1 // Copyright (C) 2006-2009 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.
29 #ifndef NO_OPENSCENEGRAPH_INTERFACE
39 /// Default constructor. Does not initialize at all.
40 /// If you need them zero initialized, SGQuat::zeros()
43 /// Initialize with nans in the debug build, that will guarantee to have
44 /// a fast uninitialized default constructor in the release but shows up
45 /// uninitialized values in the debug build very fast ...
47 for (unsigned i = 0; i < 4; ++i)
48 data()[i] = SGLimits<T>::quiet_NaN();
51 /// Constructor. Initialize by the given values
52 SGQuat(T _x, T _y, T _z, T _w)
53 { x() = _x; y() = _y; z() = _z; w() = _w; }
54 /// Constructor. Initialize by the content of a plain array,
55 /// make sure it has at least 4 elements
56 explicit SGQuat(const T* d)
57 { data()[0] = d[0]; data()[1] = d[1]; data()[2] = d[2]; data()[3] = d[3]; }
59 /// Return a unit quaternion
60 static SGQuat unit(void)
61 { return fromRealImag(1, SGVec3<T>(0, 0, 0)); }
63 /// Return a quaternion from euler angles
64 static SGQuat fromEulerRad(T z, T y, T x)
67 T zd2 = T(0.5)*z; T yd2 = T(0.5)*y; T xd2 = T(0.5)*x;
68 T Szd2 = sin(zd2); T Syd2 = sin(yd2); T Sxd2 = sin(xd2);
69 T Czd2 = cos(zd2); T Cyd2 = cos(yd2); T Cxd2 = cos(xd2);
70 T Cxd2Czd2 = Cxd2*Czd2; T Cxd2Szd2 = Cxd2*Szd2;
71 T Sxd2Szd2 = Sxd2*Szd2; T Sxd2Czd2 = Sxd2*Czd2;
72 q.w() = Cxd2Czd2*Cyd2 + Sxd2Szd2*Syd2;
73 q.x() = Sxd2Czd2*Cyd2 - Cxd2Szd2*Syd2;
74 q.y() = Cxd2Czd2*Syd2 + Sxd2Szd2*Cyd2;
75 q.z() = Cxd2Szd2*Cyd2 - Sxd2Czd2*Syd2;
79 /// Return a quaternion from euler angles in degrees
80 static SGQuat fromEulerDeg(T z, T y, T x)
82 return fromEulerRad(SGMisc<T>::deg2rad(z), SGMisc<T>::deg2rad(y),
83 SGMisc<T>::deg2rad(x));
86 /// Return a quaternion from euler angles
87 static SGQuat fromYawPitchRoll(T y, T p, T r)
88 { return fromEulerRad(y, p, r); }
90 /// Return a quaternion from euler angles
91 static SGQuat fromYawPitchRollDeg(T y, T p, T r)
92 { return fromEulerDeg(y, p, r); }
94 /// Return a quaternion from euler angles
95 static SGQuat fromHeadAttBank(T h, T a, T b)
96 { return fromEulerRad(h, a, b); }
98 /// Return a quaternion from euler angles
99 static SGQuat fromHeadAttBankDeg(T h, T a, T b)
100 { return fromEulerDeg(h, a, b); }
102 /// Return a quaternion rotation from the earth centered to the
103 /// simulation usual horizontal local frame from given
104 /// longitude and latitude.
105 /// The horizontal local frame used in simulations is the frame with x-axis
106 /// pointing north, the y-axis pointing eastwards and the z axis
107 /// pointing downwards.
108 static SGQuat fromLonLatRad(T lon, T lat)
112 T yd2 = T(-0.25)*SGMisc<T>::pi() - T(0.5)*lat;
123 /// Like the above provided for convenience
124 static SGQuat fromLonLatDeg(T lon, T lat)
125 { return fromLonLatRad(SGMisc<T>::deg2rad(lon), SGMisc<T>::deg2rad(lat)); }
126 /// Like the above provided for convenience
127 static SGQuat fromLonLat(const SGGeod& geod)
128 { return fromLonLatRad(geod.getLongitudeRad(), geod.getLatitudeRad()); }
131 /// Create a quaternion from the angle axis representation
132 static SGQuat fromAngleAxis(T angle, const SGVec3<T>& axis)
134 T angle2 = T(0.5)*angle;
135 return fromRealImag(cos(angle2), T(sin(angle2))*axis);
138 /// Create a quaternion from the angle axis representation
139 static SGQuat fromAngleAxisDeg(T angle, const SGVec3<T>& axis)
140 { return fromAngleAxis(SGMisc<T>::deg2rad(angle), axis); }
142 /// Create a quaternion from the angle axis representation where the angle
143 /// is stored in the axis' length
144 static SGQuat fromAngleAxis(const SGVec3<T>& axis)
146 T nAxis = norm(axis);
147 if (nAxis <= SGLimits<T>::min())
148 return SGQuat::unit();
149 T angle2 = T(0.5)*nAxis;
150 return fromRealImag(cos(angle2), T(sin(angle2)/nAxis)*axis);
153 /// Create a normalized quaternion just from the imaginary part.
