1 // Copyright (C) 2006 Mathias Froehlich - Mathias.Froehlich@web.de
3 // This library is free software; you can redistribute it and/or
4 // modify it under the terms of the GNU Library General Public
5 // License as published by the Free Software Foundation; either
6 // version 2 of the License, or (at your option) any later version.
8 // This library is distributed in the hope that it will be useful,
9 // but WITHOUT ANY WARRANTY; without even the implied warranty of
10 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
11 // Library General Public License for more details.
13 // You should have received a copy of the GNU General Public License
14 // along with this program; if not, write to the Free Software
15 // Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
32 struct SGQuatStorage {
33 /// Readonly raw storage interface
34 const T (&data(void) const)[4]
36 /// Readonly raw storage interface
48 struct SGQuatStorage<double> : public osg::Quat {
49 /// Access raw data by index, the index is unchecked
50 const double (&data(void) const)[4]
51 { return osg::Quat::_v; }
52 /// Access raw data by index, the index is unchecked
53 double (&data(void))[4]
54 { return osg::Quat::_v; }
56 const osg::Quat& osg() const
64 class SGQuat : protected SGQuatStorage<T> {
68 /// Default constructor. Does not initialize at all.
69 /// If you need them zero initialized, SGQuat::zeros()
72 /// Initialize with nans in the debug build, that will guarantee to have
73 /// a fast uninitialized default constructor in the release but shows up
74 /// uninitialized values in the debug build very fast ...
76 for (unsigned i = 0; i < 4; ++i)
77 data()[i] = SGLimits<T>::quiet_NaN();
80 /// Constructor. Initialize by the given values
81 SGQuat(T _x, T _y, T _z, T _w)
82 { x() = _x; y() = _y; z() = _z; w() = _w; }
83 /// Constructor. Initialize by the content of a plain array,
84 /// make sure it has at least 4 elements
85 explicit SGQuat(const T* d)
86 { data()[0] = d[0]; data()[1] = d[1]; data()[2] = d[2]; data()[3] = d[3]; }
87 explicit SGQuat(const osg::Quat& d)
88 { data()[0] = d[0]; data()[1] = d[1]; data()[2] = d[2]; data()[3] = d[3]; }
90 /// Return a unit quaternion
91 static SGQuat unit(void)
92 { return fromRealImag(1, SGVec3<T>(0, 0, 0)); }
94 /// Return a quaternion from euler angles
95 static SGQuat fromEulerRad(T z, T y, T x)
98 T zd2 = T(0.5)*z; T yd2 = T(0.5)*y; T xd2 = T(0.5)*x;
99 T Szd2 = sin(zd2); T Syd2 = sin(yd2); T Sxd2 = sin(xd2);
100 T Czd2 = cos(zd2); T Cyd2 = cos(yd2); T Cxd2 = cos(xd2);
101 T Cxd2Czd2 = Cxd2*Czd2; T Cxd2Szd2 = Cxd2*Szd2;
102 T Sxd2Szd2 = Sxd2*Szd2; T Sxd2Czd2 = Sxd2*Czd2;
103 q.w() = Cxd2Czd2*Cyd2 + Sxd2Szd2*Syd2;
104 q.x() = Sxd2Czd2*Cyd2 - Cxd2Szd2*Syd2;
105 q.y() = Cxd2Czd2*Syd2 + Sxd2Szd2*Cyd2;
106 q.z() = Cxd2Szd2*Cyd2 - Sxd2Czd2*Syd2;
110 /// Return a quaternion from euler angles in degrees
111 static SGQuat fromEulerDeg(T z, T y, T x)
113 return fromEulerRad(SGMisc<T>::deg2rad(z), SGMisc<T>::deg2rad(y),
114 SGMisc<T>::deg2rad(x));
117 /// Return a quaternion from euler angles
