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.
35 /// Default constructor. Does not initialize at all.
36 /// If you need them zero initialized, SGQuat::zeros()
39 /// Initialize with nans in the debug build, that will guarantee to have
40 /// a fast uninitialized default constructor in the release but shows up
41 /// uninitialized values in the debug build very fast ...
43 for (unsigned i = 0; i < 4; ++i)
44 data()[i] = SGLimits<T>::quiet_NaN();
47 /// Constructor. Initialize by the given values
48 SGQuat(T _x, T _y, T _z, T _w)
49 { x() = _x; y() = _y; z() = _z; w() = _w; }
50 /// Constructor. Initialize by the content of a plain array,
51 /// make sure it has at least 4 elements
52 explicit SGQuat(const T* d)
53 { data()[0] = d[0]; data()[1] = d[1]; data()[2] = d[2]; data()[3] = d[3]; }
55 /// Return a unit quaternion
56 static SGQuat unit(void)
57 { return fromRealImag(1, SGVec3<T>(0, 0, 0)); }
59 /// Return a quaternion from euler angles
60 static SGQuat fromEulerRad(T z, T y, T x)
63 T zd2 = T(0.5)*z; T yd2 = T(0.5)*y; T xd2 = T(0.5)*x;
64 T Szd2 = sin(zd2); T Syd2 = sin(yd2); T Sxd2 = sin(xd2);
65 T Czd2 = cos(zd2); T Cyd2 = cos(yd2); T Cxd2 = cos(xd2);
66 T Cxd2Czd2 = Cxd2*Czd2; T Cxd2Szd2 = Cxd2*Szd2;
67 T Sxd2Szd2 = Sxd2*Szd2; T Sxd2Czd2 = Sxd2*Czd2;
68 q.w() = Cxd2Czd2*Cyd2 + Sxd2Szd2*Syd2;
69 q.x() = Sxd2Czd2*Cyd2 - Cxd2Szd2*Syd2;
70 q.y() = Cxd2Czd2*Syd2 + Sxd2Szd2*Cyd2;
71 q.z() = Cxd2Szd2*Cyd2 - Sxd2Czd2*Syd2;
75 /// Return a quaternion from euler angles in degrees
76 static SGQuat fromEulerDeg(T z, T y, T x)
78 return fromEulerRad(SGMisc<T>::deg2rad(z), SGMisc<T>::deg2rad(y),
79 SGMisc<T>::deg2rad(x));
82 /// Return a quaternion from euler angles
83 static SGQuat fromYawPitchRoll(T y, T p, T r)
84 { return fromEulerRad(y, p, r); }
86 /// Return a quaternion from euler angles
87 static SGQuat fromYawPitchRollDeg(T y, T p, T r)
88 { return fromEulerDeg(y, p, r); }
90 /// Return a quaternion from euler angles
91 static SGQuat fromHeadAttBank(T h, T a, T b)
92 { return fromEulerRad(h, a, b); }
94 /// Return a quaternion from euler angles
95 static SGQuat fromHeadAttBankDeg(T h, T a, T b)
96 { return fromEulerDeg(h, a, b); }
98 /// Return a quaternion rotation from the earth centered to the
99 /// simulation usual horizontal local frame from given
100 /// longitude and latitude.
101 /// The horizontal local frame used in simulations is the frame with x-axis
102 /// pointing north, the y-axis pointing eastwards and the z axis
103 /// pointing downwards.
104 static SGQuat fromLonLatRad(T lon, T lat)
108 T yd2 = T(-0.25)*SGMisc<T>::pi() - T(0.5)*lat;
119 /// Like the above provided for convenience
120 static SGQuat fromLonLatDeg(T lon, T lat)
121 { return fromLonLatRad(SGMisc<T>::deg2rad(lon), SGMisc<T>::deg2rad(lat)); }
122 /// Like the above provided for convenience
123 static SGQuat fromLonLat(const SGGeod& geod)
124 { return fromLonLatRad(geod.getLongitudeRad(), geod.getLatitudeRad()); }
127 /// Create a quaternion from the angle axis representation
128 static SGQuat fromAngleAxis(T angle, const SGVec3<T>& axis)
130 T angle2 = T(0.5)*angle;
131 return fromRealImag(cos(angle2), T(sin(angle2))*axis);
134 /// Create a quaternion from the angle axis representation
135 static SGQuat fromAngleAxisDeg(T angle, const SGVec3<T>& axis)
136 { return fromAngleAxis(SGMisc<T>::deg2rad(angle), axis); }
138 /// Create a quaternion from the angle axis representation where the angle
139 /// is stored in the axis' length
140 static SGQuat fromAngleAxis(const SGVec3<T>& axis)
142 T nAxis = norm(axis);
143 if (nAxis <= SGLimits<T>::min())
144 return SGQuat::unit();
145 T angle2 = T(0.5)*nAxis;
146 return fromRealImag(cos(angle2), T(sin(angle2)/nAxis)*axis);
149 /// Create a normalized quaternion just from the imaginary part.