154 /// The imaginary part should point into that axis direction that results in
155 /// a quaternion with a positive real part.
156 /// This is the smallest numerically stable representation of an orientation
157 /// in space. See getPositiveRealImag()
158 static SGQuat fromPositiveRealImag(const SGVec3<T>& imag)
160 T r = sqrt(SGMisc<T>::max(T(0), T(1) - dot(imag, imag)));
161 return fromRealImag(r, imag);
164 /// Return a quaternion that rotates the from vector onto the to vector.
165 static SGQuat fromRotateTo(const SGVec3<T>& from, const SGVec3<T>& to)
167 T nfrom = norm(from);
169 if (nfrom <= SGLimits<T>::min() || nto <= SGLimits<T>::min())
170 return SGQuat::unit();
172 return SGQuat::fromRotateToNorm((1/nfrom)*from, (1/nto)*to);
175 /// Return a quaternion that rotates v1 onto the i1-th unit vector
176 /// and v2 into a plane that is spanned by the i2-th and i1-th unit vector.
177 static SGQuat fromRotateTo(const SGVec3<T>& v1, unsigned i1,
178 const SGVec3<T>& v2, unsigned i2)
182 if (nrmv1 <= SGLimits<T>::min() || nrmv2 <= SGLimits<T>::min())
183 return SGQuat::unit();
185 SGVec3<T> nv1 = (1/nrmv1)*v1;
186 SGVec3<T> nv2 = (1/nrmv2)*v2;
187 T dv1v2 = dot(nv1, nv2);
188 if (fabs(fabs(dv1v2)-1) <= SGLimits<T>::epsilon())
189 return SGQuat::unit();
191 // The target vector for the first rotation
192 SGVec3<T> nto1 = SGVec3<T>::zeros();
193 SGVec3<T> nto2 = SGVec3<T>::zeros();
197 // The first rotation can be done with the usual routine.
198 SGQuat q = SGQuat::fromRotateToNorm(nv1, nto1);
200 // The rotation axis for the second rotation is the
201 // target for the first one, so the rotation axis is nto1
202 // We need to get the angle.
204 // Make nv2 exactly orthogonal to nv1.
205 nv2 = normalize(nv2 - dv1v2*nv1);
207 SGVec3<T> tnv2 = q.transform(nv2);
208 T cosang = dot(nto2, tnv2);
209 T cos05ang = T(0.5)+T(0.5)*cosang;
212 cos05ang = sqrt(cos05ang);
213 T sig = dot(nto1, cross(nto2, tnv2));
214 T sin05ang = T(0.5)-T(0.5)*cosang;
217 sin05ang = copysign(sqrt(sin05ang), sig);
218 q *= SGQuat::fromRealImag(cos05ang, sin05ang*nto1);
224 // Return a quaternion which rotates the vector given by v
225 // to the vector -v. Other directions are *not* preserved.
226 static SGQuat fromChangeSign(const SGVec3<T>& v)
228 // The vector from points to the oposite direction than to.
229 // Find a vector perpendicular to the vector to.
230 T absv1 = fabs(v(0));
231 T absv2 = fabs(v(1));
232 T absv3 = fabs(v(2));
235 if (absv2 < absv1 && absv3 < absv1) {
237 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(quot, -1, 0);
238 } else if (absv1 < absv2 && absv3 < absv2) {
240 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(0, quot, -1);
241 } else if (absv1 < absv3 && absv2 < absv3) {
243 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(-1, 0, quot);
245 // The all zero case.