118 static SGQuat fromYawPitchRoll(T y, T p, T r)
119 { return fromEulerRad(y, p, r); }
121 /// Return a quaternion from euler angles
122 static SGQuat fromYawPitchRollDeg(T y, T p, T r)
123 { return fromEulerDeg(y, p, r); }
125 /// Return a quaternion from euler angles
126 static SGQuat fromHeadAttBank(T h, T a, T b)
127 { return fromEulerRad(h, a, b); }
129 /// Return a quaternion from euler angles
130 static SGQuat fromHeadAttBankDeg(T h, T a, T b)
131 { return fromEulerDeg(h, a, b); }
133 /// Return a quaternion rotation the the horizontal local frame from given
134 /// longitude and latitude
135 static SGQuat fromLonLatRad(T lon, T lat)
139 T yd2 = T(-0.25)*SGMisc<value_type>::pi() - T(0.5)*lat;
151 /// Return a quaternion rotation the the horizontal local frame from given
152 /// longitude and latitude
153 static SGQuat fromLonLatDeg(T lon, T lat)
154 { return fromLonLatRad(SGMisc<T>::deg2rad(lon), SGMisc<T>::deg2rad(lat)); }
156 /// Return a quaternion rotation the the horizontal local frame from given
157 /// longitude and latitude
158 static SGQuat fromLonLat(const SGGeod& geod)
159 { return fromLonLatRad(geod.getLongitudeRad(), geod.getLatitudeRad()); }
161 /// Create a quaternion from the angle axis representation
162 static SGQuat fromAngleAxis(T angle, const SGVec3<T>& axis)
164 T angle2 = 0.5*angle;
165 return fromRealImag(cos(angle2), T(sin(angle2))*axis);
168 /// Create a quaternion from the angle axis representation
169 static SGQuat fromAngleAxisDeg(T angle, const SGVec3<T>& axis)
170 { return fromAngleAxis(SGMisc<T>::deg2rad(angle), axis); }
172 /// Create a quaternion from the angle axis representation where the angle
173 /// is stored in the axis' length
174 static SGQuat fromAngleAxis(const SGVec3<T>& axis)
176 T nAxis = norm(axis);
177 if (nAxis <= SGLimits<T>::min())
178 return SGQuat(1, 0, 0, 0);
179 T angle2 = 0.5*nAxis;
180 return fromRealImag(cos(angle2), T(sin(angle2)/nAxis)*axis);
183 static SGQuat fromRotateTo(const SGVec3<T>& from, const SGVec3<T>& to)
185 T nfrom = norm(from);
187 if (nfrom < SGLimits<T>::min() || nto < SGLimits<T>::min())
188 return SGQuat::unit();
190 return SGQuat::fromRotateToNorm((1/nfrom)*from, (1/nto)*to);
193 // FIXME more finegrained error behavour.
194 static SGQuat fromRotateTo(const SGVec3<T>& v1, unsigned i1,
195 const SGVec3<T>& v2, unsigned i2)
199 if (nrmv1 < SGLimits<T>::min() || nrmv2 < SGLimits<T>::min())
200 return SGQuat::unit();
202 SGVec3<T> nv1 = (1/nrmv1)*v1;
203 SGVec3<T> nv2 = (1/nrmv2)*v2;
204 T dv1v2 = dot(nv1, nv2);
205 if (fabs(fabs(dv1v2)-1) < SGLimits<T>::epsilon())
206 return SGQuat::unit();
208 // The target vector for the first rotation
209 SGVec3<T> nto1 = SGVec3<T>::zeros();
210 SGVec3<T> nto2 = SGVec3<T>::zeros();
214 // The first rotation can be done with the usual routine.
215 SGQuat q = SGQuat::fromRotateToNorm(nv1, nto1);
217 // The rotation axis for the second rotation is the
218 // target for the first one, so the rotation axis is nto1
219 // We need to get the angle.
221 // Make nv2 exactly orthogonal to nv1.
