Newer
Older
#ifndef QUATERNION_HH
#define QUATERNION_HH
#include <dune/common/fvector.hh>
#include <dune/common/exceptions.hh>
template <class T>
class Quaternion : public Dune::FieldVector<T,4>
{
public:
/** \brief Default constructor */
/** \brief Constructor with the four components */
Quaternion(const T& a, const T& b, const T& c, const T& d) {
(*this)[0] = a;
(*this)[1] = b;
(*this)[2] = c;
(*this)[3] = d;
}
Quaternion(const Dune::FieldVector<T,4>& other) : Dune::FieldVector<T,4>(other) {}
/** \brief Constructor with rotation axis and angle */
Quaternion(Dune::FieldVector<T,3> axis, T angle) {
axis /= axis.two_norm();
axis *= std::sin(angle/2);
(*this)[0] = axis[0];
(*this)[1] = axis[1];
(*this)[2] = axis[2];
(*this)[3] = std::cos(angle/2);
}
/** \brief Return the identity element */
static Quaternion<T> identity() {
Quaternion<T> id;
id[0] = 0;
id[1] = 0;
id[2] = 0;
id[3] = 1;
return id;
}
/** \brief The exponential map from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$
*/
static Quaternion<T> exp(const T& v0, const T& v1, const T& v2) {
Quaternion<T> q;
T normV = std::sqrt(v0*v0 + v1*v1 + v2*v2);
// Stabilization for small |v| due to Grassia
T sin = (normV < 1e-4) ? 0.5 * (normV*normV/48) : std::sin(normV/2)/normV;
q[0] = sin * v0;
q[1] = sin * v1;
q[2] = sin * v2;
q[3] = std::cos(normV/2);
return q;
}
/** \brief Right quaternion multiplication */
Quaternion<T> mult(const Quaternion<T>& other) const {
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
Quaternion<T> q;
q[0] = (*this)[3]*other[0] - (*this)[2]*other[1] + (*this)[1]*other[2] + (*this)[0]*other[3];
q[1] = (*this)[2]*other[0] + (*this)[3]*other[1] - (*this)[0]*other[2] + (*this)[1]*other[3];
q[2] = - (*this)[1]*other[0] + (*this)[0]*other[1] + (*this)[3]*other[2] + (*this)[2]*other[3];
q[3] = - (*this)[0]*other[0] - (*this)[1]*other[1] - (*this)[2]*other[2] + (*this)[3]*other[3];
return q;
}
/** \brief Return the tripel of director vectors represented by a unit quaternion
The formulas are taken from Dichmann, Li, Maddocks, (2.6.4), (2.6.5), (2.6.6)
*/
Dune::FieldVector<T,3> director(int i) const {
Dune::FieldVector<T,3> d;
const Dune::FieldVector<T,4>& q = *this; // simpler notation
if (i==0) {
d[0] = q[0]*q[0] - q[1]*q[1] - q[2]*q[2] + q[3]*q[3];
d[1] = 2 * (q[0]*q[1] + q[2]*q[3]);
d[2] = 2 * (q[0]*q[2] - q[1]*q[3]);
} else if (i==1) {
d[0] = 2 * (q[0]*q[1] - q[2]*q[3]);
d[1] = -q[0]*q[0] + q[1]*q[1] - q[2]*q[2] + q[3]*q[3];
d[2] = 2 * (q[1]*q[2] + q[0]*q[3]);
} else if (i==2) {
d[0] = 2 * (q[0]*q[2] + q[1]*q[3]);
d[1] = 2 * (q[1]*q[2] - q[0]*q[3]);
d[2] = -q[0]*q[0] - q[1]*q[1] + q[2]*q[2] + q[3]*q[3];
} else
DUNE_THROW(Dune::Exception, "Nonexisting director " << i << " requested!");
return d;
}
/** \brief Turn quaternion into a unit quaternion by dividing by its Euclidean norm */
void normalize() {
(*this) /= this->two_norm();
}
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
Dune::FieldVector<double,3> rotate(const Dune::FieldVector<double,3>& v) const {
Dune::FieldVector<double,3> result;
Dune::FieldVector<double,3> d0 = director(0);
Dune::FieldVector<double,3> d1 = director(1);
