. For example, this SplinePath component will move a GameObject smoothly along a path using Catmull-Rom to make sure the GameObject passes through each control point. Using the interpolation utility function this component takes less than ten lines of code (minus the visualization function OnDrawGizmos).

<javascript> // SplinePath.js var path : Transform[]; // path's control points var loop : boolean; // number of nodes to generate between path nodes, to smooth out the path var betweenNodeCount : int; private var nodes : IEnumerator;

function Awake() {

 nodes = Interpolate.NewCatmullRom(path, betweenNodeCount, loop);

}

function Update() {

 if (nodes.MoveNext()) {
   transform.position = nodes.Current;
 }

}

// optional, use gizmos to draw the path in the editor function OnDrawGizmos() {

 if (path && path.length >= 2) {
   // draw control points
   for (var i = 0; i < path.length; i++) {
     Gizmos.DrawWireSphere(path[i].position, 0.15);
   }
   // draw spline curve using line segments
   var sequence = Interpolate.NewCatmullRom(path, betweenNodeCount, loop);
   var firstPoint = path[0].position;
   var segmentStart = firstPoint;
   sequence.MoveNext(); // skip the first point
   // use "for in" syntax instead of sequence.MoveNext() when convenient
   for (var segmentEnd in sequence) {
     Gizmos.DrawLine(segmentStart, segmentEnd);
     segmentStart = segmentEnd;
     // prevent infinite loop, when attribute loop == true
     if (segmentStart == firstPoint) { break; }
   }
 }

} </javascript> SplinePath.png


For the complete details on all available functions please read the doc comments. If you have any questions feel free to contact me directly at fernando@cpudreams.com.

Code

The code is organized top-down for easy reading.

Interpolate.js

<javascript>

  1. pragma strict

/**

* Interpolation utility functions: easing, bezier, and catmull-rom.
* Consider using Unity's Animation curve editor and AnimationCurve class
* before scripting the desired behaviour using this utility.
*
* Interpolation functionality available at different levels of abstraction.
* Low level access via individual easing functions (ex. EaseInOutCirc),
* Bezier(), and CatmullRom(). High level access using sequence generators,
* NewEase(), NewBezier(), and NewCatmullRom().
*
* Sequence generators are typically used as follows:
*
* var sequence = Interpolate.New[Ease|Bezier|CatmulRom](configuration);
* for (var newPoint in sequence) {
*   transform.position = newPoint;
*   yield;
* }
*
* Or:
*
* var sequence = Interpolate.New[Ease|Bezier|CatmulRom](configuration);
* function Update() {
*   if (sequence.MoveNext()) {
*     transform.position = sequence.Current;
*   }
* }
*
* The low level functions work similarly to Unity's built in Lerp and it is
* up to you to track and pass in elapsedTime and duration on every call. The
* functions take this form (or the logical equivalent for Bezier() and
* CatmullRom()).
*
* transform.position = ease(start, distance, elapsedTime, duration);
*
* For convenience in configuration you can use the Ease(EaseType) function to
* look up a concrete easing function:
*
* var easeType : EaseType; // set using Unity's property inspector
* var ease : Function; // easing of a particular EaseType
* function Awake() {
*   ease = Interpolate.Ease(easeType);
* }
*
* @author Fernando Zapata (fernando@cpudreams.com)
*/

/**

* Different methods of easing interpolation.
*/

enum EaseType {

 Linear,
 EaseInQuad,
 EaseOutQuad,
 EaseInOutQuad,
 EaseInCubic,
 EaseOutCubic,
 EaseInOutCubic,
 EaseInQuart,
 EaseOutQuart,
 EaseInOutQuart,
 EaseInQuint,
 EaseOutQuint,
 EaseInOutQuint,
 EaseInSine,
 EaseOutSine,
 EaseInOutSine,
 EaseInExpo,
 EaseOutExpo,
 EaseInOutExpo,
 EaseInCirc,
 EaseOutCirc,
 EaseInOutCirc

