Adding custom bezier easing curves to QEasingCurve

I added the possibilty to define Bezier/TCB splines and use them
as custom easing curves.

Note:
Splines have a parametric definition. This means we have a
function/polynom of t that evalutes to x and y. x/y = f(t).

For our purpose we actually need the function y = f(x).
So as a first step we have to solve the solution x = f(t) for a given
t and then in a second step we evaluate y = f(t).

f(t) is a cubic polynom so we use cardanos formula to solve this equation
directly.

For the casus irreducibilis we need 3 functions that are a combination of
arcos and cos. Instead of evaluating arcos and cos we approximate these
functions directly.

TCB splines are converted into the corresponding cubic bezier spline.

Change-Id: Id2afc15efac92e494d6358dc2e11f94e8c524da1
Reviewed-by: Aaron Kennedy <aaron.kennedy@nokia.com>
bb10
Thomas Hartmann 2011-11-02 12:33:30 +01:00 committed by Qt by Nokia
parent 7d560240fa
commit b9f0bde16e
2 changed files with 541 additions and 6 deletions

View File

@ -296,6 +296,10 @@
\omitvalue OutCurve
\omitvalue SineCurve
\omitvalue CosineCurve
\value BezierSpline Allows defining a custom easing curve using a cubic bezier spline
\sa addCubicBezierSegment()
\value TCBSpline Allows defining a custom easing curve using a TCB spline
\sa addTCBSegment
\value Custom This is returned if the user specified a custom curve type with
setCustomType(). Note that you cannot call setType() with this value,
but type() can return it.
@ -312,6 +316,7 @@
*/
#include "qeasingcurve.h"
#include <cmath>
#ifndef QT_NO_DEBUG_STREAM
#include <QtCore/qdebug.h>
@ -322,14 +327,40 @@
#include <QtCore/qdatastream.h>
#endif
#include <QtCore/qpoint.h>
#include <QtCore/qvector.h>
QT_BEGIN_NAMESPACE
static bool isConfigFunction(QEasingCurve::Type type)
{
return type >= QEasingCurve::InElastic
&& type <= QEasingCurve::OutInBounce;
return (type >= QEasingCurve::InElastic
&& type <= QEasingCurve::OutInBounce) ||
type == QEasingCurve::BezierSpline ||
type == QEasingCurve::TCBSpline;
}
struct TCBPoint {
QPointF _point;
qreal _t;
qreal _c;
qreal _b;
TCBPoint() {}
TCBPoint(QPointF point, qreal t, qreal c, qreal b) : _point(point), _t(t), _c(c), _b(b) {}
bool operator==(const TCBPoint& other)
{
return _point == other._point &&
qFuzzyCompare(_t, other._t) &&
qFuzzyCompare(_c, other._c) &&
qFuzzyCompare(_b, other._b);
}
};
typedef QVector<TCBPoint> TCBPoints;
class QEasingCurveFunction
{
public:
@ -348,6 +379,9 @@ public:
qreal _p;
qreal _a;
qreal _o;
QVector<QPointF> _bezierCurves;
TCBPoints _tcbPoints;
};
qreal QEasingCurveFunction::value(qreal t)
@ -357,7 +391,10 @@ qreal QEasingCurveFunction::value(qreal t)
QEasingCurveFunction *QEasingCurveFunction::copy() const
{
return new QEasingCurveFunction(_t, _p, _a, _o);
QEasingCurveFunction *rv = new QEasingCurveFunction(_t, _p, _a, _o);
rv->_bezierCurves = _bezierCurves;
rv->_tcbPoints = _tcbPoints;
return rv;
}
bool QEasingCurveFunction::operator==(const QEasingCurveFunction& other)