150 /// The imaginary part should point into that axis direction that results in
151 /// a quaternion with a positive real part.
152 /// This is the smallest numerically stable representation of an orientation
153 /// in space. See getPositiveRealImag()
154 static SGQuat fromPositiveRealImag(const SGVec3<T>& imag)
156 T r = sqrt(SGMisc<T>::max(T(0), T(1) - dot(imag, imag)));
157 return fromRealImag(r, imag);
160 /// Return a quaternion that rotates the from vector onto the to vector.
161 static SGQuat fromRotateTo(const SGVec3<T>& from, const SGVec3<T>& to)
163 T nfrom = norm(from);
165 if (nfrom <= SGLimits<T>::min() || nto <= SGLimits<T>::min())
166 return SGQuat::unit();
168 return SGQuat::fromRotateToNorm((1/nfrom)*from, (1/nto)*to);
171 /// Return a quaternion that rotates v1 onto the i1-th unit vector
172 /// and v2 into a plane that is spanned by the i2-th and i1-th unit vector.
173 static SGQuat fromRotateTo(const SGVec3<T>& v1, unsigned i1,
174 const SGVec3<T>& v2, unsigned i2)
178 if (nrmv1 <= SGLimits<T>::min() || nrmv2 <= SGLimits<T>::min())
179 return SGQuat::unit();
181 SGVec3<T> nv1 = (1/nrmv1)*v1;
182 SGVec3<T> nv2 = (1/nrmv2)*v2;
183 T dv1v2 = dot(nv1, nv2);
184 if (fabs(fabs(dv1v2)-1) <= SGLimits<T>::epsilon())
185 return SGQuat::unit();
187 // The target vector for the first rotation
188 SGVec3<T> nto1 = SGVec3<T>::zeros();
189 SGVec3<T> nto2 = SGVec3<T>::zeros();
193 // The first rotation can be done with the usual routine.
194 SGQuat q = SGQuat::fromRotateToNorm(nv1, nto1);
196 // The rotation axis for the second rotation is the
197 // target for the first one, so the rotation axis is nto1
198 // We need to get the angle.
200 // Make nv2 exactly orthogonal to nv1.
201 nv2 = normalize(nv2 - dv1v2*nv1);
203 SGVec3<T> tnv2 = q.transform(nv2);
204 T cosang = dot(nto2, tnv2);
205 T cos05ang = T(0.5)+T(0.5)*cosang;
208 cos05ang = sqrt(cos05ang);
209 T sig = dot(nto1, cross(nto2, tnv2));
210 T sin05ang = T(0.5)-T(0.5)*cosang;
213 sin05ang = copysign(sqrt(sin05ang), sig);
214 q *= SGQuat::fromRealImag(cos05ang, sin05ang*nto1);
220 // Return a quaternion which rotates the vector given by v
221 // to the vector -v. Other directions are *not* preserved.
222 static SGQuat fromChangeSign(const SGVec3<T>& v)
224 // The vector from points to the oposite direction than to.
225 // Find a vector perpendicular to the vector to.
226 T absv1 = fabs(v(0));
227 T absv2 = fabs(v(1));
228 T absv3 = fabs(v(2));
231 if (absv2 < absv1 && absv3 < absv1) {
233 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(quot, -1, 0);
234 } else if (absv1 < absv2 && absv3 < absv2) {
236 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(0, quot, -1);
237 } else if (absv1 < absv3 && absv2 < absv3) {
239 axis = (1/sqrt(1+quot*quot))*SGVec3<T>(-1, 0, quot);
241 // The all zero case.
