boost/math/quaternion.hpp
// boost quaternion.hpp header file
// (C) Copyright Hubert Holin 2001.
// Distributed under the Boost Software License, Version 1.0. (See
// accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
// See http://www.boost.org for updates, documentation, and revision history.
#ifndef BOOST_QUATERNION_HPP
#define BOOST_QUATERNION_HPP
#include <complex>
#include <iosfwd> // for the "<<" and ">>" operators
#include <sstream> // for the "<<" operator
#include <boost/config.hpp> // for BOOST_NO_STD_LOCALE
#include <boost/detail/workaround.hpp>
#ifndef BOOST_NO_STD_LOCALE
#include <locale> // for the "<<" operator
#endif /* BOOST_NO_STD_LOCALE */
#include <valarray>
#include <boost/math/special_functions/sinc.hpp> // for the Sinus cardinal
#include <boost/math/special_functions/sinhc.hpp> // for the Hyperbolic Sinus cardinal
namespace boost
{
namespace math
{
#define BOOST_QUATERNION_ACCESSOR_GENERATOR(type) \
type real() const \
{ \
return(a); \
} \
\
quaternion<type> unreal() const \
{ \
return(quaternion<type>(static_cast<type>(0),b,c,d)); \
} \
\
type R_component_1() const \
{ \
return(a); \
} \
\
type R_component_2() const \
{ \
return(b); \
} \
\
type R_component_3() const \
{ \
return(c); \
} \
\
type R_component_4() const \
{ \
return(d); \
} \
\
::std::complex<type> C_component_1() const \
{ \
return(::std::complex<type>(a,b)); \
} \
\
::std::complex<type> C_component_2() const \
{ \
return(::std::complex<type>(c,d)); \
}
#define BOOST_QUATERNION_MEMBER_ASSIGNMENT_GENERATOR(type) \
template<typename X> \
quaternion<type> & operator = (quaternion<X> const & a_affecter) \
{ \
a = static_cast<type>(a_affecter.R_component_1()); \
b = static_cast<type>(a_affecter.R_component_2()); \
c = static_cast<type>(a_affecter.R_component_3()); \
d = static_cast<type>(a_affecter.R_component_4()); \
\
return(*this); \
} \
\
quaternion<type> & operator = (quaternion<type> const & a_affecter) \
{ \
a = a_affecter.a; \
b = a_affecter.b; \
c = a_affecter.c; \
d = a_affecter.d; \
\
return(*this); \
} \
\
quaternion<type> & operator = (type const & a_affecter) \
{ \
a = a_affecter; \
\
b = c = d = static_cast<type>(0); \
\
return(*this); \
} \
\
quaternion<type> & operator = (::std::complex<type> const & a_affecter) \
{ \
a = a_affecter.real(); \
b = a_affecter.imag(); \
\
c = d = static_cast<type>(0); \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_DATA_GENERATOR(type) \
type a; \
type b; \
type c; \
type d;
template<typename T>
class quaternion
{
public:
typedef T value_type;
// constructor for H seen as R^4
// (also default constructor)
explicit quaternion( T const & requested_a = T(),
T const & requested_b = T(),
T const & requested_c = T(),
T const & requested_d = T())
: a(requested_a),
b(requested_b),
c(requested_c),
d(requested_d)
{
// nothing to do!
}
// constructor for H seen as C^2
explicit quaternion( ::std::complex<T> const & z0,
::std::complex<T> const & z1 = ::std::complex<T>())
: a(z0.real()),
b(z0.imag()),
c(z1.real()),
d(z1.imag())
{
// nothing to do!
}
// UNtemplated copy constructor
// (this is taken care of by the compiler itself)
// templated copy constructor
template<typename X>
explicit quaternion(quaternion<X> const & a_recopier)
: a(static_cast<T>(a_recopier.R_component_1())),
b(static_cast<T>(a_recopier.R_component_2())),
c(static_cast<T>(a_recopier.R_component_3())),
d(static_cast<T>(a_recopier.R_component_4()))
{
// nothing to do!