@ -365,7 +402,9 @@ bool QEasingCurveFunction::operator==(const QEasingCurveFunction& other)
return _t == other._t &&
qFuzzyCompare(_p, other._p) &&
qFuzzyCompare(_a, other._a) &&
qFuzzyCompare(_o, other._o);
qFuzzyCompare(_o, other._o) &&
_bezierCurves == other._bezierCurves &&
_tcbPoints == other._tcbPoints;
}
QT_BEGIN_INCLUDE_NAMESPACE
@ -388,6 +427,396 @@ public:
QEasingCurve::EasingFunction func;
};
struct BezierEase : public QEasingCurveFunction
{
struct SingleCubicBezier {
qreal p0x, p0y;
qreal p1x, p1y;
qreal p2x, p2y;
qreal p3x, p3y;
};
bool _init;
bool _valid;
QVector<SingleCubicBezier> _curves;
int _curveCount;
QVector<qreal> _intervals;
BezierEase()
: QEasingCurveFunction(InOut), _init(false), _valid(false), _curves(10), _intervals(10)
{ }
void init()
{
if (_bezierCurves.last() == QPointF(1.0, 1.0)) {
_init = true;
_curveCount = _bezierCurves.count() / 3;
for (int i=0; i < _curveCount; i++) {
_intervals[i] = _bezierCurves.at(i * 3 + 2).x();
if (i == 0) {
_curves[0].p0x = 0.0;
_curves[0].p0y = 0.0;
_curves[0].p1x = _bezierCurves.at(0).x();
_curves[0].p1y = _bezierCurves.at(0).y();
_curves[0].p2x = _bezierCurves.at(1).x();
_curves[0].p2y = _bezierCurves.at(1).y();
_curves[0].p3x = _bezierCurves.at(2).x();
_curves[0].p3y = _bezierCurves.at(2).y();
} else if (i == (_curveCount - 1)) {
_curves[i].p0x = _bezierCurves.at(_bezierCurves.count() - 4).x();
_curves[i].p0y = _bezierCurves.at(_bezierCurves.count() - 4).y();
_curves[i].p1x = _bezierCurves.at(_bezierCurves.count() - 3).x();
_curves[i].p1y = _bezierCurves.at(_bezierCurves.count() - 3).y();
_curves[i].p2x = _bezierCurves.at(_bezierCurves.count() - 2).x();
_curves[i].p2y = _bezierCurves.at(_bezierCurves.count() - 2).y();
_curves[i].p3x = _bezierCurves.at(_bezierCurves.count() - 1).x();
_curves[i].p3y = _bezierCurves.at(_bezierCurves.count() - 1).y();
} else {
_curves[i].p0x = _bezierCurves.at(i * 3 - 1).x();
_curves[i].p0y = _bezierCurves.at(i * 3 - 1).y();
_curves[i].p1x = _bezierCurves.at(i * 3).x();
_curves[i].p1y = _bezierCurves.at(i * 3).y();
_curves[i].p2x = _bezierCurves.at(i * 3 + 1).x();
_curves[i].p2y = _bezierCurves.at(i * 3 + 1).y();
_curves[i].p3x = _bezierCurves.at(i * 3 + 2).x();
_curves[i].p3y = _bezierCurves.at(i * 3 + 2).y();
}
}
_valid = true;
} else {
_valid = false;
}
}
QEasingCurveFunction *copy() const
{
BezierEase *rv = new BezierEase();
rv->_t = _t;
rv->_p = _p;
rv->_a = _a;
rv->_o = _o;
rv->_bezierCurves = _bezierCurves;
rv->_tcbPoints = _tcbPoints;
return rv;
}
void getBezierSegment(SingleCubicBezier * &singleCubicBezier, qreal x)
{
int currentSegment = 0;
while (currentSegment < _curveCount) {
if (x <= _intervals.data()[currentSegment])
break;
currentSegment++;
}
singleCubicBezier = &_curves.data()[currentSegment];
}
qreal static inline newtonIteration(const SingleCubicBezier &singleCubicBezier, qreal t, qreal x)
{
qreal currentXValue = evaluateForX(singleCubicBezier, t);
const qreal newT = t - (currentXValue - x) / evaluateDerivateForX(singleCubicBezier, t);
return newT;
}
qreal value(qreal x)
{
Q_ASSERT(_bezierCurves.count() % 3 == 0);