Dune::FieldVector<double,3> d2 = director(2);
for (int i=0; i<3; i++)
result[i] = v[0]*d0[i] + v[1]*d1[i] + v[2]*d2[i];
return result;
}
/** \brief Interpolate between two rotations */
static Quaternion<T> interpolate(const Quaternion<T>& a, const Quaternion<T>& b, double omega) {
Quaternion<T> result;
for (int i=0; i<4; i++)
result[i] = a[i]*(1-omega) + b[i]*omega;
result.normalize();
return result;
}
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
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
/** \brief Return the corresponding orthogonal matrix */
void matrix(Dune::FieldMatrix<T,3,3>& m) const {
m[0][0] = (*this)[0]*(*this)[0] - (*this)[1]*(*this)[1] - (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
m[0][1] = 2 * ( (*this)[0]*(*this)[1] - (*this)[2]*(*this)[3] );
m[0][2] = 2 * ( (*this)[0]*(*this)[2] + (*this)[1]*(*this)[3] );
m[1][0] = 2 * ( (*this)[0]*(*this)[1] + (*this)[2]*(*this)[3] );
m[1][1] = - (*this)[0]*(*this)[0] + (*this)[1]*(*this)[1] - (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
m[1][2] = 2 * ( -(*this)[0]*(*this)[3] + (*this)[1]*(*this)[2] );
m[2][0] = 2 * ( (*this)[0]*(*this)[2] - (*this)[1]*(*this)[3] );
m[2][1] = 2 * ( (*this)[0]*(*this)[3] + (*this)[1]*(*this)[2] );
m[2][2] = - (*this)[0]*(*this)[0] - (*this)[1]*(*this)[1] + (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
}
/** \brief Set unit quaternion from orthogonal matrix
We tacitly assume that the matrix really is orthogonal */
void set(const Dune::FieldMatrix<T,3,3>& m) {
// Easier writing
Dune::FieldVector<T,4>& p = (*this);
// The following equations for the derivation of a unit quaternion from a rotation
// matrix comes from 'E. Salamin, Application of Quaternions to Computation with
// Rotations, Technical Report, Stanford, 1974'
p[0] = (1 + m[0][0] - m[1][1] - m[2][2]) / 4;
p[1] = (1 - m[0][0] + m[1][1] - m[2][2]) / 4;
p[2] = (1 - m[0][0] - m[1][1] + m[2][2]) / 4;
p[3] = (1 + m[0][0] + m[1][1] + m[2][2]) / 4;
// avoid rounding problems
if (p[0] >= p[1] && p[0] >= p[2] && p[0] >= p[3]) {
p[0] = std::sqrt(p[0]);
// r_x r_y = (R_12 + R_21) / 4
p[1] = (m[0][1] + m[1][0]) / 4 / p[0];
// r_x r_z = (R_13 + R_31) / 4
p[2] = (m[0][2] + m[2][0]) / 4 / p[0];
// r_0 r_x = (R_32 - R_23) / 4
p[3] = (m[2][1] - m[1][2]) / 4 / p[0];
} else if (p[1] >= p[0] && p[1] >= p[2] && p[1] >= p[3]) {
p[1] = std::sqrt(p[1]);
// r_x r_y = (R_12 + R_21) / 4
p[0] = (m[0][1] + m[1][0]) / 4 / p[1];
// r_y r_z = (R_23 + R_32) / 4
p[2] = (m[1][2] + m[2][1]) / 4 / p[1];
// r_0 r_y = (R_13 - R_31) / 4
p[3] = (m[0][2] - m[2][0]) / 4 / p[1];
} else if (p[2] >= p[0] && p[2] >= p[1] && p[2] >= p[3]) {
p[2] = std::sqrt(p[2]);
// r_x r_z = (R_13 + R_31) / 4
p[0] = (m[0][2] + m[2][0]) / 4 / p[2];
// r_y r_z = (R_23 + R_32) / 4
p[1] = (m[1][2] + m[2][1]) / 4 / p[2];
// r_0 r_z = (R_21 - R_12) / 4
p[3] = (m[1][0] - m[0][1]) / 4 / p[2];
} else {
p[3] = std::sqrt(p[3]);
// r_0 r_x = (R_32 - R_23) / 4
p[0] = (m[2][1] - m[1][2]) / 4 / p[3];
// r_0 r_y = (R_13 - R_31) / 4
p[1] = (m[0][2] - m[2][0]) / 4 / p[3];
// r_0 r_z = (R_21 - R_12) / 4
p[2] = (m[1][0] - m[0][1]) / 4 / p[3];
}
}