222 nv2 = normalize(nv2 - dv1v2*nv1);
224 SGVec3<T> tnv2 = q.transform(nv2);
225 T cosang = dot(nto2, tnv2);
226 T cos05ang = T(0.5+0.5*cosang);
229 cos05ang = sqrt(cos05ang);
230 T sig = dot(nto1, cross(nto2, tnv2));
231 T sin05ang = T(0.5-0.5*cosang);
234 sin05ang = copysign(sqrt(sin05ang), sig);
235 q *= SGQuat::fromRealImag(cos05ang, sin05ang*nto1);
241 // Return a quaternion which rotates the vector given by v
242 // to the vector -v. Other directions are *not* preserved.
243 static SGQuat fromChangeSign(const SGVec3<T>& v)
245 // The vector from points to the oposite direction than to.
246 // Find a vector perpandicular to the vector to.
247 T absv1 = fabs(v(0));
248 T absv2 = fabs(v(1));
249 T absv3 = fabs(v(2));
252 if (absv2 < absv1 && absv3 < absv1) {
254 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(quot, -1, 0);
255 } else if (absv1 < absv2 && absv3 < absv2) {
257 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(0, quot, -1);
258 } else if (absv1 < absv3 && absv2 < absv3) {
260 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(-1, 0, quot);
262 // The all zero case.
263 return SGQuat::unit();
266 return SGQuat::fromRealImag(0, axis);
269 /// Return a quaternion from real and imaginary part
270 static SGQuat fromRealImag(T r, const SGVec3<T>& i)
280 /// Return an all zero vector
281 static SGQuat zeros(void)
282 { return SGQuat(0, 0, 0, 0); }
284 /// write the euler angles into the references
285 void getEulerRad(T& zRad, T& yRad, T& xRad) const
292 T num = 2*(y()*z() + w()*x());
293 T den = sqrQW - sqrQX - sqrQY + sqrQZ;
294 if (fabs(den) < SGLimits<T>::min() &&
295 fabs(num) < SGLimits<T>::min())
298 xRad = atan2(num, den);
300 T tmp = 2*(x()*z() - w()*y());
302 yRad = 0.5*SGMisc<T>::pi();
304 yRad = -0.5*SGMisc<T>::pi();
308 num = 2*(x()*y() + w()*z());
309 den = sqrQW + sqrQX - sqrQY - sqrQZ;
310 if (fabs(den) < SGLimits<T>::min() &&
311 fabs(num) < SGLimits<T>::min())
314 T psi = atan2(num, den);
316 psi += 2*SGMisc<T>::pi();
321 /// write the euler angles in degrees into the references
322 void getEulerDeg(T& zDeg, T& yDeg, T& xDeg) const
324 getEulerRad(zDeg, yDeg, xDeg);
325 zDeg = SGMisc<T>::rad2deg(zDeg);
326 yDeg = SGMisc<T>::rad2deg(yDeg);
327 xDeg = SGMisc<T>::rad2deg(xDeg);
330 /// write the angle axis representation into the references
331 void getAngleAxis(T& angle, SGVec3<T>& axis) const
334 if (nrm < SGLimits<T>::min()) {
336 axis = SGVec3<T>(0, 0, 0);
339 angle = acos(SGMisc<T>::max(-1, SGMisc<T>::min(1, rNrm*w())));
341 if (fabs(sAng) < SGLimits<T>::min())
342 axis = SGVec3<T>(1, 0, 0);
344 axis = (rNrm/sAng)*imag(*this);
349 /// write the angle axis representation into the references
350 void getAngleAxis(SGVec3<T>& axis) const
353 getAngleAxis(angle, axis);
357 /// Access by index, the index is unchecked
358 const T& operator()(unsigned i) const
359 { return data()[i]; }
360 /// Access by index, the index is unchecked
361 T& operator()(unsigned i)
362 { return data()[i]; }
364 /// Access raw data by index, the index is unchecked
365 const T& operator[](unsigned i) const
366 { return data()[i]; }
367 /// Access raw data by index, the index is unchecked
368 T& operator[](unsigned i)
369 { return data()[i]; }
371 /// Access the x component
372 const T& x(void) const
373 { return data()[0]; }
374 /// Access the x component
376 { return data()[0]; }
377 /// Access the y component
378 const T& y(void) const
379 { return data()[1]; }
380 /// Access the y component