if (_bezierCurves.isEmpty()) {
return x;
}
if (!_init)
init();
if (!_valid) {
qWarning("QEasingCurve: Invalid bezier curve");
return x;
}
SingleCubicBezier *singleCubicBezier = 0;
getBezierSegment(singleCubicBezier, x);
return evaluateSegmentForY(*singleCubicBezier, findTForX(*singleCubicBezier, x));
}
qreal static inline evaluateSegmentForY(const SingleCubicBezier &singleCubicBezier, qreal t)
{
const qreal p0 = singleCubicBezier.p0y;
const qreal p1 = singleCubicBezier.p1y;
const qreal p2 = singleCubicBezier.p2y;
const qreal p3 = singleCubicBezier.p3y;
const qreal s = 1 - t;
const qreal s_squared = s*s;
const qreal t_squared = t*t;
const qreal s_cubic = s_squared * s;
const qreal t_cubic = t_squared * t;
return s_cubic * p0 + 3 * s_squared * t * p1 + 3 * s * t_squared * p2 + t_cubic * p3;
}
qreal static inline evaluateForX(const SingleCubicBezier &singleCubicBezier, qreal t)
{
const qreal p0 = singleCubicBezier.p0x;
const qreal p1 = singleCubicBezier.p1x;
const qreal p2 = singleCubicBezier.p2x;
const qreal p3 = singleCubicBezier.p3x;
const qreal s = 1 - t;
const qreal s_squared = s*s;
const qreal t_squared = t*t;
const qreal s_cubic = s_squared * s;
const qreal t_cubic = t_squared * t;
return s_cubic * p0 + 3 * s_squared * t * p1 + 3 * s * t_squared * p2 + t_cubic * p3;
}
qreal static inline evaluateDerivateForX(const SingleCubicBezier &singleCubicBezier, qreal t)
{
const qreal p0 = singleCubicBezier.p0x;
const qreal p1 = singleCubicBezier.p1x;
const qreal p2 = singleCubicBezier.p2x;
const qreal p3 = singleCubicBezier.p3x;
const qreal t_squared = t*t;
return -3*p0 + 3*p1 + 6*p0*t - 12*p1*t + 6*p2*t + 3*p3*t_squared - 3*p0*t_squared + 9*p1*t_squared - 9*p2*t_squared;
}
qreal static inline _cbrt(qreal d)
{
qreal sign = 1;
if (d < 0)
sign = -1;
d = d * sign;
qreal t_i = _fast_cbrt(d);
//one step of Halley's Method to get a better approximation
const qreal t_i_cubic = t_i * t_i * t_i;
qreal t = t_i * (t_i_cubic + d + d) / (t_i_cubic + t_i_cubic + d);
//another step
/*t_i = t;
t_i_cubic = pow(t_i, 3);
t = t_i * (t_i_cubic + d + d) / (t_i_cubic + t_i_cubic + d);*/
return t * sign;
}
float static inline _fast_cbrt(float x)
{
int& i = (int&) x;
i = (i - (127<<23)) / 3 + (127<<23);
return x;
}
double static inline _fast_cbrt(double d)
{
const unsigned int B1 = 715094163;
double t = 0.0;
unsigned int* pt = (unsigned int*) &t;
unsigned int* px = (unsigned int*) &d;
pt[1]=px[1]/3+B1;
return t;
}
qreal static inline _acos(qreal x)
{
return sqrt(1-x)*(1.5707963267948966192313216916398f + x*(-0.213300989f + x*(0.077980478f + x*-0.02164095f)));
}
qreal static inline _cos(qreal x) //super fast _cos
{
const qreal pi_times2 = 2 * M_PI;
const qreal pi_neg = -1 * M_PI;
const qreal pi_by2 = M_PI / 2.0;
x += pi_by2; //the polynom is for sin
if (x < pi_neg)
x += pi_times2;
else if (x > M_PI)
x -= pi_times2;
const qreal a = 0.405284735;
const qreal b = 1.27323954;
const qreal x_squared = x * x;
if (x < 0) {
qreal cos = b * x + a * x_squared;
if (cos < 0)
return 0.225 * (cos * -1 * cos - cos) + cos;