242 return SGQuat::unit();
245 return SGQuat::fromRealImag(0, axis);
248 /// Return a quaternion from real and imaginary part
249 static SGQuat fromRealImag(T r, const SGVec3<T>& i)
259 /// Return an all zero vector
260 static SGQuat zeros(void)
261 { return SGQuat(0, 0, 0, 0); }
263 /// write the euler angles into the references
264 void getEulerRad(T& zRad, T& yRad, T& xRad) const
271 T num = 2*(y()*z() + w()*x());
272 T den = sqrQW - sqrQX - sqrQY + sqrQZ;
273 if (fabs(den) <= SGLimits<T>::min() &&
274 fabs(num) <= SGLimits<T>::min())
277 xRad = atan2(num, den);
279 T tmp = 2*(x()*z() - w()*y());
281 yRad = T(0.5)*SGMisc<T>::pi();
283 yRad = -T(0.5)*SGMisc<T>::pi();
287 num = 2*(x()*y() + w()*z());
288 den = sqrQW + sqrQX - sqrQY - sqrQZ;
289 if (fabs(den) <= SGLimits<T>::min() &&
290 fabs(num) <= SGLimits<T>::min())
293 T psi = atan2(num, den);
295 psi += 2*SGMisc<T>::pi();
300 /// write the euler angles in degrees into the references
301 void getEulerDeg(T& zDeg, T& yDeg, T& xDeg) const
303 getEulerRad(zDeg, yDeg, xDeg);
304 zDeg = SGMisc<T>::rad2deg(zDeg);
305 yDeg = SGMisc<T>::rad2deg(yDeg);
306 xDeg = SGMisc<T>::rad2deg(xDeg);
309 /// write the angle axis representation into the references
310 void getAngleAxis(T& angle, SGVec3<T>& axis) const
313 if (nrm <= SGLimits<T>::min()) {
315 axis = SGVec3<T>(0, 0, 0);
318 angle = acos(SGMisc<T>::max(-1, SGMisc<T>::min(1, rNrm*w())));
320 if (fabs(sAng) <= SGLimits<T>::min())
321 axis = SGVec3<T>(1, 0, 0);
323 axis = (rNrm/sAng)*imag(*this);
328 /// write the angle axis representation into the references
329 void getAngleAxis(SGVec3<T>& axis) const
332 getAngleAxis(angle, axis);
336 /// Get the imaginary part of the quaternion.
337 /// The imaginary part should point into that axis direction that results in
338 /// a quaternion with a positive real part.
339 /// This is the smallest numerically stable representation of an orientation
340 /// in space. See fromPositiveRealImag()
341 SGVec3<T> getPositiveRealImag() const
343 if (real(*this) < T(0))
344 return (T(-1)/norm(*this))*imag(*this);
346 return (T(1)/norm(*this))*imag(*this);
349 /// Access by index, the index is unchecked
350 const T& operator()(unsigned i) const
351 { return data()[i]; }
352 /// Access by index, the index is unchecked
353 T& operator()(unsigned i)
354 { return data()[i]; }
356 /// Access raw data by index, the index is unchecked
357 const T& operator[](unsigned i) const
358 { return data()[i]; }
359 /// Access raw data by index, the index is unchecked
360 T& operator[](unsigned i)
361 { return data()[i]; }
363 /// Access the x component
364 const T& x(void) const
365 { return data()[0]; }
366 /// Access the x component
368 { return data()[0]; }
369 /// Access the y component
370 const T& y(void) const
371 { return data()[1]; }
372 /// Access the y component
374 { return data()[1]; }
375 /// Access the z component
376 const T& z(void) const
377 { return data()[2]; }
378 /// Access the z component
380 { return data()[2]; }
381 /// Access the w component
382 const T& w(void) const
383 { return data()[3]; }
384 /// Access the w component
386 { return data()[3]; }
388 /// Get the data pointer
389 const T (&data(void) const)[4]
391 /// Get the data pointer
396 SGQuat& operator+=(const SGQuat& v)
397 { data()[0]+=v(0);data()[1]+=v(1);data()[2]+=v(2);data()[3]+=v(3);return *this; }
398 /// Inplace subtraction
399 SGQuat& operator-=(const SGQuat& v)