246 return SGQuat::unit();
249 return SGQuat::fromRealImag(0, axis);
252 /// Return a quaternion from real and imaginary part
253 static SGQuat fromRealImag(T r, const SGVec3<T>& i)
263 /// Return an all zero vector
264 static SGQuat zeros(void)
265 { return SGQuat(0, 0, 0, 0); }
267 /// write the euler angles into the references
268 void getEulerRad(T& zRad, T& yRad, T& xRad) const
275 T num = 2*(y()*z() + w()*x());
276 T den = sqrQW - sqrQX - sqrQY + sqrQZ;
277 if (fabs(den) <= SGLimits<T>::min() &&
278 fabs(num) <= SGLimits<T>::min())
281 xRad = atan2(num, den);
283 T tmp = 2*(x()*z() - w()*y());
285 yRad = T(0.5)*SGMisc<T>::pi();
287 yRad = -T(0.5)*SGMisc<T>::pi();
291 num = 2*(x()*y() + w()*z());
292 den = sqrQW + sqrQX - sqrQY - sqrQZ;
293 if (fabs(den) <= SGLimits<T>::min() &&
294 fabs(num) <= SGLimits<T>::min())
297 T psi = atan2(num, den);
299 psi += 2*SGMisc<T>::pi();
304 /// write the euler angles in degrees into the references
305 void getEulerDeg(T& zDeg, T& yDeg, T& xDeg) const
307 getEulerRad(zDeg, yDeg, xDeg);
308 zDeg = SGMisc<T>::rad2deg(zDeg);
309 yDeg = SGMisc<T>::rad2deg(yDeg);
310 xDeg = SGMisc<T>::rad2deg(xDeg);
313 /// write the angle axis representation into the references
314 void getAngleAxis(T& angle, SGVec3<T>& axis) const
317 if (nrm <= SGLimits<T>::min()) {
319 axis = SGVec3<T>(0, 0, 0);
322 angle = acos(SGMisc<T>::max(-1, SGMisc<T>::min(1, rNrm*w())));
324 if (fabs(sAng) <= SGLimits<T>::min())
325 axis = SGVec3<T>(1, 0, 0);
327 axis = (rNrm/sAng)*imag(*this);
332 /// write the angle axis representation into the references
333 void getAngleAxis(SGVec3<T>& axis) const
336 getAngleAxis(angle, axis);
340 /// Get the imaginary part of the quaternion.
341 /// The imaginary part should point into that axis direction that results in
342 /// a quaternion with a positive real part.
343 /// This is the smallest numerically stable representation of an orientation
344 /// in space. See fromPositiveRealImag()
345 SGVec3<T> getPositiveRealImag() const
347 if (real(*this) < T(0))
348 return (T(-1)/norm(*this))*imag(*this);
350 return (T(1)/norm(*this))*imag(*this);
353 /// Access by index, the index is unchecked
354 const T& operator()(unsigned i) const
355 { return data()[i]; }
356 /// Access by index, the index is unchecked
357 T& operator()(unsigned i)
358 { return data()[i]; }
360 /// Access raw data by index, the index is unchecked
361 const T& operator[](unsigned i) const
362 { return data()[i]; }
363 /// Access raw data by index, the index is unchecked
364 T& operator[](unsigned i)
365 { return data()[i]; }
367 /// Access the x component
368 const T& x(void) const
369 { return data()[0]; }
370 /// Access the x component
372 { return data()[0]; }
373 /// Access the y component
374 const T& y(void) const
375 { return data()[1]; }
376 /// Access the y component
378 { return data()[1]; }
379 /// Access the z component
380 const T& z(void) const
381 { return data()[2]; }
382 /// Access the z component
384 { return data()[2]; }
385 /// Access the w component
386 const T& w(void) const
387 { return data()[3]; }
388 /// Access the w component
390 { return data()[3]; }
392 /// Get the data pointer
393 const T (&data(void) const)[4]
395 /// Get the data pointer
400 SGQuat& operator+=(const SGQuat& v)
401 { data()[0]+=v(0);data()[1]+=v(1);data()[2]+=v(2);data()[3]+=v(3);return *this; }
402 /// Inplace subtraction
403 SGQuat& operator-=(const SGQuat& v)
404 { data()[0]-=v(0);data()[1]-=v(1);data()[2]-=v(2);data()[3]-=v(3);return *this; }
405 /// Inplace scalar multiplication
407 SGQuat& operator*=(S s)