}

/**

* Returns sequence generator from start to end over duration using the
* given easing function. The sequence is generated as it is accessed
* using the Time.deltaTime to calculate the portion of duration that has
* elapsed.
*/

static function NewEase(ease : Function, start : Vector3,

                       end : Vector3, duration : float) : IEnumerator {
 var timer = NewTimer(duration);
 return NewEase(ease, start, end, duration, timer);

}

/**

* Instead of easing based on time, generate n interpolated points (slices)
* between the start and end positions.
*/

static function NewEase(ease : Function, start : Vector3,

                       end : Vector3, slices : int) : IEnumerator {
 var counter = NewCounter(0, slices + 1, 1);
 return NewEase(ease, start, end, slices + 1, counter);

}

/**

* Generic easing sequence generator used to implement the time and
* slice variants. Normally you would not use this function directly.
*/

static function NewEase(ease : Function, start : Vector3, end : Vector3,

                       total : float, driver : IEnumerator) : IEnumerator {
 var distance = end - start;
 for (var i in driver) {
   yield Ease(ease, start, distance, i, total);
 }

}

/**

* Vector3 interpolation using given easing method. Easing is done independently
* on all three vector axis.
*/

static function Ease(ease : Function,

                    start : Vector3, distance : Vector3,
                    elapsedTime : float, duration : float) : Vector3 {
 start.x = ease(start.x, distance.x, elapsedTime, duration);
 start.y = ease(start.y, distance.y, elapsedTime, duration);
 start.z = ease(start.z, distance.z, elapsedTime, duration);
 return start;

}

/**

* Returns the static method that implements the given easing type for scalars.
* Use this method to easily switch between easing interpolation types.
*
* All easing methods clamp elapsedTime so that it is always <= duration.
*
* var ease = Interpolate.Ease(EaseType.EaseInQuad);
* i = ease(start, distance, elapsedTime, duration);
*/

static function Ease(type : EaseType) : Function {

 // Source Flash easing functions:
 // http://gizma.com/easing/
 // http://www.robertpenner.com/easing/easing_demo.html
 //
 // Changed to use more friendly variable names, that follow my Lerp
 // conventions:
 // start = b (start value)
 // distance = c (change in value)
 // elapsedTime = t (current time)
 // duration = d (time duration)
 var f : Function;
 switch (type) {
 case EaseType.Linear: f = Interpolate.Linear; break;
 case EaseType.EaseInQuad: f = Interpolate.EaseInQuad; break;
 case EaseType.EaseOutQuad: f = Interpolate.EaseOutQuad; break;
 case EaseType.EaseInOutQuad: f = Interpolate.EaseInOutQuad; break;
 case EaseType.EaseInCubic: f = Interpolate.EaseInCubic; break;
 case EaseType.EaseOutCubic: f = Interpolate.EaseOutCubic; break;
 case EaseType.EaseInOutCubic: f = Interpolate.EaseInOutCubic; break;
 case EaseType.EaseInQuart: f = Interpolate.EaseInQuart; break;
 case EaseType.EaseOutQuart: f = Interpolate.EaseOutQuart; break;
 case EaseType.EaseInOutQuart: f = Interpolate.EaseInOutQuart; break;
 case EaseType.EaseInQuint: f = Interpolate.EaseInQuint; break;
 case EaseType.EaseOutQuint: f = Interpolate.EaseOutQuint; break;
 case EaseType.EaseInOutQuint: f = Interpolate.EaseInOutQuint; break;
 case EaseType.EaseInSine: f = Interpolate.EaseInSine; break;
 case EaseType.EaseOutSine: f = Interpolate.EaseOutSine; break;
 case EaseType.EaseInOutSine: f = Interpolate.EaseInOutSine; break;
 case EaseType.EaseInExpo: f = Interpolate.EaseInExpo; break;
 case EaseType.EaseOutExpo: f = Interpolate.EaseOutExpo; break;
 case EaseType.EaseInOutExpo: f = Interpolate.EaseInOutExpo; break;
 case EaseType.EaseInCirc: f = Interpolate.EaseInCirc; break;
 case EaseType.EaseOutCirc: f = Interpolate.EaseOutCirc; break;
 case EaseType.EaseInOutCirc: f = Interpolate.EaseInOutCirc; break;
 }
 return f;