}
// destructor
// (this is taken care of by the compiler itself)
// accessors
//
// Note: Like complex number, quaternions do have a meaningful notion of "real part",
// but unlike them there is no meaningful notion of "imaginary part".
// Instead there is an "unreal part" which itself is a quaternion, and usually
// nothing simpler (as opposed to the complex number case).
// However, for practicallity, there are accessors for the other components
// (these are necessary for the templated copy constructor, for instance).
BOOST_QUATERNION_ACCESSOR_GENERATOR(T)
// assignment operators
BOOST_QUATERNION_MEMBER_ASSIGNMENT_GENERATOR(T)
// other assignment-related operators
//
// NOTE: Quaternion multiplication is *NOT* commutative;
// symbolically, "q *= rhs;" means "q = q * rhs;"
// and "q /= rhs;" means "q = q * inverse_of(rhs);"
quaternion<T> & operator += (T const & rhs)
{
T at = a + rhs; // exception guard
a = at;
return(*this);
}
quaternion<T> & operator += (::std::complex<T> const & rhs)
{
T at = a + rhs.real(); // exception guard
T bt = b + rhs.imag(); // exception guard
a = at;
b = bt;
return(*this);
}
template<typename X>
quaternion<T> & operator += (quaternion<X> const & rhs)
{
T at = a + static_cast<T>(rhs.R_component_1()); // exception guard
T bt = b + static_cast<T>(rhs.R_component_2()); // exception guard
T ct = c + static_cast<T>(rhs.R_component_3()); // exception guard
T dt = d + static_cast<T>(rhs.R_component_4()); // exception guard
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
quaternion<T> & operator -= (T const & rhs)
{
T at = a - rhs; // exception guard
a = at;
return(*this);
}
quaternion<T> & operator -= (::std::complex<T> const & rhs)
{
T at = a - rhs.real(); // exception guard
T bt = b - rhs.imag(); // exception guard
a = at;
b = bt;
return(*this);
}
template<typename X>
quaternion<T> & operator -= (quaternion<X> const & rhs)
{
T at = a - static_cast<T>(rhs.R_component_1()); // exception guard
T bt = b - static_cast<T>(rhs.R_component_2()); // exception guard
T ct = c - static_cast<T>(rhs.R_component_3()); // exception guard
T dt = d - static_cast<T>(rhs.R_component_4()); // exception guard
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
quaternion<T> & operator *= (T const & rhs)
{
T at = a * rhs; // exception guard
T bt = b * rhs; // exception guard
T ct = c * rhs; // exception guard
T dt = d * rhs; // exception guard
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
quaternion<T> & operator *= (::std::complex<T> const & rhs)
{
T ar = rhs.real();
T br = rhs.imag();
T at = +a*ar-b*br;
T bt = +a*br+b*ar;
T ct = +c*ar+d*br;
T dt = -c*br+d*ar;
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
template<typename X>
quaternion<T> & operator *= (quaternion<X> const & rhs)
{
T ar = static_cast<T>(rhs.R_component_1());
T br = static_cast<T>(rhs.R_component_2());
T cr = static_cast<T>(rhs.R_component_3());
T dr = static_cast<T>(rhs.R_component_4());
T at = +a*ar-b*br-c*cr-d*dr;
T bt = +a*br+b*ar+c*dr-d*cr; //(a*br+ar*b)+(c*dr-cr*d);
T ct = +a*cr-b*dr+c*ar+d*br; //(a*cr+ar*c)+(d*br-dr*b);
T dt = +a*dr+b*cr-c*br+d*ar; //(a*dr+ar*d)+(b*cr-br*c);
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
quaternion<T> & operator /= (T const & rhs)
{
T at = a / rhs; // exception guard
T bt = b / rhs; // exception guard
