[136] | 1 | // This file is part of Eigen, a lightweight C++ template library
|
---|
| 2 | // for linear algebra.
|
---|
| 3 | //
|
---|
| 4 | // Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
|
---|
| 5 | //
|
---|
| 6 | // This Source Code Form is subject to the terms of the Mozilla
|
---|
| 7 | // Public License v. 2.0. If a copy of the MPL was not distributed
|
---|
| 8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
|
---|
| 9 |
|
---|
| 10 | // no include guard, we'll include this twice from All.h from Eigen2Support, and it's internal anyway
|
---|
| 11 |
|
---|
| 12 | namespace Eigen {
|
---|
| 13 |
|
---|
| 14 | template<typename Other,
|
---|
| 15 | int OtherRows=Other::RowsAtCompileTime,
|
---|
| 16 | int OtherCols=Other::ColsAtCompileTime>
|
---|
| 17 | struct ei_quaternion_assign_impl;
|
---|
| 18 |
|
---|
| 19 | /** \geometry_module \ingroup Geometry_Module
|
---|
| 20 | *
|
---|
| 21 | * \class Quaternion
|
---|
| 22 | *
|
---|
| 23 | * \brief The quaternion class used to represent 3D orientations and rotations
|
---|
| 24 | *
|
---|
| 25 | * \param _Scalar the scalar type, i.e., the type of the coefficients
|
---|
| 26 | *
|
---|
| 27 | * This class represents a quaternion \f$ w+xi+yj+zk \f$ that is a convenient representation of
|
---|
| 28 | * orientations and rotations of objects in three dimensions. Compared to other representations
|
---|
| 29 | * like Euler angles or 3x3 matrices, quatertions offer the following advantages:
|
---|
| 30 | * \li \b compact storage (4 scalars)
|
---|
| 31 | * \li \b efficient to compose (28 flops),
|
---|
| 32 | * \li \b stable spherical interpolation
|
---|
| 33 | *
|
---|
| 34 | * The following two typedefs are provided for convenience:
|
---|
| 35 | * \li \c Quaternionf for \c float
|
---|
| 36 | * \li \c Quaterniond for \c double
|
---|
| 37 | *
|
---|
| 38 | * \sa class AngleAxis, class Transform
|
---|
| 39 | */
|
---|
| 40 |
|
---|
| 41 | template<typename _Scalar> struct ei_traits<Quaternion<_Scalar> >
|
---|
| 42 | {
|
---|
| 43 | typedef _Scalar Scalar;
|
---|
| 44 | };
|
---|
| 45 |
|
---|
| 46 | template<typename _Scalar>
|
---|
| 47 | class Quaternion : public RotationBase<Quaternion<_Scalar>,3>
|
---|
| 48 | {
|
---|
| 49 | typedef RotationBase<Quaternion<_Scalar>,3> Base;
|
---|
| 50 |
|
---|
| 51 | public:
|
---|
| 52 | EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(_Scalar,4)
|
---|
| 53 |
|
---|
| 54 | using Base::operator*;
|
---|
| 55 |
|
---|
| 56 | /** the scalar type of the coefficients */
|
---|
| 57 | typedef _Scalar Scalar;
|
---|
| 58 |
|
---|
| 59 | /** the type of the Coefficients 4-vector */
|
---|
| 60 | typedef Matrix<Scalar, 4, 1> Coefficients;
|
---|
| 61 | /** the type of a 3D vector */
|
---|
| 62 | typedef Matrix<Scalar,3,1> Vector3;
|
---|
| 63 | /** the equivalent rotation matrix type */
|
---|
| 64 | typedef Matrix<Scalar,3,3> Matrix3;
|
---|
| 65 | /** the equivalent angle-axis type */
|
---|
| 66 | typedef AngleAxis<Scalar> AngleAxisType;