400 { data()[0]-=v(0);data()[1]-=v(1);data()[2]-=v(2);data()[3]-=v(3);return *this; }
401 /// Inplace scalar multiplication
403 SGQuat& operator*=(S s)
404 { data()[0] *= s; data()[1] *= s; data()[2] *= s; data()[3] *= s; return *this; }
405 /// Inplace scalar multiplication by 1/s
407 SGQuat& operator/=(S s)
408 { return operator*=(1/T(s)); }
409 /// Inplace quaternion multiplication
410 SGQuat& operator*=(const SGQuat& v);
412 /// Transform a vector from the current coordinate frame to a coordinate
413 /// frame rotated with the quaternion
414 SGVec3<T> transform(const SGVec3<T>& v) const
416 T r = 2/dot(*this, *this);
417 SGVec3<T> qimag = imag(*this);
419 return (r*qr*qr - 1)*v + (r*dot(qimag, v))*qimag - (r*qr)*cross(qimag, v);
421 /// Transform a vector from the coordinate frame rotated with the quaternion
422 /// to the current coordinate frame
423 SGVec3<T> backTransform(const SGVec3<T>& v) const
425 T r = 2/dot(*this, *this);
426 SGVec3<T> qimag = imag(*this);
428 return (r*qr*qr - 1)*v + (r*dot(qimag, v))*qimag + (r*qr)*cross(qimag, v);
431 /// Rotate a given vector with the quaternion
432 SGVec3<T> rotate(const SGVec3<T>& v) const
433 { return backTransform(v); }
434 /// Rotate a given vector with the inverse quaternion
435 SGVec3<T> rotateBack(const SGVec3<T>& v) const
436 { return transform(v); }
438 /// Return the time derivative of the quaternion given the angular velocity
440 derivative(const SGVec3<T>& angVel) const
444 deriv.w() = T(0.5)*(-x()*angVel(0) - y()*angVel(1) - z()*angVel(2));
445 deriv.x() = T(0.5)*( w()*angVel(0) - z()*angVel(1) + y()*angVel(2));
446 deriv.y() = T(0.5)*( z()*angVel(0) + w()*angVel(1) - x()*angVel(2));
447 deriv.z() = T(0.5)*(-y()*angVel(0) + x()*angVel(1) + w()*angVel(2));
452 /// Return the angular velocity w that makes q0 translate to q1 using
453 /// an explicit euler step with stepsize h.
454 /// That is, look for an w where
455 /// q1 = normalize(q0 + h*q0.derivative(w))
457 forwardDifferenceVelocity(const SGQuat& q0, const SGQuat& q1, const T& h)
459 // Let D_q0*w = q0.derivative(w), D_q0 the above 4x3 matrix.
460 // Then D_q0^t*D_q0 = 0.25*Id and D_q0*q0 = 0.
461 // Let lambda be a nonzero normailzation factor, then
462 // q1 = normalize(q0 + h*q0.derivative(w))
464 // lambda*q1 = q0 + h*D_q0*w.
465 // Multiply left by the transpose D_q0^t and reorder gives
466 // 4*lambda/h*D_q0^t*q1 = w.
467 // Now compute lambda by substitution of w into the original
469 // lambda*q1 = q0 + 4*lambda*D_q0*D_q0^t*q1,
470 // multiply by q1^t from the left
471 // lambda*<q1,q1> = <q0,q1> + 4*lambda*<D_q0^t*q1,D_q0^t*q1>
472 // and solving for lambda gives
473 // lambda = <q0,q1>/(1 - 4*<D_q0^t*q1,D_q0^t*q1>).
475 // The transpose of the derivative matrix
476 // the 0.5 factor is handled below
477 // also note that the initializer uses x, y, z, w instead of w, x, y, z
478 SGQuat d0(q0.w(), q0.z(), -q0.y(), -q0.x());
479 SGQuat d1(-q0.z(), q0.w(), q0.x(), -q0.y());
480 SGQuat d2(q0.y(), -q0.x(), q0.w(), -q0.z());
482 SGVec3<T> Dq(dot(d0, q1), dot(d1, q1), dot(d2, q1));
483 // Like above, but take into account that Dq = 2*D_q0^t*q1
484 T lambda = dot(q0, q1)/(T(1) - dot(Dq, Dq));
485 return (2*lambda/h)*Dq;
490 // Private because it assumes normalized inputs.
492 fromRotateToSmaller90Deg(T cosang,
493 const SGVec3<T>& from, const SGVec3<T>& to)
495 // In this function we assume that the angle required to rotate from
496 // the vector from to the vector to is <= 90 deg.