T ct = c / rhs; // exception guard
T dt = d / rhs; // exception guard
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
quaternion<T> & operator /= (::std::complex<T> const & rhs)
{
T ar = rhs.real();
T br = rhs.imag();
T denominator = ar*ar+br*br;
T at = (+a*ar+b*br)/denominator; //(a*ar+b*br)/denominator;
T bt = (-a*br+b*ar)/denominator; //(ar*b-a*br)/denominator;
T ct = (+c*ar-d*br)/denominator; //(ar*c-d*br)/denominator;
T dt = (+c*br+d*ar)/denominator; //(ar*d+br*c)/denominator;
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
template<typename X>
quaternion<T> & operator /= (quaternion<X> const & rhs)
{
T ar = static_cast<T>(rhs.R_component_1());
T br = static_cast<T>(rhs.R_component_2());
T cr = static_cast<T>(rhs.R_component_3());
T dr = static_cast<T>(rhs.R_component_4());
T denominator = ar*ar+br*br+cr*cr+dr*dr;
T at = (+a*ar+b*br+c*cr+d*dr)/denominator; //(a*ar+b*br+c*cr+d*dr)/denominator;
T bt = (-a*br+b*ar-c*dr+d*cr)/denominator; //((ar*b-a*br)+(cr*d-c*dr))/denominator;
T ct = (-a*cr+b*dr+c*ar-d*br)/denominator; //((ar*c-a*cr)+(dr*b-d*br))/denominator;
T dt = (-a*dr-b*cr+c*br+d*ar)/denominator; //((ar*d-a*dr)+(br*c-b*cr))/denominator;
a = at;
b = bt;
c = ct;
d = dt;
return(*this);
}
protected:
BOOST_QUATERNION_MEMBER_DATA_GENERATOR(T)
private:
};
// declaration of quaternion specialization
template<> class quaternion<float>;
template<> class quaternion<double>;
template<> class quaternion<long double>;
// helper templates for converting copy constructors (declaration)
namespace detail
{
template< typename T,
typename U
>
quaternion<T> quaternion_type_converter(quaternion<U> const & rhs);
}
// implementation of quaternion specialization
#define BOOST_QUATERNION_CONSTRUCTOR_GENERATOR(type) \
explicit quaternion( type const & requested_a = static_cast<type>(0), \
type const & requested_b = static_cast<type>(0), \
type const & requested_c = static_cast<type>(0), \
type const & requested_d = static_cast<type>(0)) \
: a(requested_a), \
b(requested_b), \
c(requested_c), \
d(requested_d) \
{ \
} \
\
explicit quaternion( ::std::complex<type> const & z0, \
::std::complex<type> const & z1 = ::std::complex<type>()) \
: a(z0.real()), \
b(z0.imag()), \
c(z1.real()), \
d(z1.imag()) \
{ \
}
#define BOOST_QUATERNION_MEMBER_ADD_GENERATOR_1(type) \
quaternion<type> & operator += (type const & rhs) \
{ \
a += rhs; \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_ADD_GENERATOR_2(type) \
quaternion<type> & operator += (::std::complex<type> const & rhs) \
{ \
a += rhs.real(); \
b += rhs.imag(); \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_ADD_GENERATOR_3(type) \
template<typename X> \
quaternion<type> & operator += (quaternion<X> const & rhs) \
{ \
a += static_cast<type>(rhs.R_component_1()); \
b += static_cast<type>(rhs.R_component_2()); \
c += static_cast<type>(rhs.R_component_3()); \
d += static_cast<type>(rhs.R_component_4()); \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_SUB_GENERATOR_1(type) \
quaternion<type> & operator -= (type const & rhs) \
{ \
a -= rhs; \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_SUB_GENERATOR_2(type) \
quaternion<type> & operator -= (::std::complex<type> const & rhs) \
{ \
a -= rhs.real(); \
b -= rhs.imag(); \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_SUB_GENERATOR_3(type) \