|
---|
| 67 |
|
---|
| 68 | /** \returns the \c x coefficient */
|
---|
| 69 | inline Scalar x() const { return m_coeffs.coeff(0); }
|
---|
| 70 | /** \returns the \c y coefficient */
|
---|
| 71 | inline Scalar y() const { return m_coeffs.coeff(1); }
|
---|
| 72 | /** \returns the \c z coefficient */
|
---|
| 73 | inline Scalar z() const { return m_coeffs.coeff(2); }
|
---|
| 74 | /** \returns the \c w coefficient */
|
---|
| 75 | inline Scalar w() const { return m_coeffs.coeff(3); }
|
---|
| 76 |
|
---|
| 77 | /** \returns a reference to the \c x coefficient */
|
---|
| 78 | inline Scalar& x() { return m_coeffs.coeffRef(0); }
|
---|
| 79 | /** \returns a reference to the \c y coefficient */
|
---|
| 80 | inline Scalar& y() { return m_coeffs.coeffRef(1); }
|
---|
| 81 | /** \returns a reference to the \c z coefficient */
|
---|
| 82 | inline Scalar& z() { return m_coeffs.coeffRef(2); }
|
---|
| 83 | /** \returns a reference to the \c w coefficient */
|
---|
| 84 | inline Scalar& w() { return m_coeffs.coeffRef(3); }
|
---|
| 85 |
|
---|
| 86 | /** \returns a read-only vector expression of the imaginary part (x,y,z) */
|
---|
| 87 | inline const Block<const Coefficients,3,1> vec() const { return m_coeffs.template start<3>(); }
|
---|
| 88 |
|
---|
| 89 | /** \returns a vector expression of the imaginary part (x,y,z) */
|
---|
| 90 | inline Block<Coefficients,3,1> vec() { return m_coeffs.template start<3>(); }
|
---|
| 91 |
|
---|
| 92 | /** \returns a read-only vector expression of the coefficients (x,y,z,w) */
|
---|
| 93 | inline const Coefficients& coeffs() const { return m_coeffs; }
|
---|
| 94 |
|
---|
| 95 | /** \returns a vector expression of the coefficients (x,y,z,w) */
|
---|
| 96 | inline Coefficients& coeffs() { return m_coeffs; }
|
---|
| 97 |
|
---|
| 98 | /** Default constructor leaving the quaternion uninitialized. */
|
---|
| 99 | inline Quaternion() {}
|
---|
| 100 |
|
---|
| 101 | /** Constructs and initializes the quaternion \f$ w+xi+yj+zk \f$ from
|
---|
| 102 | * its four coefficients \a w, \a x, \a y and \a z.
|
---|
| 103 | *
|
---|
| 104 | * \warning Note the order of the arguments: the real \a w coefficient first,
|
---|
| 105 | * while internally the coefficients are stored in the following order:
|
---|
| 106 | * [\c x, \c y, \c z, \c w]
|
---|
| 107 | */
|
---|
| 108 | inline Quaternion(Scalar w, Scalar x, Scalar y, Scalar z)
|
---|
| 109 | { m_coeffs << x, y, z, w; }
|
---|
| 110 |
|
---|
| 111 | /** Copy constructor */
|
---|
| 112 | inline Quaternion(const Quaternion& other) { m_coeffs = other.m_coeffs; }
|
---|
| 113 |
|
---|
| 114 | /** Constructs and initializes a quaternion from the angle-axis \a aa */
|
---|
| 115 | explicit inline Quaternion(const AngleAxisType& aa) { *this = aa; }
|
---|
| 116 |
|
---|
| 117 | /** Constructs and initializes a quaternion from either:
|
---|
| 118 | * - a rotation matrix expression,
|
---|
| 119 | * - a 4D vector expression representing quaternion coefficients.