497 // That is done so because of possible instabilities when we rotate more
500 // Note that the next comment does actually cover a *more* *general* case
501 // than we need in this function. That shows that this formula is even
502 // valid for rotations up to 180deg.
504 // Because of the signs in the axis, it is sufficient to care for angles
505 // in the interval [-pi,pi]. That means that 0.5*angle is in the interval
506 // [-pi/2,pi/2]. But in that range the cosine is allways >= 0.
507 // So we do not need to care for egative roots in the following equation:
508 T cos05ang = sqrt(T(0.5)+T(0.5)*cosang);
511 // Now our assumption of angles <= 90 deg comes in play.
512 // For that reason, we know that cos05ang is not zero.
513 // It is even more, we can see from the above formula that
514 // sqrt(0.5) < cos05ang.
517 // Compute the rotation axis, that is
518 // sin(angle)*normalized rotation axis
519 SGVec3<T> axis = cross(to, from);
521 // We need sin(0.5*angle)*normalized rotation axis.
522 // So rescale with sin(0.5*x)/sin(x).
523 // To do that we use the equation:
524 // sin(x) = 2*sin(0.5*x)*cos(0.5*x)
525 return SGQuat::fromRealImag( cos05ang, (1/(2*cos05ang))*axis);
528 // Private because it assumes normalized inputs.
530 fromRotateToNorm(const SGVec3<T>& from, const SGVec3<T>& to)
532 // To avoid instabilities with roundoff, we distinguish between rotations
533 // with more then 90deg and rotations with less than 90deg.
535 // Compute the cosine of the angle.
536 T cosang = dot(from, to);
538 // For the small ones do direct computation
539 if (T(-0.5) < cosang)
540 return SGQuat::fromRotateToSmaller90Deg(cosang, from, to);
542 // For larger rotations. first rotate from to -from.
543 // Past that we will have a smaller angle again.
544 SGQuat q1 = SGQuat::fromChangeSign(from);
545 SGQuat q2 = SGQuat::fromRotateToSmaller90Deg(-cosang, -from, to);
552 /// Unary +, do nothing ...
556 operator+(const SGQuat<T>& v)
559 /// Unary -, do nearly nothing
563 operator-(const SGQuat<T>& v)
564 { return SGQuat<T>(-v(0), -v(1), -v(2), -v(3)); }
570 operator+(const SGQuat<T>& v1, const SGQuat<T>& v2)
571 { return SGQuat<T>(v1(0)+v2(0), v1(1)+v2(1), v1(2)+v2(2), v1(3)+v2(3)); }
577 operator-(const SGQuat<T>& v1, const SGQuat<T>& v2)
578 { return SGQuat<T>(v1(0)-v2(0), v1(1)-v2(1), v1(2)-v2(2), v1(3)-v2(3)); }
580 /// Scalar multiplication
581 template<typename S, typename T>
584 operator*(S s, const SGQuat<T>& v)
585 { return SGQuat<T>(s*v(0), s*v(1), s*v(2), s*v(3)); }
587 /// Scalar multiplication
588 template<typename S, typename T>
591 operator*(const SGQuat<T>& v, S s)
592 { return SGQuat<T>(s*v(0), s*v(1), s*v(2), s*v(3)); }
594 /// Quaterion multiplication
598 operator*(const SGQuat<T>& v1, const SGQuat<T>& v2)
601 v.x() = v1.w()*v2.x() + v1.x()*v2.w() + v1.y()*v2.z() - v1.z()*v2.y();
602 v.y() = v1.w()*v2.y() - v1.x()*v2.z() + v1.y()*v2.w() + v1.z()*v2.x();
603 v.z() = v1.w()*v2.z() + v1.x()*v2.y() - v1.y()*v2.x() + v1.z()*v2.w();
604 v.w() = v1.w()*v2.w() - v1.x()*v2.x() - v1.y()*v2.y() - v1.z()*v2.z();
608 /// Now define the inplace multiplication
612 SGQuat<T>::operator*=(const SGQuat& v)
613 { (*this) = (*this)*v; return *this; }
615 /// The conjugate of the quaternion, this is also the
616 /// inverse for normalized quaternions
620 conj(const SGQuat<T>& v)
621 { return SGQuat<T>(-v(0), -v(1), -v(2), v(3)); }
623 /// The conjugate of the quaternion, this is also the