template<typename X> \
quaternion<type> & operator -= (quaternion<X> const & rhs) \
{ \
a -= static_cast<type>(rhs.R_component_1()); \
b -= static_cast<type>(rhs.R_component_2()); \
c -= static_cast<type>(rhs.R_component_3()); \
d -= static_cast<type>(rhs.R_component_4()); \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_MUL_GENERATOR_1(type) \
quaternion<type> & operator *= (type const & rhs) \
{ \
a *= rhs; \
b *= rhs; \
c *= rhs; \
d *= rhs; \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_MUL_GENERATOR_2(type) \
quaternion<type> & operator *= (::std::complex<type> const & rhs) \
{ \
type ar = rhs.real(); \
type br = rhs.imag(); \
\
type at = +a*ar-b*br; \
type bt = +a*br+b*ar; \
type ct = +c*ar+d*br; \
type dt = -c*br+d*ar; \
\
a = at; \
b = bt; \
c = ct; \
d = dt; \
\
return(*this); \
}
#define BOOST_QUATERNION_MEMBER_MUL_GENERATOR_3(type) \
template<typename X> \
quaternion<type> & operator *= (quaternion<X> const & rhs) \
{ \
type ar = static_cast<type>(rhs.R_component_1()); \
type br = static_cast<type>(rhs.R_component_2()); \
type cr = static_cast<type>(rhs.R_component_3()); \
type dr = static_cast<type>(rhs.R_component_4()); \
\
type at = +a*ar-b*br-c*cr-d*dr; \
type bt = +a*br+b*ar+c*dr-d*cr; \
type ct = +a*cr-b*dr+c*ar+d*br; \
type dt = +a*dr+b*cr-c*br+d*ar; \
\
a = at; \
b = bt; \
c = ct; \
d = dt; \
\
return(*this); \
}
// There is quite a lot of repetition in the code below. This is intentional.
// The last conditional block is the normal form, and the others merely
// consist of workarounds for various compiler deficiencies. Hopefuly, when
// more compilers are conformant and we can retire support for those that are
// not, we will be able to remove the clutter. This is makes the situation
// (painfully) explicit.
#define BOOST_QUATERNION_MEMBER_DIV_GENERATOR_1(type) \
quaternion<type> & operator /= (type const & rhs) \
{ \
a /= rhs; \
b /= rhs; \
c /= rhs; \
d /= rhs; \
\
return(*this); \
}
#if defined(BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP)
#define BOOST_QUATERNION_MEMBER_DIV_GENERATOR_2(type) \
quaternion<type> & operator /= (::std::complex<type> const & rhs) \
{ \
using ::std::valarray; \
using ::std::abs; \
\
valarray<type> tr(2); \
\
tr[0] = rhs.real(); \
tr[1] = rhs.imag(); \
\
type mixam = static_cast<type>(1)/(abs(tr).max)(); \
\
tr *= mixam; \
\
valarray<type> tt(4); \
\
tt[0] = +a*tr[0]+b*tr[1]; \
tt[1] = -a*tr[1]+b*tr[0]; \
tt[2] = +c*tr[0]-d*tr[1]; \
tt[3] = +c*tr[1]+d*tr[0]; \
\
tr *= tr; \
\
tt *= (mixam/tr.sum()); \
\
a = tt[0]; \
b = tt[1]; \
c = tt[2]; \
d = tt[3]; \
\
return(*this); \
}
#else
#define BOOST_QUATERNION_MEMBER_DIV_GENERATOR_2(type) \
quaternion<type> & operator /= (::std::complex<type> const & rhs) \
{ \
using ::std::valarray; \
\
valarray<type> tr(2); \
\
tr[0] = rhs.real(); \
tr[1] = rhs.imag(); \
\
type mixam = static_cast<type>(1)/(abs(tr).max)(); \
\
tr *= mixam; \
\
valarray<type> tt(4); \
\
tt[0] = +a*tr[0]+b*tr[1]; \
tt[1] = -a*tr[1]+b*tr[0]; \
tt[2] = +c*tr[0]-d*tr[1]; \
tt[3] = +c*tr[1]+d*tr[0]; \
\
tr *= tr; \
\
tt *= (mixam/tr.sum()); \
\
a = tt[0]; \
b = tt[1]; \
c = tt[2]; \
d = tt[3]; \
\
return(*this); \
}
#endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */
#if defined(BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP)
