1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
|
/****************************************************************************
**
** 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.
**
** $QT_END_LICENSE$
**
****************************************************************************/
#include <vector>
#include <QtCore/qglobal.h>
#include <QtCore/qmath.h>
namespace Q3DStudio {
class CubicPolynomial
{
public:
CubicPolynomial(double p0, double p1, double p2, double p3);
std::vector<double> extrema() const;
std::vector<double> roots() const;
private:
double m_a;
double m_b;
double m_c;
double m_d;
};
CubicPolynomial::CubicPolynomial(double p0, double p1, double p2, double p3)
: m_a(p3 - 3.0 * p2 + 3.0 * p1 - p0)
, m_b(3.0 * p2 - 6.0 * p1 + 3.0 * p0)
, m_c(3.0 * p1 - 3.0 * p0)
, m_d(p0)
{}
std::vector<double> CubicPolynomial::extrema() const
{
std::vector<double> out;
auto addValidValue = [&out](double value) {
if (!std::isnan(value) && !std::isinf(value))
out.push_back(qBound(0.0, value, 1.0));
};
// Find the roots of the first derivative of y.
auto pd2 = (2.0 * m_b) / (3.0 * m_a) / 2.0;
auto q = m_c / (3.0 * m_a);
auto radi = std::pow(pd2, 2.0) - q;
auto x1 = -pd2 + std::sqrt(radi);
auto x2 = -pd2 - std::sqrt(radi);
addValidValue(x1);
addValidValue(x2);
return out;
}
std::vector<double> CubicPolynomial::roots() const
{
std::vector<double> out;
auto addValidValue = [&out](double value) {
if (!(std::isnan(value) || std::isinf(value)))
out.push_back(value);
};
if (m_a == 0.0) {
if (m_b == 0.0) { // Linear
if (m_c != 0.0)
out.push_back(-m_d / m_c);
} else { // Quadratic
const double p = m_c / m_b / 2.0;
const double q = m_d / m_b;
const double sqrt = std::sqrt(std::pow(p, 2.0) - q);
addValidValue(-p + sqrt);
addValidValue(-p - sqrt);
}
} else { // Cubic
const double p = 3.0 * m_a * m_c - std::pow(m_b, 2.0);
const double q = 2.0 * std::pow(m_b, 3.0) - 9.0 * m_a * m_b * m_c
+ 27.0 * std::pow(m_a, 2.0) * m_d;
auto disc = std::pow(q, 2.0) + 4.0 * std::pow(p, 3.0);
auto toX = [&](double y) { return (y - m_b) / (3.0 * m_a); };
if (disc >= 0) { // One real solution.
double sqrt = 4.0 * std::sqrt(std::pow(q, 2.0) + 4.0 * std::pow(p, 3.0));
double u = .5 * std::cbrt(-4.0 * q + sqrt);
double v = .5 * std::cbrt(-4.0 * q - sqrt);
addValidValue(toX(u + v));
} else { // Three real solutions.
double phi = acos(-q / (2 * std::sqrt(-std::pow(p, 3.0))));
double sqrt = std::sqrt(-p) * 2.0;
double y1 = sqrt * cos(phi / 3.0);
double y2 = sqrt * cos((phi / 3.0) + (2.0 * M_PI / 3.0));
double y3 = sqrt * cos((phi / 3.0) + (4.0 * M_PI / 3.0));
addValidValue(toX(y1));
addValidValue(toX(y2));
addValidValue(toX(y3));
}
}
return out;
}
double evaluateForT(double t, double p0, double p1, double p2, double p3)
{
assert(t >= 0. && t <= 1.);
const double it = 1.0 - t;
return p0 * std::pow(it, 3.0) + p1 * 3.0 * std::pow(it, 2.0) * t
+ p2 * 3.0 * it * std::pow(t, 2.0) + p3 * std::pow(t, 3.0);
}
inline float EvaluateBezierKeyframe(long inTime, long inTime1, float inValue1, long inC1Time,
float inC1Value, long inC2Time, float inC2Value, long inTime2,
float inValue2)
{
if (inTime == inTime1)
return inValue1;
if (inTime == inTime2)
return inValue2;
auto polynomial = CubicPolynomial(inTime1 - inTime, inC1Time - inTime, inC2Time - inTime,
inTime2 - inTime);
for (double t : polynomial.roots()) {
if (t < 0.0 || t > 1.0)
continue;
// the curve handles are confined so that only 1 solution should ever exist for a give time
return evaluateForT(t, inValue1, inC1Value, inC2Value, inValue2);
}
assert(false); // no solution found?!
return 0.0;
}
} // namespace Qt3DStudio
|