624 /// inverse for normalized quaternions
628 inverse(const SGQuat<T>& v)
629 { return (1/dot(v, v))*SGQuat<T>(-v(0), -v(1), -v(2), v(3)); }
631 /// The imagniary part of the quaternion
635 real(const SGQuat<T>& v)
638 /// The imagniary part of the quaternion
642 imag(const SGQuat<T>& v)
643 { return SGVec3<T>(v.x(), v.y(), v.z()); }
645 /// Scalar dot product
649 dot(const SGQuat<T>& v1, const SGQuat<T>& v2)
650 { return v1(0)*v2(0) + v1(1)*v2(1) + v1(2)*v2(2) + v1(3)*v2(3); }
652 /// The euclidean norm of the vector, that is what most people call length
656 norm(const SGQuat<T>& v)
657 { return sqrt(dot(v, v)); }
659 /// The euclidean norm of the vector, that is what most people call length
663 length(const SGQuat<T>& v)
664 { return sqrt(dot(v, v)); }
666 /// The 1-norm of the vector, this one is the fastest length function we
667 /// can implement on modern cpu's
671 norm1(const SGQuat<T>& v)
672 { return fabs(v(0)) + fabs(v(1)) + fabs(v(2)) + fabs(v(3)); }
674 /// The euclidean norm of the vector, that is what most people call length
678 normalize(const SGQuat<T>& q)
679 { return (1/norm(q))*q; }
681 /// Return true if exactly the same
685 operator==(const SGQuat<T>& v1, const SGQuat<T>& v2)
686 { return v1(0)==v2(0) && v1(1)==v2(1) && v1(2)==v2(2) && v1(3)==v2(3); }
688 /// Return true if not exactly the same
692 operator!=(const SGQuat<T>& v1, const SGQuat<T>& v2)
693 { return ! (v1 == v2); }
695 /// Return true if equal to the relative tolerance tol
696 /// Note that this is not the same than comparing quaternions to represent
697 /// the same rotation
701 equivalent(const SGQuat<T>& v1, const SGQuat<T>& v2, T tol)
702 { return norm1(v1 - v2) < tol*(norm1(v1) + norm1(v2)); }
704 /// Return true if about equal to roundoff of the underlying type
705 /// Note that this is not the same than comparing quaternions to represent
706 /// the same rotation
710 equivalent(const SGQuat<T>& v1, const SGQuat<T>& v2)
711 { return equivalent(v1, v2, 100*SGLimits<T>::epsilon()); }
717 isNaN(const SGQuat<T>& v)
719 return SGMisc<T>::isNaN(v(0)) || SGMisc<T>::isNaN(v(1))
720 || SGMisc<T>::isNaN(v(2)) || SGMisc<T>::isNaN(v(3));
724 /// quaternion interpolation for t in [0,1] interpolate between src (=0)
729 interpolate(T t, const SGQuat<T>& src, const SGQuat<T>& dst)
731 T cosPhi = dot(src, dst);
732 // need to take the shorter way ...
733 int signCosPhi = SGMisc<T>::sign(cosPhi);
734 // cosPhi must be corrected for that sign
735 cosPhi = fabs(cosPhi);
737 // first opportunity to fail - make sure acos will succeed later -
742 // now the half angle between the orientations
745 // need the scales now, if the angle is very small, do linear interpolation
746 // to avoid instabilities
748 if (fabs(o) <= SGLimits<T>::epsilon()) {
752 // note that we can give a positive lower bound for sin(o) here
755 scale0 = sin((1 - t)*o)*so;
756 scale1 = sin(t*o)*so;
759 return scale0*src + signCosPhi*scale1*dst;
762 /// Output to an ostream
763 template<typename char_type, typename traits_type, typename T>
765 std::basic_ostream<char_type, traits_type>&
766 operator<<(std::basic_ostream<char_type, traits_type>& s, const SGQuat<T>& v)
767 { return s << "[ " << v(0) << ", " << v(1) << ", " << v(2) << ", " << v(3) << " ]"; }
771 toQuatf(const SGQuatd& v)
772 { return SGQuatf((float)v(0), (float)v(1), (float)v(2), (float)v(3)); }
776 toQuatd(const SGQuatf& v)
777 { return SGQuatd(v(0), v(1), v(2), v(3)); }