#define BOOST_QUATERNION_MEMBER_DIV_GENERATOR_3(type) \
template<typename X> \
quaternion<type> & operator /= (quaternion<X> const & rhs) \
{ \
using ::std::valarray; \
using ::std::abs; \
\
valarray<type> tr(4); \
\
tr[0] = static_cast<type>(rhs.R_component_1()); \
tr[1] = static_cast<type>(rhs.R_component_2()); \
tr[2] = static_cast<type>(rhs.R_component_3()); \
tr[3] = static_cast<type>(rhs.R_component_4()); \
\
type mixam = static_cast<type>(1)/(abs(tr).max)(); \
\
tr *= mixam; \
\
valarray<type> tt(4); \
\
tt[0] = +a*tr[0]+b*tr[1]+c*tr[2]+d*tr[3]; \
tt[1] = -a*tr[1]+b*tr[0]-c*tr[3]+d*tr[2]; \
tt[2] = -a*tr[2]+b*tr[3]+c*tr[0]-d*tr[1]; \
tt[3] = -a*tr[3]-b*tr[2]+c*tr[1]+d*tr[0]; \
\
tr *= tr; \
\
tt *= (mixam/tr.sum()); \
\
a = tt[0]; \
b = tt[1]; \
c = tt[2]; \
d = tt[3]; \
\
return(*this); \
}
#else
#define BOOST_QUATERNION_MEMBER_DIV_GENERATOR_3(type) \
template<typename X> \
quaternion<type> & operator /= (quaternion<X> const & rhs) \
{ \
using ::std::valarray; \
\
valarray<type> tr(4); \
\
tr[0] = static_cast<type>(rhs.R_component_1()); \
tr[1] = static_cast<type>(rhs.R_component_2()); \
tr[2] = static_cast<type>(rhs.R_component_3()); \
tr[3] = static_cast<type>(rhs.R_component_4()); \
\
type mixam = static_cast<type>(1)/(abs(tr).max)(); \
\
tr *= mixam; \
\
valarray<type> tt(4); \
\
tt[0] = +a*tr[0]+b*tr[1]+c*tr[2]+d*tr[3]; \
tt[1] = -a*tr[1]+b*tr[0]-c*tr[3]+d*tr[2]; \
tt[2] = -a*tr[2]+b*tr[3]+c*tr[0]-d*tr[1]; \
tt[3] = -a*tr[3]-b*tr[2]+c*tr[1]+d*tr[0]; \
\
tr *= tr; \
\
tt *= (mixam/tr.sum()); \
\
a = tt[0]; \
b = tt[1]; \
c = tt[2]; \
d = tt[3]; \
\
return(*this); \
}
#endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */
#define BOOST_QUATERNION_MEMBER_ADD_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_ADD_GENERATOR_1(type) \
BOOST_QUATERNION_MEMBER_ADD_GENERATOR_2(type) \
BOOST_QUATERNION_MEMBER_ADD_GENERATOR_3(type)
#define BOOST_QUATERNION_MEMBER_SUB_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_SUB_GENERATOR_1(type) \
BOOST_QUATERNION_MEMBER_SUB_GENERATOR_2(type) \
BOOST_QUATERNION_MEMBER_SUB_GENERATOR_3(type)
#define BOOST_QUATERNION_MEMBER_MUL_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_MUL_GENERATOR_1(type) \
BOOST_QUATERNION_MEMBER_MUL_GENERATOR_2(type) \
BOOST_QUATERNION_MEMBER_MUL_GENERATOR_3(type)
#define BOOST_QUATERNION_MEMBER_DIV_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_DIV_GENERATOR_1(type) \
BOOST_QUATERNION_MEMBER_DIV_GENERATOR_2(type) \
BOOST_QUATERNION_MEMBER_DIV_GENERATOR_3(type)
#define BOOST_QUATERNION_MEMBER_ALGEBRAIC_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_ADD_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_SUB_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_MUL_GENERATOR(type) \
BOOST_QUATERNION_MEMBER_DIV_GENERATOR(type)
template<>
class quaternion<float>
{
public:
typedef float value_type;
BOOST_QUATERNION_CONSTRUCTOR_GENERATOR(float)
// UNtemplated copy constructor
// (this is taken care of by the compiler itself)
// explicit copy constructors (precision-loosing converters)
explicit quaternion(quaternion<double> const & a_recopier)
{
*this = detail::quaternion_type_converter<float, double>(a_recopier);
}
explicit quaternion(quaternion<long double> const & a_recopier)
{
*this = detail::quaternion_type_converter<float, long double>(a_recopier);