}

/**

* Returns sequence generator from the first node to the last node over
* duration time using the points in-between the first and last node
* as control points of a bezier curve used to generate the interpolated points
* in the sequence. If there are no control points (ie. only two nodes, first
* and last) then this behaves exactly the same as NewEase(). In other words
* a zero-degree bezier spline curve is just the easing method. The sequence
* is generated as it is accessed using the Time.deltaTime to calculate the
* portion of duration that has elapsed.
*/

static function NewBezier(ease : Function, nodes : Transform[],

                         duration : float) : IEnumerator {
 var timer = NewTimer(duration);
 return NewBezier(ease, nodes, TransformDotPosition, duration, timer);

}

/**

* Instead of interpolating based on time, generate n interpolated points
* (slices) between the first and last node.
*/

static function NewBezier(ease : Function, nodes : Transform[],

                         slices : int) : IEnumerator {
 var counter = NewCounter(0, slices + 1, 1);
 return NewBezier(ease, nodes, TransformDotPosition, slices + 1, counter);

}

/**

* A Vector3[] variation of the Transform[] NewBezier() function.
* Same functionality but using Vector3s to define bezier curve.
*/

static function NewBezier(ease : Function, points : Vector3[],

                         duration : float) : IEnumerator {
 var timer = NewTimer(duration);
 return NewBezier(ease, points, Identity, duration, timer);

}

/**

* A Vector3[] variation of the Transform[] NewBezier() function.
* Same functionality but using Vector3s to define bezier curve.
*/

static function NewBezier(ease : Function, points : Vector3[],

                         slices : int) : IEnumerator {
 var counter = NewCounter(0, slices + 1, 1);
 return NewBezier(ease, points, Identity, slices + 1, counter);

}

/**

* Generic bezier spline sequence generator used to implement the time and
* slice variants. Normally you would not use this function directly.
*/

static function NewBezier(ease : Function, nodes : IList, toVector3 : Function,

                         maxStep : float, steps : IEnumerator) : IEnumerator {
 // need at least two nodes to spline between
 if (nodes.Count >= 2) {
   // copy nodes array since Bezier is destructive
   var points = new Vector3[nodes.Count];
   for (var step in steps) {
     // re-initialize copy before each destructive call to Bezier
     for (var i = 0; i < nodes.Count; i++) {
       points[i] = toVector3(nodes[i]);
     }
     yield Bezier(ease, points, step, maxStep);
     // make sure last value is always generated
   }
 }

}

/**

* A Vector3 n-degree bezier spline.
*
* WARNING: The points array is modified by Bezier. See NewBezier() for a
* safe and user friendly alternative.
*
* You can pass zero control points, just the start and end points, for just
* plain easing. In other words a zero-degree bezier spline curve is just the
* easing method.
*
* @param points start point, n control points, end point
*/

static function Bezier(ease : Function, points : Vector3[],

                       elapsedTime : float, duration : float) : Vector3 {
 // Reference: http://ibiblio.org/e-notes/Splines/Bezier.htm
 // Interpolate the n starting points to generate the next j = (n - 1) points,
 // then interpolate those n - 1 points to generate the next n - 2 points,
 // continue this until we have generated the last point (n - (n - 1)), j = 1.
 // We store the next set of output points in the same array as the
 // input points used to generate them. This works because we store the
 // result in the slot of the input point that is no longer used for this
 // iteration.
 for (var j = points.length - 1; j > 0; j--) {
   for (var i = 0; i < j; i++) {
     points[i].x = ease(points[i].x, points[i + 1].x - points[i].x,
                        elapsedTime, duration);
     points[i].y = ease(points[i].y, points[i + 1].y - points[i].y,
                        elapsedTime, duration);
     points[i].z = ease(points[i].z, points[i + 1].z - points[i].z,
                        elapsedTime, duration);
   }
 }
 return points[0];