|
---|
| 120 | * \sa operator=(MatrixBase<Derived>)
|
---|
| 121 | */
|
---|
| 122 | template<typename Derived>
|
---|
| 123 | explicit inline Quaternion(const MatrixBase<Derived>& other) { *this = other; }
|
---|
| 124 |
|
---|
| 125 | Quaternion& operator=(const Quaternion& other);
|
---|
| 126 | Quaternion& operator=(const AngleAxisType& aa);
|
---|
| 127 | template<typename Derived>
|
---|
| 128 | Quaternion& operator=(const MatrixBase<Derived>& m);
|
---|
| 129 |
|
---|
| 130 | /** \returns a quaternion representing an identity rotation
|
---|
| 131 | * \sa MatrixBase::Identity()
|
---|
| 132 | */
|
---|
| 133 | static inline Quaternion Identity() { return Quaternion(1, 0, 0, 0); }
|
---|
| 134 |
|
---|
| 135 | /** \sa Quaternion::Identity(), MatrixBase::setIdentity()
|
---|
| 136 | */
|
---|
| 137 | inline Quaternion& setIdentity() { m_coeffs << 0, 0, 0, 1; return *this; }
|
---|
| 138 |
|
---|
| 139 | /** \returns the squared norm of the quaternion's coefficients
|
---|
| 140 | * \sa Quaternion::norm(), MatrixBase::squaredNorm()
|
---|
| 141 | */
|
---|
| 142 | inline Scalar squaredNorm() const { return m_coeffs.squaredNorm(); }
|
---|
| 143 |
|
---|
| 144 | /** \returns the norm of the quaternion's coefficients
|
---|
| 145 | * \sa Quaternion::squaredNorm(), MatrixBase::norm()
|
---|
| 146 | */
|
---|
| 147 | inline Scalar norm() const { return m_coeffs.norm(); }
|
---|
| 148 |
|
---|
| 149 | /** Normalizes the quaternion \c *this
|
---|
| 150 | * \sa normalized(), MatrixBase::normalize() */
|
---|
| 151 | inline void normalize() { m_coeffs.normalize(); }
|
---|
| 152 | /** \returns a normalized version of \c *this
|
---|
| 153 | * \sa normalize(), MatrixBase::normalized() */
|
---|
| 154 | inline Quaternion normalized() const { return Quaternion(m_coeffs.normalized()); }
|
---|
| 155 |
|
---|
| 156 | /** \returns the dot product of \c *this and \a other
|
---|
| 157 | * Geometrically speaking, the dot product of two unit quaternions
|
---|
| 158 | * corresponds to the cosine of half the angle between the two rotations.
|
---|
| 159 | * \sa angularDistance()
|
---|
| 160 | */
|
---|
| 161 | inline Scalar eigen2_dot(const Quaternion& other) const { return m_coeffs.eigen2_dot(other.m_coeffs); }
|
---|
| 162 |
|
---|
| 163 | inline Scalar angularDistance(const Quaternion& other) const;
|
---|
| 164 |
|
---|
| 165 | Matrix3 toRotationMatrix(void) const;
|
---|
| 166 |
|
---|
| 167 | template<typename Derived1, typename Derived2>
|
---|
| 168 | Quaternion& setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b);
|
---|
| 169 |
|
---|
| 170 | inline Quaternion operator* (const Quaternion& q) const;
|
---|
| 171 | inline Quaternion& operator*= (const Quaternion& q);
|
---|
| 172 |
|
---|
| 173 | Quaternion inverse(void) const;
|
---|
| 174 | Quaternion conjugate(void) const;
|
---|
| 175 |
|
---|
| 176 | Quaternion slerp(Scalar t, const Quaternion& other) const;
|
---|
| 177 |
|
---|
| 178 | template<typename Derived>
|
---|
| 179 | Vector3 operator* (const MatrixBase<Derived>& vec) const;
|
---|
| 180 |
|
---|
| 181 | /** \returns \c *this with scalar type casted to \a NewScalarType
|
---|
| 182 | *
|
---|
| 183 | * Note that if \a NewScalarType is equal to the current scalar type of \c *this
|
---|
| 184 | * then this function smartly returns a const reference to \c *this.
|
---|
| 185 | */
|
---|
| 186 | template<typename NewScalarType>
|
---|
| 187 | inline typename internal::cast_return_type<Quaternion,Quaternion<NewScalarType> >::type cast() const
|
---|
| 188 | { return typename internal::cast_return_type<Quaternion,Quaternion<NewScalarType> >::type(*this); }
|
---|
| 189 |
|
---|
| 190 | /** Copy constructor with scalar type conversion */
|
---|
| 191 | template<typename OtherScalarType>
|
---|
| 192 | inline explicit Quaternion(const Quaternion<OtherScalarType>& other)
|
---|
| 193 | { m_coeffs = other.coeffs().template cast<Scalar>(); }
|
---|
| 194 |
|
---|
| 195 | /** \returns \c true if \c *this is approximately equal to \a other, within the precision
|
---|
| 196 | * determined by \a prec.