}
// destructor
// (this is taken care of by the compiler itself)
// accessors
//
// Note: Like complex number, quaternions do have a meaningful notion of "real part",
// but unlike them there is no meaningful notion of "imaginary part".
// Instead there is an "unreal part" which itself is a quaternion, and usually
// nothing simpler (as opposed to the complex number case).
// However, for practicallity, there are accessors for the other components
// (these are necessary for the templated copy constructor, for instance).
BOOST_QUATERNION_ACCESSOR_GENERATOR(float)
// assignment operators
BOOST_QUATERNION_MEMBER_ASSIGNMENT_GENERATOR(float)
// other assignment-related operators
//
// NOTE: Quaternion multiplication is *NOT* commutative;
// symbolically, "q *= rhs;" means "q = q * rhs;"
// and "q /= rhs;" means "q = q * inverse_of(rhs);"
BOOST_QUATERNION_MEMBER_ALGEBRAIC_GENERATOR(float)
protected:
BOOST_QUATERNION_MEMBER_DATA_GENERATOR(float)
private:
};
template<>
class quaternion<double>
{
public:
typedef double value_type;
BOOST_QUATERNION_CONSTRUCTOR_GENERATOR(double)
// UNtemplated copy constructor
// (this is taken care of by the compiler itself)
// converting copy constructor
explicit quaternion(quaternion<float> const & a_recopier)
{
*this = detail::quaternion_type_converter<double, float>(a_recopier);
}
// explicit copy constructors (precision-loosing converters)
explicit quaternion(quaternion<long double> const & a_recopier)
{
*this = detail::quaternion_type_converter<double, long double>(a_recopier);
}
// destructor
// (this is taken care of by the compiler itself)
// accessors
//
// Note: Like complex number, quaternions do have a meaningful notion of "real part",
// but unlike them there is no meaningful notion of "imaginary part".
// Instead there is an "unreal part" which itself is a quaternion, and usually
// nothing simpler (as opposed to the complex number case).
// However, for practicallity, there are accessors for the other components
// (these are necessary for the templated copy constructor, for instance).
BOOST_QUATERNION_ACCESSOR_GENERATOR(double)
// assignment operators
BOOST_QUATERNION_MEMBER_ASSIGNMENT_GENERATOR(double)
// other assignment-related operators
//
// NOTE: Quaternion multiplication is *NOT* commutative;
// symbolically, "q *= rhs;" means "q = q * rhs;"
// and "q /= rhs;" means "q = q * inverse_of(rhs);"
BOOST_QUATERNION_MEMBER_ALGEBRAIC_GENERATOR(double)
protected:
BOOST_QUATERNION_MEMBER_DATA_GENERATOR(double)
private:
};
template<>
class quaternion<long double>
{
public:
typedef long double value_type;
BOOST_QUATERNION_CONSTRUCTOR_GENERATOR(long double)
// UNtemplated copy constructor
// (this is taken care of by the compiler itself)
// converting copy constructors
explicit quaternion(quaternion<float> const & a_recopier)
{
*this = detail::quaternion_type_converter<long double, float>(a_recopier);
}
explicit quaternion(quaternion<double> const & a_recopier)
{
*this = detail::quaternion_type_converter<long double, double>(a_recopier);
}
// destructor
// (this is taken care of by the compiler itself)
// accessors
//
// Note: Like complex number, quaternions do have a meaningful notion of "real part",
// but unlike them there is no meaningful notion of "imaginary part".