return 0.225 * (cos * cos - cos) + cos;
} //else
qreal cos = b * x - a * x_squared;
if (cos < 0)
return 0.225 * (cos * 1 *-cos - cos) + cos;
return 0.225 * (cos * cos - cos) + cos;
}
bool static inline inRange(qreal f)
{
return (f >= -0.01 && f <= 1.01);
}
void static inline cosacos(qreal x, qreal &s1, qreal &s2, qreal &s3 )
{
//This function has no proper algebraic representation in real numbers.
//We use approximations instead
const qreal x_squared = x * x;
const qreal x_plus_one_sqrt = sqrt(1.0 + x);
const qreal one_minus_x_sqrt = sqrt(1.0 - x);
//cos(acos(x) / 3)
//s1 = _cos(_acos(x) / 3);
s1 = 0.463614 - 0.0347815 * x + 0.00218245 * x_squared + 0.402421 * x_plus_one_sqrt;
//cos(acos((x) - M_PI) / 3)
//s3 = _cos((_acos(x) - M_PI) / 3);
s3 = 0.463614 + 0.402421 * one_minus_x_sqrt + 0.0347815 * x + 0.00218245 * x_squared;
//cos((acos(x) + M_PI) / 3)
//s2 = _cos((_acos(x) + M_PI) / 3);
s2 = -0.401644 * one_minus_x_sqrt - 0.0686804 * x + 0.401644 * x_plus_one_sqrt;
}
qreal static inline singleRealSolutionForCubic(qreal a, qreal b, qreal c)
{
//returns the real solutiuon in [0..1]
//We use the Cardano formula
//substituiton: x = z - a/3
// z^3+pz+q=0
if (c < 0.000001 && c > -0.000001)
return 0;
const qreal a_by3 = a / 3.0;
const qreal a_cubic = a * a * a;
const qreal p = b - a * a_by3;
const qreal q = 2.0 * a_cubic / 27.0 - a * b / 3.0 + c;
const qreal q_squared = q * q;
const qreal p_cubic = p * p * p;
const qreal D = 0.25 * q_squared + p_cubic / 27.0;
if (D >= 0) {
const qreal D_sqrt = sqrt(D);
qreal u = _cbrt( -q * 0.5 + D_sqrt);
qreal v = _cbrt( -q * 0.5 - D_sqrt);
qreal z1 = u + v;
qreal t1 = z1 - a_by3;
if (inRange(t1))
return t1;
qreal z2 = -1 *u;
qreal t2 = z2 - a_by3;
return t2;
}
//casus irreducibilis
const qreal p_minus_sqrt = sqrt(-p);
//const qreal f = sqrt(4.0 / 3.0 * -p);
const qreal f = sqrt(4.0 / 3.0) * p_minus_sqrt;
//const qreal sqrtP = sqrt(27.0 / -p_cubic);
const qreal sqrtP = -3.0*sqrt(3.0) / (p_minus_sqrt * p);
const qreal g = -q * 0.5 * sqrtP;
qreal s1;
qreal s2;
qreal s3;
cosacos(g, s1, s2, s3);
qreal z1 = -1* f * s2;
qreal t1 = z1 - a_by3;
if (inRange(t1))
return t1;
qreal z2 = f * s1;
qreal t2 = z2 - a_by3;
if (inRange(t2))
return t2;
qreal z3 = -1 * f * s3;
qreal t3 = z3 - a_by3;
return t3;
}
qreal static inline findTForX(const SingleCubicBezier &singleCubicBezier, qreal x)
{
const qreal p0 = singleCubicBezier.p0x;
const qreal p1 = singleCubicBezier.p1x;
const qreal p2 = singleCubicBezier.p2x;
const qreal p3 = singleCubicBezier.p3x;
const qreal factorT3 = p3 - p0 + 3 * p1 - 3 * p2;
const qreal factorT2 = 3 * p0 - 6 * p1 + 3 * p2;
const qreal factorT1 = -3 * p0 + 3 * p1;
const qreal factorT0 = p0 - x;
const qreal a = factorT2 / factorT3;
const qreal b = factorT1 / factorT3;
const qreal c = factorT0 / factorT3;
return singleRealSolutionForCubic(a, b, c);
//one new iteration to increase numeric stability
//return newtonIteration(singleCubicBezier, t, x);
}
};
struct TCBEase : public BezierEase
{
qreal value(qreal x)
{
Q_ASSERT(_bezierCurves.count() % 3 == 0);
if (_bezierCurves.isEmpty()) {