408 { data()[0] *= s; data()[1] *= s; data()[2] *= s; data()[3] *= s; return *this; }
409 /// Inplace scalar multiplication by 1/s
411 SGQuat& operator/=(S s)
412 { return operator*=(1/T(s)); }
413 /// Inplace quaternion multiplication
414 SGQuat& operator*=(const SGQuat& v);
416 /// Transform a vector from the current coordinate frame to a coordinate
417 /// frame rotated with the quaternion
418 SGVec3<T> transform(const SGVec3<T>& v) const
420 T r = 2/dot(*this, *this);
421 SGVec3<T> qimag = imag(*this);
423 return (r*qr*qr - 1)*v + (r*dot(qimag, v))*qimag - (r*qr)*cross(qimag, v);
425 /// Transform a vector from the coordinate frame rotated with the quaternion
426 /// to the current coordinate frame
427 SGVec3<T> backTransform(const SGVec3<T>& v) const
429 T r = 2/dot(*this, *this);
430 SGVec3<T> qimag = imag(*this);
432 return (r*qr*qr - 1)*v + (r*dot(qimag, v))*qimag + (r*qr)*cross(qimag, v);
435 /// Rotate a given vector with the quaternion
436 SGVec3<T> rotate(const SGVec3<T>& v) const
437 { return backTransform(v); }
438 /// Rotate a given vector with the inverse quaternion
439 SGVec3<T> rotateBack(const SGVec3<T>& v) const
440 { return transform(v); }
442 /// Return the time derivative of the quaternion given the angular velocity
444 derivative(const SGVec3<T>& angVel) const
448 deriv.w() = T(0.5)*(-x()*angVel(0) - y()*angVel(1) - z()*angVel(2));
449 deriv.x() = T(0.5)*( w()*angVel(0) - z()*angVel(1) + y()*angVel(2));
450 deriv.y() = T(0.5)*( z()*angVel(0) + w()*angVel(1) - x()*angVel(2));
451 deriv.z() = T(0.5)*(-y()*angVel(0) + x()*angVel(1) + w()*angVel(2));
456 /// Return the angular velocity w that makes q0 translate to q1 using
457 /// an explicit euler step with stepsize h.
458 /// That is, look for an w where
459 /// q1 = normalize(q0 + h*q0.derivative(w))
461 forwardDifferenceVelocity(const SGQuat& q0, const SGQuat& q1, const T& h)
463 // Let D_q0*w = q0.derivative(w), D_q0 the above 4x3 matrix.
464 // Then D_q0^t*D_q0 = 0.25*Id and D_q0*q0 = 0.
465 // Let lambda be a nonzero normailzation factor, then
466 // q1 = normalize(q0 + h*q0.derivative(w))
468 // lambda*q1 = q0 + h*D_q0*w.
469 // Multiply left by the transpose D_q0^t and reorder gives
470 // 4*lambda/h*D_q0^t*q1 = w.
471 // Now compute lambda by substitution of w into the original
473 // lambda*q1 = q0 + 4*lambda*D_q0*D_q0^t*q1,
474 // multiply by q1^t from the left
475 // lambda*<q1,q1> = <q0,q1> + 4*lambda*<D_q0^t*q1,D_q0^t*q1>
476 // and solving for lambda gives
477 // lambda = <q0,q1>/(1 - 4*<D_q0^t*q1,D_q0^t*q1>).
479 // The transpose of the derivative matrix
480 // the 0.5 factor is handled below
481 // also note that the initializer uses x, y, z, w instead of w, x, y, z
482 SGQuat d0(q0.w(), q0.z(), -q0.y(), -q0.x());
483 SGQuat d1(-q0.z(), q0.w(), q0.x(), -q0.y());
484 SGQuat d2(q0.y(), -q0.x(), q0.w(), -q0.z());
486 SGVec3<T> Dq(dot(d0, q1), dot(d1, q1), dot(d2, q1));
487 // Like above, but take into account that Dq = 2*D_q0^t*q1
488 T lambda = dot(q0, q1)/(T(1) - dot(Dq, Dq));
489 return (2*lambda/h)*Dq;
494 // Private because it assumes normalized inputs.
496 fromRotateToSmaller90Deg(T cosang,
497 const SGVec3<T>& from, const SGVec3<T>& to)
499 // In this function we assume that the angle required to rotate from
500 // the vector from to the vector to is <= 90 deg.
501 // That is done so because of possible instabilities when we rotate more
504 // Note that the next comment does actually cover a *more* *general* case
505 // than we need in this function. That shows that this formula is even
506 // valid for rotations up to 180deg.