}

/**

* Returns sequence generator from the first node, through each control point,
* and to the last node. N points are generated between each node (slices)
* using Catmull-Rom.
*/

static function NewCatmullRom(nodes : Transform[], slices : int,

                             loop : boolean) : IEnumerator {
 return NewCatmullRom(nodes, TransformDotPosition, slices, loop);

}

/**

* A Vector3[] variation of the Transform[] NewCatmullRom() function.
* Same functionality but using Vector3s to define curve.
*/

static function NewCatmullRom(points : Vector3[], slices : int,

                             loop : boolean) : IEnumerator {
 return NewCatmullRom(points, Identity, slices, loop);

}

/**

* Generic catmull-rom spline sequence generator used to implement the
* Vector3[] and Transform[] variants. Normally you would not use this
* function directly.
*/

static function NewCatmullRom(nodes : IList, toVector3 : Function,

                             slices : int, loop : boolean) : IEnumerator {
 // need at least two nodes to spline between
 if (nodes.Count >= 2) {
   // yield the first point explicitly, if looping the first point
   // will be generated again in the step for loop when interpolating
   // from last point back to the first point
   yield toVector3(nodes[0]);
   var last = nodes.Count - 1;
   for (var current = 0; loop || current < last; current++) {
     // wrap around when looping
     if (loop && current > last) {
       current = 0;
     }
     // handle edge cases for looping and non-looping scenarios
     // when looping we wrap around, when not looping use start for previous
     // and end for next when you at the ends of the nodes array
     var previous = (current == 0) ? ((loop) ? last : current) : current - 1;
     var start = current;
     var end = (current == last) ? ((loop) ? 0 : current) : current + 1;
     var next = (end == last) ? ((loop) ? 0 : end) : end + 1;
     // adding one guarantees yielding at least the end point
     var stepCount = slices + 1;
     for (var step = 1; step <= stepCount; step++) {
       yield CatmullRom(toVector3(nodes[previous]),
                        toVector3(nodes[start]),
                        toVector3(nodes[end]),
                        toVector3(nodes[next]),
                        step, stepCount);
     }
   }
 }

}

/**

* A Vector3 Catmull-Rom spline. Catmull-Rom splines are similar to bezier
* splines but have the useful property that the generated curve will go
* through each of the control points.
*
* NOTE: The NewCatmullRom() functions are an easier to use alternative to this
* raw Catmull-Rom implementation.
*
* @param previous the point just before the start point or the start point
*                 itself if no previous point is available
* @param start generated when elapsedTime == 0
* @param end generated when elapsedTime >= duration
* @param next the point just after the end point or the end point itself if no
*             next point is available
*/

static function CatmullRom(previous : Vector3, start : Vector3, end :

                          Vector3, next : Vector3, elapsedTime : float,
                          duration : float) : Vector3 {
 // References used:
 // p.266 GemsV1
 //
 // tension is often set to 0.5 but you can use any reasonable value:
 // http://www.cs.cmu.edu/~462/projects/assn2/assn2/catmullRom.pdf
 //
 // bias and tension controls:
 // http://local.wasp.uwa.edu.au/~pbourke/miscellaneous/interpolation/
 var percentComplete = elapsedTime / duration;
 var percentCompleteSquared = percentComplete * percentComplete;
 var percentCompleteCubed = percentCompleteSquared * percentComplete;
 return previous * (-0.5*percentCompleteCubed +
                    percentCompleteSquared -
                    0.5*percentComplete) +
   start * (1.5*percentCompleteCubed +
            -2.5*percentCompleteSquared + 1.0) +
   end * (-1.5*percentCompleteCubed +
          2.0*percentCompleteSquared +
          0.5*percentComplete) +
   next * (0.5*percentCompleteCubed -
           0.5*percentCompleteSquared);