|
---|
| 197 | *
|
---|
| 198 | * \sa MatrixBase::isApprox() */
|
---|
| 199 | bool isApprox(const Quaternion& other, typename NumTraits<Scalar>::Real prec = precision<Scalar>()) const
|
---|
| 200 | { return m_coeffs.isApprox(other.m_coeffs, prec); }
|
---|
| 201 |
|
---|
| 202 | protected:
|
---|
| 203 | Coefficients m_coeffs;
|
---|
| 204 | };
|
---|
| 205 |
|
---|
| 206 | /** \ingroup Geometry_Module
|
---|
| 207 | * single precision quaternion type */
|
---|
| 208 | typedef Quaternion<float> Quaternionf;
|
---|
| 209 | /** \ingroup Geometry_Module
|
---|
| 210 | * double precision quaternion type */
|
---|
| 211 | typedef Quaternion<double> Quaterniond;
|
---|
| 212 |
|
---|
| 213 | // Generic Quaternion * Quaternion product
|
---|
| 214 | template<typename Scalar> inline Quaternion<Scalar>
|
---|
| 215 | ei_quaternion_product(const Quaternion<Scalar>& a, const Quaternion<Scalar>& b)
|
---|
| 216 | {
|
---|
| 217 | return Quaternion<Scalar>
|
---|
| 218 | (
|
---|
| 219 | a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
|
---|
| 220 | a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
|
---|
| 221 | a.w() * b.y() + a.y() * b.w() + a.z() * b.x() - a.x() * b.z(),
|
---|
| 222 | a.w() * b.z() + a.z() * b.w() + a.x() * b.y() - a.y() * b.x()
|
---|
| 223 | );
|
---|
| 224 | }
|
---|
| 225 |
|
---|
| 226 | /** \returns the concatenation of two rotations as a quaternion-quaternion product */
|
---|
| 227 | template <typename Scalar>
|
---|
| 228 | inline Quaternion<Scalar> Quaternion<Scalar>::operator* (const Quaternion& other) const
|
---|
| 229 | {
|
---|
| 230 | return ei_quaternion_product(*this,other);
|
---|
| 231 | }
|
---|
| 232 |
|
---|
| 233 | /** \sa operator*(Quaternion) */
|
---|
| 234 | template <typename Scalar>
|
---|
| 235 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator*= (const Quaternion& other)
|
---|
| 236 | {
|
---|
| 237 | return (*this = *this * other);
|
---|
| 238 | }
|
---|
| 239 |
|
---|
| 240 | /** Rotation of a vector by a quaternion.
|
---|
| 241 | * \remarks If the quaternion is used to rotate several points (>1)
|
---|
| 242 | * then it is much more efficient to first convert it to a 3x3 Matrix.
|
---|
| 243 | * Comparison of the operation cost for n transformations:
|
---|
| 244 | * - Quaternion: 30n
|
---|
| 245 | * - Via a Matrix3: 24 + 15n
|
---|
| 246 | */
|
---|
| 247 | template <typename Scalar>
|
---|
| 248 | template<typename Derived>
|
---|
| 249 | inline typename Quaternion<Scalar>::Vector3
|
---|
| 250 | Quaternion<Scalar>::operator* (const MatrixBase<Derived>& v) const
|
---|
| 251 | {
|
---|
| 252 | // Note that this algorithm comes from the optimization by hand
|
---|
| 253 | // of the conversion to a Matrix followed by a Matrix/Vector product.
|
---|
| 254 | // It appears to be much faster than the common algorithm found
|
---|
| 255 | // in the litterature (30 versus 39 flops). It also requires two
|
---|
| 256 | // Vector3 as temporaries.
|
---|
| 257 | Vector3 uv;
|
---|
| 258 | uv = 2 * this->vec().cross(v);
|
---|
| 259 | return v + this->w() * uv + this->vec().cross(uv);
|
---|
| 260 | }
|
---|
| 261 |
|
---|
| 262 | template<typename Scalar>
|
---|
| 263 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const Quaternion& other)
|
---|
| 264 | {
|
---|
| 265 | m_coeffs = other.m_coeffs;
|
---|
| 266 | return *this;
|
---|
| 267 | }
|
---|
| 268 |
|
---|
| 269 | /** Set \c *this from an angle-axis \a aa and returns a reference to \c *this
|
---|
| 270 | */
|
---|
| 271 | template<typename Scalar>
|
---|
| 272 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const AngleAxisType& aa)
|
---|
| 273 | {
|
---|
| 274 | Scalar ha = Scalar(0.5)*aa.angle(); // Scalar(0.5) to suppress precision loss warnings
|
---|
| 275 | this->w() = ei_cos(ha);
|
---|
| 276 | this->vec() = ei_sin(ha) * aa.axis();
|
---|
| 277 | return *this;
|
---|
| 278 | }
|
---|
| 279 |
|
---|
| 280 | /** Set \c *this from the expression \a xpr:
|
---|
| 281 | * - if \a xpr is a 4x1 vector, then \a xpr is assumed to be a quaternion
|
---|
| 282 | * - if \a xpr is a 3x3 matrix, then \a xpr is assumed to be rotation matrix
|
---|
| 283 | * and \a xpr is converted to a quaternion
|
---|
| 284 | */
|
---|
| 285 | template<typename Scalar>
|
---|
| 286 | template<typename Derived>
|
---|
| 287 | inline Quaternion<Scalar>& Quaternion<Scalar>::operator=(const MatrixBase<Derived>& xpr)
|
---|
| 288 | {
|
---|
| 289 | ei_quaternion_assign_impl<Derived>::run(*this, xpr.derived());
|
---|
| 290 | return *this;
|
---|
| 291 | }
|
---|
| 292 |
|
---|
| 293 | /** Convert the quaternion to a 3x3 rotation matrix */
|
---|
| 294 | template<typename Scalar>
|
---|
| 295 | inline typename Quaternion<Scalar>::Matrix3
|
---|
| 296 | Quaternion<Scalar>::toRotationMatrix(void) const
|
---|
| 297 | {
|
---|
| 298 | // NOTE if inlined, then gcc 4.2 and 4.4 get rid of the temporary (not gcc 4.3 !!)
