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/****************************************************************************
**
** Copyright (C) 1993-2009 NVIDIA Corporation.
** Copyright (C) 2017 The Qt Company Ltd.
** Contact: https://www.qt.io/licensing/
**
** This file is part of Qt 3D Studio.
**
** $QT_BEGIN_LICENSE:GPL$
** Commercial License Usage
** Licensees holding valid commercial Qt licenses may use this file in
** accordance with the commercial license agreement provided with the
** Software or, alternatively, in accordance with the terms contained in
** a written agreement between you and The Qt Company. For licensing terms
** and conditions see https://www.qt.io/terms-conditions. For further
** information use the contact form at https://www.qt.io/contact-us.
**
** GNU General Public License Usage
** Alternatively, this file may be used under the terms of the GNU
** General Public License version 3 or (at your option) any later version
** approved by the KDE Free Qt Foundation. The licenses are as published by
** the Free Software Foundation and appearing in the file LICENSE.GPL3
** included in the packaging of this file. Please review the following
** information to ensure the GNU General Public License requirements will
** be met: https://www.gnu.org/licenses/gpl-3.0.html.
**
** $QT_END_LICENSE$
**
****************************************************************************/
//==============================================================================
// Namespace
//==============================================================================
namespace Q3DStudio {
//==============================================================================
// Constants used by the functions
//==============================================================================
const FLOAT M_PHI = 3.14159265358979323846f;
/* epsilon surrounding for near zero values */
const FLOAT EQN_EPS = 1e-9f;
#define ISZERO(x) ((x) > -EQN_EPS && (x) < EQN_EPS)
// cube-root
#ifndef CUBEROOT
#define CUBEROOT(x) \
((x) > 0.0 ? static_cast<FLOAT>(pow((x), 1.0f / 3.0f)) \
: ((x) < 0.0 ? -static_cast<FLOAT>(pow(-(x), 1.0f / 3.0f)) : 0.0f))
#endif
//==============================================================================
/**
* Find the root/s to a quadratic equation.
*
* The equation is of the form: c[2]*x^2 + c[1]*x + c[0] = 0
* Note that this will fail if c[2] is zero, or this is a linear equation.
*
* @param inConstants The array of constants to the equation; see form above
* @param outSolutions The array of solutions to the equation
* @return the number of solutions for the equation
*/
INT32 CCubicRoots::SolveQuadric(FLOAT inConstants[3], FLOAT outSolutions[2])
{
#ifdef _PERF_LOG
PerfLogMathEvent1 thePerfLog(DATALOGGER_CUBICROOT);
#endif
FLOAT theP, theQ, theD;
INT32 theResult = 0;
/* normal form: x^2 + px + theQ = 0 */
theP = inConstants[1] / (2 * inConstants[2]);
theQ = inConstants[0] / inConstants[2];
theD = theP * theP - theQ;
if (ISZERO(theD)) {
outSolutions[0] = -theP;
theResult = 1;
} else if (theD < 0.0f) {
theResult = 0;
} else if (theD > 0.0f) {
FLOAT theSquareRootD = sqrtf(theD);
outSolutions[0] = theSquareRootD - theP;
outSolutions[1] = -theSquareRootD - theP;
theResult = 2;
}
return theResult;
}
//==============================================================================
/**
* Find the root/s to a cubic equation.
*
* The equation is of the form: c[3]*x^3 + c[2]*x^2 + c[1]*x + c[0] = 0
* Note that this will fail if c[3] is zero, or this is a quadratic equation.
*
* @param inConstants The array of constants to the equation; see form above
* @param outSolutions The array of solutions to the equation
* @return the number of solutions for the equation
*/
INT32 CCubicRoots::SolveCubic(FLOAT inConstants[4], FLOAT outSolutions[3])
{
#ifdef _PERF_LOG
PerfLogMathEvent1 thePerfLog(DATALOGGER_CUBICROOT);
#endif
INT32 theResult;
FLOAT theSubstitute;
FLOAT theVariableA, theVariableB, theVariableC;
FLOAT theASquared, theVariableP, theVariableQ;
FLOAT thePCubed, theVariableD;
/* normal form: x^3 + Ax^2 + Bx + C = 0 */
theVariableA = inConstants[2] / inConstants[3];
theVariableB = inConstants[1] / inConstants[3];
theVariableC = inConstants[0] / inConstants[3];
/* substitute x = y - A/3 to eliminate quadric term:
x^3 +px + q = 0 */
theASquared = theVariableA * theVariableA;
theVariableP = 1.0f / 3.0f * (-1.0f / 3.0f * theASquared + theVariableB);
theVariableQ = 1.0f / 2.0f * (2.0f / 27.0f * theVariableA * theASquared
- 1.0f / 3.0f * theVariableA * theVariableB + theVariableC);
/* use Cardano's formula */
thePCubed = theVariableP * theVariableP * theVariableP;
theVariableD = theVariableQ * theVariableQ + thePCubed;
if (ISZERO(theVariableD)) {
if (ISZERO(theVariableQ)) /* one triple solution */
{
outSolutions[0] = 0;
theResult = 1;
} else /* one single and one double solution */
{
FLOAT theVariableU = CUBEROOT(-theVariableQ);
outSolutions[0] = 2.0f * theVariableU;
outSolutions[1] = -theVariableU;
theResult = 2;
}
} else if (theVariableD < 0.0f) /* Casus irreducibilis: three real solutions */
{
FLOAT thePhi = 1.0f / 3.0f * static_cast<FLOAT>(acos(-theVariableQ / sqrtf(-thePCubed)));
FLOAT theVariableT = 2.0f * sqrtf(-theVariableP);
outSolutions[0] = theVariableT * static_cast<FLOAT>(cos(thePhi));
outSolutions[1] = -theVariableT * static_cast<FLOAT>(cos(thePhi + M_PHI / 3.0f));
outSolutions[2] = -theVariableT * static_cast<FLOAT>(cos(thePhi - M_PHI / 3.0f));
theResult = 3;
} else /* one real solution */
{
FLOAT theSquareRootD = sqrtf(theVariableD);
FLOAT theVariableU = CUBEROOT(theSquareRootD - theVariableQ);
FLOAT theVariableV = -CUBEROOT(theSquareRootD + theVariableQ);
outSolutions[0] = theVariableU + theVariableV;
theResult = 1;
}
/* resubstitute */
theSubstitute = 1.0f / 3.0f * theVariableA;
for (INT32 theIndex = 0; theIndex < theResult; ++theIndex)
outSolutions[theIndex] -= theSubstitute;
return theResult;
}
}; // namespace Q3DStudio
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