}

/**

* Sequence of eleapsedTimes until elapsedTime is >= duration.
*
* Note: elapsedTimes are calculated using the value of Time.deltatTime each
* time a value is requested.
*/

static function NewTimer(duration : float) : IEnumerator {

 var elapsedTime = 0.0;
 while (elapsedTime < duration) {
   yield elapsedTime;
   elapsedTime += Time.deltaTime;
   // make sure last value is never skipped
   if (elapsedTime >= duration) {
     yield elapsedTime;
   }
 }

}

/**

* Generates sequence of integers from start to end (inclusive) one step
* at a time.
*/

static function NewCounter(start : int, end : int, step : int) : IEnumerator {

 for (var i = start; i <= end; i += step) {
   yield i;
 }

}

static function Identity(v : Vector3) : Vector3 {

 return v;

}

static function TransformDotPosition(t : Transform) : Vector3 {

 return t.position;

}

/**

* Linear interpolation (same as Mathf.Lerp)
*/

static function Linear(start : float, distance : float,

                      elapsedTime : float, duration : float) : float {
 // clamp elapsedTime to be <= duration
 if (elapsedTime > duration) { elapsedTime = duration; }
 return distance * (elapsedTime / duration) + start;

}

/**

* quadratic easing in - accelerating from zero velocity
*/

static function EaseInQuad(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 return distance*elapsedTime*elapsedTime + start;

}

/**

* quadratic easing out - decelerating to zero velocity
*/

static function EaseOutQuad(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 return -distance * elapsedTime*(elapsedTime-2) + start;

}

/**

* quadratic easing in/out - acceleration until halfway, then deceleration
*/

static function EaseInOutQuad(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 2.0 : elapsedTime / (duration / 2);
 if (elapsedTime < 1) return distance/2*elapsedTime*elapsedTime + start;
 elapsedTime--;
 return -distance/2 * (elapsedTime*(elapsedTime-2) - 1) + start;

}

/**

* cubic easing in - accelerating from zero velocity
*/

static function EaseInCubic(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 return distance*elapsedTime*elapsedTime*elapsedTime + start;

}

/**

* cubic easing out - decelerating to zero velocity
*/

static function EaseOutCubic(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 elapsedTime--;
 return distance*(elapsedTime*elapsedTime*elapsedTime + 1) + start;

}

/**

* cubic easing in/out - acceleration until halfway, then deceleration
*/

static function EaseInOutCubic(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 2.0 : elapsedTime / (duration / 2);
 if (elapsedTime < 1) return distance/2*elapsedTime*elapsedTime*elapsedTime +
                        start;
 elapsedTime -= 2;
 return distance/2*(elapsedTime*elapsedTime*elapsedTime + 2) + start;

}

/**

* quartic easing in - accelerating from zero velocity
*/

static function EaseInQuart(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 return distance*elapsedTime*elapsedTime*elapsedTime*elapsedTime + start;

}

/**

* quartic easing out - decelerating to zero velocity
*/

static function EaseOutQuart(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 elapsedTime--;
 return -distance * (elapsedTime*elapsedTime*elapsedTime*elapsedTime - 1) +
                        start;

}

/**

* quartic easing in/out - acceleration until halfway, then deceleration
*/

static function EaseInOutQuart(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 2.0 : elapsedTime / (duration / 2);
 if (elapsedTime < 1) return distance/2*
                        elapsedTime*elapsedTime*elapsedTime*elapsedTime +
                        start;
 elapsedTime -= 2;
 return -distance/2 * (elapsedTime*elapsedTime*elapsedTime*elapsedTime - 2) +
                        start;

}


/**

* quintic easing in - accelerating from zero velocity
*/

static function EaseInQuint(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 return distance*elapsedTime*elapsedTime*elapsedTime*elapsedTime*elapsedTime +
                        start;