|
---|
| 299 | // if not inlined then the cost of the return by value is huge ~ +35%,
|
---|
| 300 | // however, not inlining this function is an order of magnitude slower, so
|
---|
| 301 | // it has to be inlined, and so the return by value is not an issue
|
---|
| 302 | Matrix3 res;
|
---|
| 303 |
|
---|
| 304 | const Scalar tx = Scalar(2)*this->x();
|
---|
| 305 | const Scalar ty = Scalar(2)*this->y();
|
---|
| 306 | const Scalar tz = Scalar(2)*this->z();
|
---|
| 307 | const Scalar twx = tx*this->w();
|
---|
| 308 | const Scalar twy = ty*this->w();
|
---|
| 309 | const Scalar twz = tz*this->w();
|
---|
| 310 | const Scalar txx = tx*this->x();
|
---|
| 311 | const Scalar txy = ty*this->x();
|
---|
| 312 | const Scalar txz = tz*this->x();
|
---|
| 313 | const Scalar tyy = ty*this->y();
|
---|
| 314 | const Scalar tyz = tz*this->y();
|
---|
| 315 | const Scalar tzz = tz*this->z();
|
---|
| 316 |
|
---|
| 317 | res.coeffRef(0,0) = Scalar(1)-(tyy+tzz);
|
---|
| 318 | res.coeffRef(0,1) = txy-twz;
|
---|
| 319 | res.coeffRef(0,2) = txz+twy;
|
---|
| 320 | res.coeffRef(1,0) = txy+twz;
|
---|
| 321 | res.coeffRef(1,1) = Scalar(1)-(txx+tzz);
|
---|
| 322 | res.coeffRef(1,2) = tyz-twx;
|
---|
| 323 | res.coeffRef(2,0) = txz-twy;
|
---|
| 324 | res.coeffRef(2,1) = tyz+twx;
|
---|
| 325 | res.coeffRef(2,2) = Scalar(1)-(txx+tyy);
|
---|
| 326 |
|
---|
| 327 | return res;
|
---|
| 328 | }
|
---|
| 329 |
|
---|
| 330 | /** Sets *this to be a quaternion representing a rotation sending the vector \a a to the vector \a b.
|
---|
| 331 | *
|
---|
| 332 | * \returns a reference to *this.
|
---|
| 333 | *
|
---|
| 334 | * Note that the two input vectors do \b not have to be normalized.