// Instead there is an "unreal part" which itself is a quaternion, and usually
// nothing simpler (as opposed to the complex number case).
// However, for practicallity, there are accessors for the other components
// (these are necessary for the templated copy constructor, for instance).
BOOST_QUATERNION_ACCESSOR_GENERATOR(long double)
// assignment operators
BOOST_QUATERNION_MEMBER_ASSIGNMENT_GENERATOR(long double)
// other assignment-related operators
//
// NOTE: Quaternion multiplication is *NOT* commutative;
// symbolically, "q *= rhs;" means "q = q * rhs;"
// and "q /= rhs;" means "q = q * inverse_of(rhs);"
BOOST_QUATERNION_MEMBER_ALGEBRAIC_GENERATOR(long double)
protected:
BOOST_QUATERNION_MEMBER_DATA_GENERATOR(long double)
private:
};
#undef BOOST_QUATERNION_MEMBER_ALGEBRAIC_GENERATOR
#undef BOOST_QUATERNION_MEMBER_ADD_GENERATOR
#undef BOOST_QUATERNION_MEMBER_SUB_GENERATOR
#undef BOOST_QUATERNION_MEMBER_MUL_GENERATOR
#undef BOOST_QUATERNION_MEMBER_DIV_GENERATOR
#undef BOOST_QUATERNION_MEMBER_ADD_GENERATOR_1
#undef BOOST_QUATERNION_MEMBER_ADD_GENERATOR_2
#undef BOOST_QUATERNION_MEMBER_ADD_GENERATOR_3
#undef BOOST_QUATERNION_MEMBER_SUB_GENERATOR_1
#undef BOOST_QUATERNION_MEMBER_SUB_GENERATOR_2
#undef BOOST_QUATERNION_MEMBER_SUB_GENERATOR_3
#undef BOOST_QUATERNION_MEMBER_MUL_GENERATOR_1
#undef BOOST_QUATERNION_MEMBER_MUL_GENERATOR_2
#undef BOOST_QUATERNION_MEMBER_MUL_GENERATOR_3
#undef BOOST_QUATERNION_MEMBER_DIV_GENERATOR_1
#undef BOOST_QUATERNION_MEMBER_DIV_GENERATOR_2
#undef BOOST_QUATERNION_MEMBER_DIV_GENERATOR_3
#undef BOOST_QUATERNION_CONSTRUCTOR_GENERATOR
#undef BOOST_QUATERNION_MEMBER_ASSIGNMENT_GENERATOR
#undef BOOST_QUATERNION_MEMBER_DATA_GENERATOR
#undef BOOST_QUATERNION_ACCESSOR_GENERATOR
// operators
#define BOOST_QUATERNION_OPERATOR_GENERATOR_BODY(op) \
{ \
quaternion<T> res(lhs); \
res op##= rhs; \
return(res); \
}
#define BOOST_QUATERNION_OPERATOR_GENERATOR_1_L(op) \
template<typename T> \
inline quaternion<T> operator op (T const & lhs, quaternion<T> const & rhs) \
BOOST_QUATERNION_OPERATOR_GENERATOR_BODY(op)
#define BOOST_QUATERNION_OPERATOR_GENERATOR_1_R(op) \
template<typename T> \
inline quaternion<T> operator op (quaternion<T> const & lhs, T const & rhs) \
BOOST_QUATERNION_OPERATOR_GENERATOR_BODY(op)
#define BOOST_QUATERNION_OPERATOR_GENERATOR_2_L(op) \
template<typename T> \
inline quaternion<T> operator op (::std::complex<T> const & lhs, quaternion<T> const & rhs) \
BOOST_QUATERNION_OPERATOR_GENERATOR_BODY(op)