382 { return data()[1]; }
383 /// Access the z component
384 const T& z(void) const
385 { return data()[2]; }
386 /// Access the z component
388 { return data()[2]; }
389 /// Access the w component
390 const T& w(void) const
391 { return data()[3]; }
392 /// Access the w component
394 { return data()[3]; }
396 /// Get the data pointer
397 using SGQuatStorage<T>::data;
399 /// Readonly interface function to ssg's sgQuat/sgdQuat
400 const T (&sg(void) const)[4]
402 /// Interface function to ssg's sgQuat/sgdQuat
406 /// Interface function to osg's Quat*
407 using SGQuatStorage<T>::osg;
410 SGQuat& operator+=(const SGQuat& v)
411 { data()[0]+=v(0);data()[1]+=v(1);data()[2]+=v(2);data()[3]+=v(3);return *this; }
412 /// Inplace subtraction
413 SGQuat& operator-=(const SGQuat& v)
414 { data()[0]-=v(0);data()[1]-=v(1);data()[2]-=v(2);data()[3]-=v(3);return *this; }
415 /// Inplace scalar multiplication
417 SGQuat& operator*=(S s)
418 { data()[0] *= s; data()[1] *= s; data()[2] *= s; data()[3] *= s; return *this; }
419 /// Inplace scalar multiplication by 1/s
421 SGQuat& operator/=(S s)
422 { return operator*=(1/T(s)); }
423 /// Inplace quaternion multiplication
424 SGQuat& operator*=(const SGQuat& v);
426 /// Transform a vector from the current coordinate frame to a coordinate
427 /// frame rotated with the quaternion
428 SGVec3<T> transform(const SGVec3<T>& v) const
430 T r = 2/dot(*this, *this);
431 SGVec3<T> qimag = imag(*this);
433 return (r*qr*qr - 1)*v + (r*dot(qimag, v))*qimag - (r*qr)*cross(qimag, v);
435 /// Transform a vector from the coordinate frame rotated with the quaternion
436 /// to the current coordinate frame
437 SGVec3<T> backTransform(const SGVec3<T>& v) const
439 T r = 2/dot(*this, *this);
440 SGVec3<T> qimag = imag(*this);
442 return (r*qr*qr - 1)*v + (r*dot(qimag, v))*qimag + (r*qr)*cross(qimag, v);
445 /// Rotate a given vector with the quaternion
446 SGVec3<T> rotate(const SGVec3<T>& v) const
447 { return backTransform(v); }
448 /// Rotate a given vector with the inverse quaternion
449 SGVec3<T> rotateBack(const SGVec3<T>& v) const
450 { return transform(v); }
452 /// Return the time derivative of the quaternion given the angular velocity
454 derivative(const SGVec3<T>& angVel)
458 deriv.w() = 0.5*(-x()*angVel(0) - y()*angVel(1) - z()*angVel(2));
459 deriv.x() = 0.5*( w()*angVel(0) - z()*angVel(1) + y()*angVel(2));
460 deriv.y() = 0.5*( z()*angVel(0) + w()*angVel(1) - x()*angVel(2));
461 deriv.z() = 0.5*(-y()*angVel(0) + x()*angVel(1) + w()*angVel(2));
468 // Private because it assumes normalized inputs.
470 fromRotateToSmaller90Deg(T cosang,
471 const SGVec3<T>& from, const SGVec3<T>& to)
473 // In this function we assume that the angle required to rotate from
474 // the vector from to the vector to is <= 90 deg.
475 // That is done so because of possible instabilities when we rotate more
478 // Note that the next comment does actually cover a *more* *general* case
479 // than we need in this function. That shows that this formula is even
480 // valid for rotations up to 180deg.
482 // Because of the signs in the axis, it is sufficient to care for angles
483 // in the interval [-pi,pi]. That means that 0.5*angle is in the interval
484 // [-pi/2,pi/2]. But in that range the cosine is allways >= 0.
485 // So we do not need to care for egative roots in the following equation:
486 T cos05ang = sqrt(0.5+0.5*cosang);
489 // Now our assumption of angles <= 90 deg comes in play.