|
---|
| 335 | */
|
---|
| 336 | template<typename Scalar>
|
---|
| 337 | template<typename Derived1, typename Derived2>
|
---|
| 338 | inline Quaternion<Scalar>& Quaternion<Scalar>::setFromTwoVectors(const MatrixBase<Derived1>& a, const MatrixBase<Derived2>& b)
|
---|
| 339 | {
|
---|
| 340 | Vector3 v0 = a.normalized();
|
---|
| 341 | Vector3 v1 = b.normalized();
|
---|
| 342 | Scalar c = v0.eigen2_dot(v1);
|
---|
| 343 |
|
---|
| 344 | // if dot == 1, vectors are the same
|
---|
| 345 | if (ei_isApprox(c,Scalar(1)))
|
---|
| 346 | {
|
---|
| 347 | // set to identity
|
---|
| 348 | this->w() = 1; this->vec().setZero();
|
---|
| 349 | return *this;
|
---|
| 350 | }
|
---|
| 351 | // if dot == -1, vectors are opposites
|
---|
| 352 | if (ei_isApprox(c,Scalar(-1)))
|
---|
| 353 | {
|
---|
| 354 | this->vec() = v0.unitOrthogonal();
|
---|
| 355 | this->w() = 0;
|
---|
| 356 | return *this;
|
---|
| 357 | }
|
---|
| 358 |
|
---|
| 359 | Vector3 axis = v0.cross(v1);
|
---|
| 360 | Scalar s = ei_sqrt((Scalar(1)+c)*Scalar(2));
|
---|
| 361 | Scalar invs = Scalar(1)/s;
|
---|
| 362 | this->vec() = axis * invs;
|
---|
| 363 | this->w() = s * Scalar(0.5);
|
---|
| 364 |
|
---|
| 365 | return *this;
|
---|
| 366 | }
|
---|
| 367 |
|
---|
| 368 | /** \returns the multiplicative inverse of \c *this
|
---|
| 369 | * Note that in most cases, i.e., if you simply want the opposite rotation,
|
---|
| 370 | * and/or the quaternion is normalized, then it is enough to use the conjugate.
|
---|
| 371 | *
|
---|
| 372 | * \sa Quaternion::conjugate()
|
---|
| 373 | */
|
---|
| 374 | template <typename Scalar>
|
---|
| 375 | inline Quaternion<Scalar> Quaternion<Scalar>::inverse() const
|
---|
| 376 | {
|
---|
| 377 | // FIXME should this function be called multiplicativeInverse and conjugate() be called inverse() or opposite() ??
|
---|
| 378 | Scalar n2 = this->squaredNorm();
|
---|
| 379 | if (n2 > 0)
|
---|
| 380 | return Quaternion(conjugate().coeffs() / n2);
|
---|
| 381 | else
|
---|
| 382 | {
|
---|
| 383 | // return an invalid result to flag the error
|
---|
| 384 | return Quaternion(Coefficients::Zero());
|
---|
| 385 | }
|
---|
| 386 | }
|
---|
| 387 |
|
---|
| 388 | /** \returns the conjugate of the \c *this which is equal to the multiplicative inverse
|
---|
| 389 | * if the quaternion is normalized.
|
---|
| 390 | * The conjugate of a quaternion represents the opposite rotation.
|
---|
| 391 | *
|
---|
| 392 | * \sa Quaternion::inverse()
|
---|
| 393 | */
|
---|
| 394 | template <typename Scalar>
|
---|
| 395 | inline Quaternion<Scalar> Quaternion<Scalar>::conjugate() const
|
---|
| 396 | {
|
---|
| 397 | return Quaternion(this->w(),-this->x(),-this->y(),-this->z());
|
---|
| 398 | }
|
---|
| 399 |
|
---|
| 400 | /** \returns the angle (in radian) between two rotations
|
---|
| 401 | * \sa eigen2_dot()
|
---|
| 402 | */
|
---|
| 403 | template <typename Scalar>
|
---|
| 404 | inline Scalar Quaternion<Scalar>::angularDistance(const Quaternion& other) const
|
---|
| 405 | {
|
---|
| 406 | double d = ei_abs(this->eigen2_dot(other));
|
---|
| 407 | if (d>=1.0)
|
---|
| 408 | return 0;
|
---|
| 409 | return Scalar(2) * std::acos(d);
|
---|
| 410 | }
|
---|
| 411 |
|
---|
| 412 | /** \returns the spherical linear interpolation between the two quaternions