qWarning("QEasingCurve: Invalid tcb curve");
return x;
}
return BezierEase::value(x);
}
};
struct ElasticEase : public QEasingCurveFunction
{
ElasticEase(Type type)
@ -399,6 +828,8 @@ struct ElasticEase : public QEasingCurveFunction
ElasticEase *rv = new ElasticEase(_t);
rv->_p = _p;
rv->_a = _a;
rv->_bezierCurves = _bezierCurves;
rv->_tcbPoints = _tcbPoints;
return rv;
}
@ -431,6 +862,8 @@ struct BounceEase : public QEasingCurveFunction
{
BounceEase *rv = new BounceEase(_t);
rv->_a = _a;
rv->_bezierCurves = _bezierCurves;
rv->_tcbPoints = _tcbPoints;
return rv;
}
@ -462,6 +895,8 @@ struct BackEase : public QEasingCurveFunction
{
BackEase *rv = new BackEase(_t);
rv->_o = _o;
rv->_bezierCurves = _bezierCurves;
rv->_tcbPoints = _tcbPoints;
return rv;
}
@ -598,6 +1033,12 @@ static QEasingCurveFunction *curveToFunctionObject(QEasingCurve::Type type)
case QEasingCurve::OutInBack:
curveFunc = new BackEase(BackEase::OutIn);
break;
case QEasingCurve::BezierSpline:
curveFunc = new BezierEase();
break;
case QEasingCurve::TCBSpline:
curveFunc = new TCBEase();
break;
default:
curveFunc = new QEasingCurveFunction(QEasingCurveFunction::In, qreal(0.3), qreal(1.0), qreal(1.70158));
}
@ -758,6 +1199,88 @@ void QEasingCurve::setOvershoot(qreal overshoot)
d_ptr->config->_o = overshoot;
}
/*!
Adds a segment of a cubic bezier spline to define a custom easing curve.
It is only applicable if type() is QEasingCurve::BezierSpline.
Note that the spline implicitly starts at (0.0, 0.0) and has to end at (1.0, 1.0) to
be a valid easing curve.
*/
void QEasingCurve::addCubicBezierSegment(const QPointF & c1, const QPointF & c2, const QPointF & endPoint)
{
if (!d_ptr->config)
d_ptr->config = curveToFunctionObject(d_ptr->type);
d_ptr->config->_bezierCurves << c1 << c2 << endPoint;
}
QVector<QPointF> static inline tcbToBezier(const TCBPoints &tcbPoints)
{
const int count = tcbPoints.count();
QVector<QPointF> bezierPoints;
for (int i = 1; i < count; i++) {
const qreal t_0 = tcbPoints.at(i - 1)._t;
const qreal c_0 = tcbPoints.at(i - 1)._c;
qreal b_0 = -1;
qreal const t_1 = tcbPoints.at(i)._t;
qreal const c_1 = tcbPoints.at(i)._c;
qreal b_1 = 1;
QPointF c_minusOne; //P1 last segment - not available for the first point
const QPointF c0(tcbPoints.at(i - 1)._point); //P0 Hermite/TBC
const QPointF c3(tcbPoints.at(i)._point); //P1 Hermite/TBC
QPointF c4; //P0 next segment - not available for the last point
if (i > 1) { //first point no left tangent
c_minusOne = tcbPoints.at(i - 2)._point;
b_0 = tcbPoints.at(i - 1)._b;
}
if (i < (count - 1)) { //last point no right tangent
c4 = tcbPoints.at(i + 1)._point;
b_1 = tcbPoints.at(i)._b;
}
const qreal dx_0 = 0.5 * (1-t_0) * ((1 + b_0) * (1 + c_0) * (c0.x() - c_minusOne.x()) + (1- b_0) * (1 - c_0) * (c3.x() - c0.x()));
const qreal dy_0 = 0.5 * (1-t_0) * ((1 + b_0) * (1 + c_0) * (c0.y() - c_minusOne.y()) + (1- b_0) * (1 - c_0) * (c3.y() - c0.y()));
const qreal dx_1 = 0.5 * (1-t_1) * ((1 + b_1) * (1 - c_1) * (c3.x() - c0.x()) + (1 - b_1) * (1 + c_1) * (c4.x() - c3.x()));