490 // For that reason, we know that cos05ang is not zero.
491 // It is even more, we can see from the above formula that
492 // sqrt(0.5) < cos05ang.
495 // Compute the rotation axis, that is
496 // sin(angle)*normalized rotation axis
497 SGVec3<T> axis = cross(to, from);
499 // We need sin(0.5*angle)*normalized rotation axis.
500 // So rescale with sin(0.5*x)/sin(x).
501 // To do that we use the equation:
502 // sin(x) = 2*sin(0.5*x)*cos(0.5*x)
503 return SGQuat::fromRealImag( cos05ang, (1/(2*cos05ang))*axis);
506 // Private because it assumes normalized inputs.
508 fromRotateToNorm(const SGVec3<T>& from, const SGVec3<T>& to)
510 // To avoid instabilities with roundoff, we distinguish between rotations
511 // with more then 90deg and rotations with less than 90deg.
513 // Compute the cosine of the angle.
514 T cosang = dot(from, to);
516 // For the small ones do direct computation
517 if (T(-0.5) < cosang)
518 return SGQuat::fromRotateToSmaller90Deg(cosang, from, to);
520 // For larger rotations. first rotate from to -from.
521 // Past that we will have a smaller angle again.
522 SGQuat q1 = SGQuat::fromChangeSign(from);
523 SGQuat q2 = SGQuat::fromRotateToSmaller90Deg(-cosang, -from, to);
528 /// Unary +, do nothing ...
532 operator+(const SGQuat<T>& v)
535 /// Unary -, do nearly nothing
539 operator-(const SGQuat<T>& v)
540 { return SGQuat<T>(-v(0), -v(1), -v(2), -v(3)); }
546 operator+(const SGQuat<T>& v1, const SGQuat<T>& v2)
547 { return SGQuat<T>(v1(0)+v2(0), v1(1)+v2(1), v1(2)+v2(2), v1(3)+v2(3)); }
553 operator-(const SGQuat<T>& v1, const SGQuat<T>& v2)
554 { return SGQuat<T>(v1(0)-v2(0), v1(1)-v2(1), v1(2)-v2(2), v1(3)-v2(3)); }
556 /// Scalar multiplication
557 template<typename S, typename T>
560 operator*(S s, const SGQuat<T>& v)
561 { return SGQuat<T>(s*v(0), s*v(1), s*v(2), s*v(3)); }
563 /// Scalar multiplication
564 template<typename S, typename T>
567 operator*(const SGQuat<T>& v, S s)
568 { return SGQuat<T>(s*v(0), s*v(1), s*v(2), s*v(3)); }
570 /// Quaterion multiplication
574 operator*(const SGQuat<T>& v1, const SGQuat<T>& v2)
577 v.x() = v1.w()*v2.x() + v1.x()*v2.w() + v1.y()*v2.z() - v1.z()*v2.y();
578 v.y() = v1.w()*v2.y() - v1.x()*v2.z() + v1.y()*v2.w() + v1.z()*v2.x();
579 v.z() = v1.w()*v2.z() + v1.x()*v2.y() - v1.y()*v2.x() + v1.z()*v2.w();
580 v.w() = v1.w()*v2.w() - v1.x()*v2.x() - v1.y()*v2.y() - v1.z()*v2.z();
584 /// Now define the inplace multiplication
588 SGQuat<T>::operator*=(const SGQuat& v)
589 { (*this) = (*this)*v; return *this; }
591 /// The conjugate of the quaternion, this is also the
592 /// inverse for normalized quaternions
596 conj(const SGQuat<T>& v)
597 { return SGQuat<T>(-v(0), -v(1), -v(2), v(3)); }
599 /// The conjugate of the quaternion, this is also the
600 /// inverse for normalized quaternions
604 inverse(const SGQuat<T>& v)
605 { return (1/dot(v, v))*SGQuat<T>(-v(0), -v(1), -v(2), v(3)); }
607 /// The imagniary part of the quaternion
611 real(const SGQuat<T>& v)
614 /// The imagniary part of the quaternion
618 imag(const SGQuat<T>& v)
619 { return SGVec3<T>(v.x(), v.y(), v.z()); }