508 // Because of the signs in the axis, it is sufficient to care for angles
509 // in the interval [-pi,pi]. That means that 0.5*angle is in the interval
510 // [-pi/2,pi/2]. But in that range the cosine is allways >= 0.
511 // So we do not need to care for egative roots in the following equation:
512 T cos05ang = sqrt(T(0.5)+T(0.5)*cosang);
515 // Now our assumption of angles <= 90 deg comes in play.
516 // For that reason, we know that cos05ang is not zero.
517 // It is even more, we can see from the above formula that
518 // sqrt(0.5) < cos05ang.
521 // Compute the rotation axis, that is
522 // sin(angle)*normalized rotation axis
523 SGVec3<T> axis = cross(to, from);
525 // We need sin(0.5*angle)*normalized rotation axis.
526 // So rescale with sin(0.5*x)/sin(x).
527 // To do that we use the equation:
528 // sin(x) = 2*sin(0.5*x)*cos(0.5*x)
529 return SGQuat::fromRealImag( cos05ang, (1/(2*cos05ang))*axis);
532 // Private because it assumes normalized inputs.
534 fromRotateToNorm(const SGVec3<T>& from, const SGVec3<T>& to)
536 // To avoid instabilities with roundoff, we distinguish between rotations
537 // with more then 90deg and rotations with less than 90deg.
539 // Compute the cosine of the angle.
540 T cosang = dot(from, to);
542 // For the small ones do direct computation
543 if (T(-0.5) < cosang)
544 return SGQuat::fromRotateToSmaller90Deg(cosang, from, to);
546 // For larger rotations. first rotate from to -from.
547 // Past that we will have a smaller angle again.
548 SGQuat q1 = SGQuat::fromChangeSign(from);
549 SGQuat q2 = SGQuat::fromRotateToSmaller90Deg(-cosang, -from, to);
556 /// Unary +, do nothing ...
560 operator+(const SGQuat<T>& v)
563 /// Unary -, do nearly nothing
567 operator-(const SGQuat<T>& v)
568 { return SGQuat<T>(-v(0), -v(1), -v(2), -v(3)); }
574 operator+(const SGQuat<T>& v1, const SGQuat<T>& v2)
575 { return SGQuat<T>(v1(0)+v2(0), v1(1)+v2(1), v1(2)+v2(2), v1(3)+v2(3)); }
581 operator-(const SGQuat<T>& v1, const SGQuat<T>& v2)
582 { return SGQuat<T>(v1(0)-v2(0), v1(1)-v2(1), v1(2)-v2(2), v1(3)-v2(3)); }
584 /// Scalar multiplication
585 template<typename S, typename T>
588 operator*(S s, const SGQuat<T>& v)
589 { return SGQuat<T>(s*v(0), s*v(1), s*v(2), s*v(3)); }
591 /// Scalar multiplication
592 template<typename S, typename T>
595 operator*(const SGQuat<T>& v, S s)
596 { return SGQuat<T>(s*v(0), s*v(1), s*v(2), s*v(3)); }
598 /// Quaterion multiplication
602 operator*(const SGQuat<T>& v1, const SGQuat<T>& v2)
605 v.x() = v1.w()*v2.x() + v1.x()*v2.w() + v1.y()*v2.z() - v1.z()*v2.y();
606 v.y() = v1.w()*v2.y() - v1.x()*v2.z() + v1.y()*v2.w() + v1.z()*v2.x();
607 v.z() = v1.w()*v2.z() + v1.x()*v2.y() - v1.y()*v2.x() + v1.z()*v2.w();
608 v.w() = v1.w()*v2.w() - v1.x()*v2.x() - v1.y()*v2.y() - v1.z()*v2.z();
612 /// Now define the inplace multiplication
616 SGQuat<T>::operator*=(const SGQuat& v)
617 { (*this) = (*this)*v; return *this; }
619 /// The conjugate of the quaternion, this is also the
620 /// inverse for normalized quaternions
624 conj(const SGQuat<T>& v)
625 { return SGQuat<T>(-v(0), -v(1), -v(2), v(3)); }
627 /// The conjugate of the quaternion, this is also the
628 /// inverse for normalized quaternions
632 inverse(const SGQuat<T>& v)
633 { return (1/dot(v, v))*SGQuat<T>(-v(0), -v(1), -v(2), v(3)); }
635 /// The imagniary part of the quaternion
639 real(const SGQuat<T>& v)