}

/**

* quintic easing out - decelerating to zero velocity
*/

static function EaseOutQuint(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 elapsedTime--;
 return distance * (elapsedTime * elapsedTime * elapsedTime * elapsedTime *
                    elapsedTime + 1) + start;

}

/**

* quintic easing in/out - acceleration until halfway, then deceleration
*/

static function EaseInOutQuint(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 2.0 : elapsedTime / (duration / 2);
 if (elapsedTime < 1) return distance/2 * elapsedTime * elapsedTime *
                        elapsedTime * elapsedTime * elapsedTime + start;
 elapsedTime -= 2;
 return distance/2 * (elapsedTime * elapsedTime * elapsedTime * elapsedTime *
                      elapsedTime + 2) + start;

}

/**

* sinusoidal easing in - accelerating from zero velocity
*/

static function EaseInSine(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime to be <= duration
 if (elapsedTime > duration) { elapsedTime = duration; }
 return -distance * Mathf.Cos(elapsedTime/duration * (Mathf.PI/2)) +
                        distance + start;

}

/**

* sinusoidal easing out - decelerating to zero velocity
*/

static function EaseOutSine(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime to be <= duration
 if (elapsedTime > duration) { elapsedTime = duration; }
 return distance * Mathf.Sin(elapsedTime/duration * (Mathf.PI/2)) + start;

}

/**

* sinusoidal easing in/out - accelerating until halfway, then decelerating
*/

static function EaseInOutSine(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime to be <= duration
 if (elapsedTime > duration) { elapsedTime = duration; }
 return -distance/2 * (Mathf.Cos(Mathf.PI*elapsedTime/duration) - 1) + start;

}

/**

* exponential easing in - accelerating from zero velocity
*/

static function EaseInExpo(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime to be <= duration
 if (elapsedTime > duration) { elapsedTime = duration; }
 return distance * Mathf.Pow( 2, 10 * (elapsedTime/duration - 1) ) + start;

}

/**

* exponential easing out - decelerating to zero velocity
*/

static function EaseOutExpo(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime to be <= duration
 if (elapsedTime > duration) { elapsedTime = duration; }
 return distance * ( -Mathf.Pow( 2, -10 * elapsedTime/duration ) + 1 ) + start;

}

/**

* exponential easing in/out - accelerating until halfway, then decelerating
*/

static function EaseInOutExpo(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 2.0 : elapsedTime / (duration / 2);
 if (elapsedTime < 1) return distance/2 *
                        Mathf.Pow( 2, 10 * (elapsedTime - 1) ) + start;
 elapsedTime--;
 return distance/2 * ( -Mathf.Pow( 2, -10 * elapsedTime) + 2 ) + start;

}

/**

* circular easing in - accelerating from zero velocity
*/

static function EaseInCirc(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 return -distance * (Mathf.Sqrt(1 - elapsedTime*elapsedTime) - 1) + start;

}

/**

* circular easing out - decelerating to zero velocity
*/

static function EaseOutCirc(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 1.0 : elapsedTime / duration;
 elapsedTime--;
 return distance * Mathf.Sqrt(1 - elapsedTime*elapsedTime) + start;

}

/**

* circular easing in/out - acceleration until halfway, then deceleration
*/

static function EaseInOutCirc(start : float, distance : float,

                    elapsedTime : float, duration : float) : float {
 // clamp elapsedTime so that it cannot be greater than duration
 elapsedTime = (elapsedTime > duration) ? 2.0 : elapsedTime / (duration / 2);
 if (elapsedTime < 1) return -distance/2 *
                        (Mathf.Sqrt(1 - elapsedTime*elapsedTime) - 1) + start;
 elapsedTime -= 2;
 return distance/2 * (Mathf.Sqrt(1 - elapsedTime*elapsedTime) + 1) + start;

} </javascript>

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