621 /// Scalar dot product
625 dot(const SGQuat<T>& v1, const SGQuat<T>& v2)
626 { return v1(0)*v2(0) + v1(1)*v2(1) + v1(2)*v2(2) + v1(3)*v2(3); }
628 /// The euclidean norm of the vector, that is what most people call length
632 norm(const SGQuat<T>& v)
633 { return sqrt(dot(v, v)); }
635 /// The euclidean norm of the vector, that is what most people call length
639 length(const SGQuat<T>& v)
640 { return sqrt(dot(v, v)); }
642 /// The 1-norm of the vector, this one is the fastest length function we
643 /// can implement on modern cpu's
647 norm1(const SGQuat<T>& v)
648 { return fabs(v(0)) + fabs(v(1)) + fabs(v(2)) + fabs(v(3)); }
650 /// The euclidean norm of the vector, that is what most people call length
654 normalize(const SGQuat<T>& q)
655 { return (1/norm(q))*q; }
657 /// Return true if exactly the same
661 operator==(const SGQuat<T>& v1, const SGQuat<T>& v2)
662 { return v1(0)==v2(0) && v1(1)==v2(1) && v1(2)==v2(2) && v1(3)==v2(3); }
664 /// Return true if not exactly the same
668 operator!=(const SGQuat<T>& v1, const SGQuat<T>& v2)
669 { return ! (v1 == v2); }
671 /// Return true if equal to the relative tolerance tol
672 /// Note that this is not the same than comparing quaternions to represent
673 /// the same rotation
677 equivalent(const SGQuat<T>& v1, const SGQuat<T>& v2, T tol)
678 { return norm1(v1 - v2) < tol*(norm1(v1) + norm1(v2)); }
680 /// Return true if about equal to roundoff of the underlying type
681 /// Note that this is not the same than comparing quaternions to represent
682 /// the same rotation
686 equivalent(const SGQuat<T>& v1, const SGQuat<T>& v2)
687 { return equivalent(v1, v2, 100*SGLimits<T>::epsilon()); }
693 isNaN(const SGQuat<T>& v)
695 return SGMisc<T>::isNaN(v(0)) || SGMisc<T>::isNaN(v(1))
696 || SGMisc<T>::isNaN(v(2)) || SGMisc<T>::isNaN(v(3));
700 /// quaternion interpolation for t in [0,1] interpolate between src (=0)
705 interpolate(T t, const SGQuat<T>& src, const SGQuat<T>& dst)
707 T cosPhi = dot(src, dst);
708 // need to take the shorter way ...
709 int signCosPhi = SGMisc<T>::sign(cosPhi);
710 // cosPhi must be corrected for that sign
711 cosPhi = fabs(cosPhi);
713 // first opportunity to fail - make sure acos will succeed later -
718 // now the half angle between the orientations
721 // need the scales now, if the angle is very small, do linear interpolation
722 // to avoid instabilities
724 if (fabs(o) < SGLimits<T>::epsilon()) {
728 // note that we can give a positive lower bound for sin(o) here
731 scale0 = sin((1 - t)*o)*so;
732 scale1 = sin(t*o)*so;
735 return scale0*src + signCosPhi*scale1*dst;
738 /// Output to an ostream
739 template<typename char_type, typename traits_type, typename T>
741 std::basic_ostream<char_type, traits_type>&
742 operator<<(std::basic_ostream<char_type, traits_type>& s, const SGQuat<T>& v)
743 { return s << "[ " << v(0) << ", " << v(1) << ", " << v(2) << ", " << v(3) << " ]"; }
747 toQuatf(const SGQuatd& v)
748 { return SGQuatf((float)v(0), (float)v(1), (float)v(2), (float)v(3)); }
752 toQuatd(const SGQuatf& v)
753 { return SGQuatd(v(0), v(1), v(2), v(3)); }