const qreal dy_1 = 0.5 * (1-t_1) * ((1 + b_1) * (1 - c_1) * (c3.y() - c0.y()) + (1 - b_1) * (1 + c_1) * (c4.y() - c3.y()));
const QPointF d_0 = QPointF(dx_0, dy_0);
const QPointF d_1 = QPointF(dx_1, dy_1);
QPointF c1 = (3 * c0 + d_0) / 3;
QPointF c2 = (3 * c3 - d_1) / 3;
bezierPoints << c1 << c2 << c3;
}
return bezierPoints;
}
/*!
Adds a segment of a TCB bezier spline to define a custom easing curve.
It is only applicable if type() is QEasingCurve::TCBSpline.
The spline has to start explitly at (0.0, 0.0) and has to end at (1.0, 1.0) to
be a valid easing curve.
The three parameters are called tension, continuity and bias. All three parameters are
valid between -1 and 1 and define the tangent of the control point.
If all three parameters are 0 the resulting spline is a Catmull-Rom spline.
The begin and endpoint always have a bias of -1 and 1, since the outer tangent is not defined.
*/
void QEasingCurve::addTCBSegment(const QPointF &nextPoint, qreal t, qreal c, qreal b)
{
if (!d_ptr->config)
d_ptr->config = curveToFunctionObject(d_ptr->type);
d_ptr->config->_tcbPoints.append(TCBPoint(nextPoint, t, c ,b));
if (nextPoint == QPointF(1.0, 1.0)) {
d_ptr->config->_bezierCurves = tcbToBezier(d_ptr->config->_tcbPoints);
d_ptr->config->_tcbPoints.clear();
}
}
/*!
Returns the type of the easing curve.
*/
@ -771,16 +1294,22 @@ void QEasingCurvePrivate::setType_helper(QEasingCurve::Type newType)
qreal amp = -1.0;
qreal period = -1.0;
qreal overshoot = -1.0;
QVector<QPointF> bezierCurves;
QVector<TCBPoint> tcbPoints;
if (config) {
amp = config->_a;
period = config->_p;
overshoot = config->_o;
bezierCurves = config->_bezierCurves;
tcbPoints = config->_tcbPoints;
delete config;
config = 0;
}
if (isConfigFunction(newType) || (amp != -1.0) || (period != -1.0) || (overshoot != -1.0)) {
if (isConfigFunction(newType) || (amp != -1.0) || (period != -1.0) || (overshoot != -1.0) ||
!bezierCurves.isEmpty()) {
config = curveToFunctionObject(newType);
if (amp != -1.0)
config->_a = amp;
@ -788,6 +1317,8 @@ void QEasingCurvePrivate::setType_helper(QEasingCurve::Type newType)
config->_p = period;
if (overshoot != -1.0)
config->_o = overshoot;
config->_bezierCurves = bezierCurves;
config->_tcbPoints = tcbPoints;
func = 0;
} else if (newType != QEasingCurve::Custom) {
func = curveToFunc(newType);

View File

@ -52,6 +52,7 @@ QT_BEGIN_NAMESPACE
QT_MODULE(Core)
class QEasingCurvePrivate;
class QPointF;
class Q_CORE_EXPORT QEasingCurve
{
Q_GADGET
@ -70,7 +71,7 @@ public:
InBack, OutBack, InOutBack, OutInBack,
InBounce, OutBounce, InOutBounce, OutInBounce,
InCurve, OutCurve, SineCurve, CosineCurve,
Custom, NCurveTypes
BezierSpline, TCBSpline, Custom, NCurveTypes
};
QEasingCurve(Type type = Linear);
@ -91,6 +92,9 @@ public:
qreal overshoot() const;
void setOvershoot(qreal overshoot);
void addCubicBezierSegment(const QPointF & c1, const QPointF & c2, const QPointF & endPoint);
void addTCBSegment(const QPointF &nextPoint, qreal t, qreal c, qreal b);
Type type() const;
void setType(Type type);
typedef qreal (*EasingFunction)(qreal progress);