642 /// The imagniary part of the quaternion
646 imag(const SGQuat<T>& v)
647 { return SGVec3<T>(v.x(), v.y(), v.z()); }
649 /// Scalar dot product
653 dot(const SGQuat<T>& v1, const SGQuat<T>& v2)
654 { return v1(0)*v2(0) + v1(1)*v2(1) + v1(2)*v2(2) + v1(3)*v2(3); }
656 /// The euclidean norm of the vector, that is what most people call length
660 norm(const SGQuat<T>& v)
661 { return sqrt(dot(v, v)); }
663 /// The euclidean norm of the vector, that is what most people call length
667 length(const SGQuat<T>& v)
668 { return sqrt(dot(v, v)); }
670 /// The 1-norm of the vector, this one is the fastest length function we
671 /// can implement on modern cpu's
675 norm1(const SGQuat<T>& v)
676 { return fabs(v(0)) + fabs(v(1)) + fabs(v(2)) + fabs(v(3)); }
678 /// The euclidean norm of the vector, that is what most people call length
682 normalize(const SGQuat<T>& q)
683 { return (1/norm(q))*q; }
685 /// Return true if exactly the same
689 operator==(const SGQuat<T>& v1, const SGQuat<T>& v2)
690 { return v1(0)==v2(0) && v1(1)==v2(1) && v1(2)==v2(2) && v1(3)==v2(3); }
692 /// Return true if not exactly the same
696 operator!=(const SGQuat<T>& v1, const SGQuat<T>& v2)
697 { return ! (v1 == v2); }
699 /// Return true if equal to the relative tolerance tol
700 /// Note that this is not the same than comparing quaternions to represent
701 /// the same rotation
705 equivalent(const SGQuat<T>& v1, const SGQuat<T>& v2, T tol)
706 { return norm1(v1 - v2) < tol*(norm1(v1) + norm1(v2)); }
708 /// Return true if about equal to roundoff of the underlying type
709 /// Note that this is not the same than comparing quaternions to represent
710 /// the same rotation
714 equivalent(const SGQuat<T>& v1, const SGQuat<T>& v2)
715 { return equivalent(v1, v2, 100*SGLimits<T>::epsilon()); }
721 isNaN(const SGQuat<T>& v)
723 return SGMisc<T>::isNaN(v(0)) || SGMisc<T>::isNaN(v(1))
724 || SGMisc<T>::isNaN(v(2)) || SGMisc<T>::isNaN(v(3));
728 /// quaternion interpolation for t in [0,1] interpolate between src (=0)
733 interpolate(T t, const SGQuat<T>& src, const SGQuat<T>& dst)
735 T cosPhi = dot(src, dst);
736 // need to take the shorter way ...
737 int signCosPhi = SGMisc<T>::sign(cosPhi);
738 // cosPhi must be corrected for that sign
739 cosPhi = fabs(cosPhi);
741 // first opportunity to fail - make sure acos will succeed later -
746 // now the half angle between the orientations
749 // need the scales now, if the angle is very small, do linear interpolation
750 // to avoid instabilities
752 if (fabs(o) <= SGLimits<T>::epsilon()) {
756 // note that we can give a positive lower bound for sin(o) here
759 scale0 = sin((1 - t)*o)*so;
760 scale1 = sin(t*o)*so;
763 return scale0*src + signCosPhi*scale1*dst;
766 /// Output to an ostream
767 template<typename char_type, typename traits_type, typename T>
769 std::basic_ostream<char_type, traits_type>&
770 operator<<(std::basic_ostream<char_type, traits_type>& s, const SGQuat<T>& v)
771 { return s << "[ " << v(0) << ", " << v(1) << ", " << v(2) << ", " << v(3) << " ]"; }
775 toQuatf(const SGQuatd& v)
776 { return SGQuatf((float)v(0), (float)v(1), (float)v(2), (float)v(3)); }
780 toQuatd(const SGQuatf& v)
781 { return SGQuatd(v(0), v(1), v(2), v(3)); }
783 #ifndef NO_OPENSCENEGRAPH_INTERFACE
786 toSG(const osg::Quat& q)
787 { return SGQuatd(q[0], q[1], q[2], q[3]); }
791 toOsg(const SGQuatd& q)
792 { return osg::Quat(q[0], q[1], q[2], q[3]); }