|
---|
| 413 | * \c *this and \a other at the parameter \a t
|
---|
| 414 | */
|
---|
| 415 | template <typename Scalar>
|
---|
| 416 | Quaternion<Scalar> Quaternion<Scalar>::slerp(Scalar t, const Quaternion& other) const
|
---|
| 417 | {
|
---|
| 418 | static const Scalar one = Scalar(1) - machine_epsilon<Scalar>();
|
---|
| 419 | Scalar d = this->eigen2_dot(other);
|
---|
| 420 | Scalar absD = ei_abs(d);
|
---|
| 421 |
|
---|
| 422 | Scalar scale0;
|
---|
| 423 | Scalar scale1;
|
---|
| 424 |
|
---|
| 425 | if (absD>=one)
|
---|
| 426 | {
|
---|
| 427 | scale0 = Scalar(1) - t;
|
---|
| 428 | scale1 = t;
|
---|
| 429 | }
|
---|
| 430 | else
|
---|
| 431 | {
|
---|
| 432 | // theta is the angle between the 2 quaternions
|
---|
| 433 | Scalar theta = std::acos(absD);
|
---|
| 434 | Scalar sinTheta = ei_sin(theta);
|
---|
| 435 |
|
---|
| 436 | scale0 = ei_sin( ( Scalar(1) - t ) * theta) / sinTheta;
|
---|
| 437 | scale1 = ei_sin( ( t * theta) ) / sinTheta;
|
---|
| 438 | if (d<0)
|
---|
| 439 | scale1 = -scale1;
|
---|
| 440 | }
|
---|
| 441 |
|
---|
| 442 | return Quaternion<Scalar>(scale0 * coeffs() + scale1 * other.coeffs());
|
---|
| 443 | }
|
---|
| 444 |
|
---|
| 445 | // set from a rotation matrix
|
---|
| 446 | template<typename Other>
|
---|
| 447 | struct ei_quaternion_assign_impl<Other,3,3>
|
---|
| 448 | {
|
---|
| 449 | typedef typename Other::Scalar Scalar;
|
---|
| 450 | static inline void run(Quaternion<Scalar>& q, const Other& mat)
|
---|
| 451 | {
|
---|
| 452 | // This algorithm comes from "Quaternion Calculus and Fast Animation",
|
---|
| 453 | // Ken Shoemake, 1987 SIGGRAPH course notes
|
---|
| 454 | Scalar t = mat.trace();
|
---|
| 455 | if (t > 0)
|
---|
| 456 | {
|
---|
| 457 | t = ei_sqrt(t + Scalar(1.0));
|
---|
| 458 | q.w() = Scalar(0.5)*t;
|
---|
| 459 | t = Scalar(0.5)/t;
|
---|
| 460 | q.x() = (mat.coeff(2,1) - mat.coeff(1,2)) * t;
|
---|
| 461 | q.y() = (mat.coeff(0,2) - mat.coeff(2,0)) * t;
|
---|
| 462 | q.z() = (mat.coeff(1,0) - mat.coeff(0,1)) * t;
|
---|
| 463 | }
|
---|
| 464 | else
|
---|
| 465 | {
|
---|
| 466 | int i = 0;
|
---|
| 467 | if (mat.coeff(1,1) > mat.coeff(0,0))
|
---|
| 468 | i = 1;
|
---|
| 469 | if (mat.coeff(2,2) > mat.coeff(i,i))
|
---|
| 470 | i = 2;
|
---|
| 471 | int j = (i+1)%3;
|
---|
| 472 | int k = (j+1)%3;
|
---|
| 473 |
|
---|
| 474 | t = ei_sqrt(mat.coeff(i,i)-mat.coeff(j,j)-mat.coeff(k,k) + Scalar(1.0));
|
---|
| 475 | q.coeffs().coeffRef(i) = Scalar(0.5) * t;
|
---|
| 476 | t = Scalar(0.5)/t;
|
---|
| 477 | q.w() = (mat.coeff(k,j)-mat.coeff(j,k))*t;
|
---|
| 478 | q.coeffs().coeffRef(j) = (mat.coeff(j,i)+mat.coeff(i,j))*t;
|
---|
| 479 | q.coeffs().coeffRef(k) = (mat.coeff(k,i)+mat.coeff(i,k))*t;
|
---|
| 480 | }
|
---|
| 481 | }
|
---|
| 482 | };
|
---|
| 483 |
|
---|
| 484 | // set from a vector of coefficients assumed to be a quaternion
|
---|
| 485 | template<typename Other>
|
---|
| 486 | struct ei_quaternion_assign_impl<Other,4,1>
|
---|
| 487 | {
|
---|
| 488 | typedef typename Other::Scalar Scalar;
|
---|
| 489 | static inline void run(Quaternion<Scalar>& q, const Other& vec)
|
---|
| 490 | {
|
---|
| 491 | q.coeffs() = vec;
|
---|
| 492 | }
|
---|
| 493 | };
|
---|
| 494 |
|
---|
| 495 | } // end namespace Eigen
|
---|