#define BOOST_QUATERNION_OPERATOR_GENERATOR_2_R(op) \
template<typename T> \
inline quaternion<T> operator op (quaternion<T> const & lhs, ::std::complex<T> const & rhs) \
BOOST_QUATERNION_OPERATOR_GENERATOR_BODY(op)
#define BOOST_QUATERNION_OPERATOR_GENERATOR_3(op) \
template<typename T> \
inline quaternion<T> operator op (quaternion<T> const & lhs, quaternion<T> const & rhs) \
BOOST_QUATERNION_OPERATOR_GENERATOR_BODY(op)
#define BOOST_QUATERNION_OPERATOR_GENERATOR(op) \
BOOST_QUATERNION_OPERATOR_GENERATOR_1_L(op) \
BOOST_QUATERNION_OPERATOR_GENERATOR_1_R(op) \
BOOST_QUATERNION_OPERATOR_GENERATOR_2_L(op) \
BOOST_QUATERNION_OPERATOR_GENERATOR_2_R(op) \
BOOST_QUATERNION_OPERATOR_GENERATOR_3(op)
BOOST_QUATERNION_OPERATOR_GENERATOR(+)
BOOST_QUATERNION_OPERATOR_GENERATOR(-)
BOOST_QUATERNION_OPERATOR_GENERATOR(*)
BOOST_QUATERNION_OPERATOR_GENERATOR(/)
#undef BOOST_QUATERNION_OPERATOR_GENERATOR
#undef BOOST_QUATERNION_OPERATOR_GENERATOR_1_L
#undef BOOST_QUATERNION_OPERATOR_GENERATOR_1_R
#undef BOOST_QUATERNION_OPERATOR_GENERATOR_2_L
#undef BOOST_QUATERNION_OPERATOR_GENERATOR_2_R
#undef BOOST_QUATERNION_OPERATOR_GENERATOR_3
#undef BOOST_QUATERNION_OPERATOR_GENERATOR_BODY
template<typename T>
inline quaternion<T> operator + (quaternion<T> const & q)
{
return(q);
}
template<typename T>
inline quaternion<T> operator - (quaternion<T> const & q)
{
return(quaternion<T>(-q.R_component_1(),-q.R_component_2(),-q.R_component_3(),-q.R_component_4()));
}
template<typename T>
inline bool operator == (T const & lhs, quaternion<T> const & rhs)
{
return (
(rhs.R_component_1() == lhs)&&
(rhs.R_component_2() == static_cast<T>(0))&&
(rhs.R_component_3() == static_cast<T>(0))&&
(rhs.R_component_4() == static_cast<T>(0))
);
}
template<typename T>
inline bool operator == (quaternion<T> const & lhs, T const & rhs)
{
return (
(lhs.R_component_1() == rhs)&&
(lhs.R_component_2() == static_cast<T>(0))&&
(lhs.R_component_3() == static_cast<T>(0))&&
(lhs.R_component_4() == static_cast<T>(0))
);
}
template<typename T>
inline bool operator == (::std::complex<T> const & lhs, quaternion<T> const & rhs)
{
return (
(rhs.R_component_1() == lhs.real())&&
(rhs.R_component_2() == lhs.imag())&&
(rhs.R_component_3() == static_cast<T>(0))&&
(rhs.R_component_4() == static_cast<T>(0))
);
}
template<typename T>
inline bool operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs)
{
return (
(lhs.R_component_1() == rhs.real())&&
(lhs.R_component_2() == rhs.imag())&&
(lhs.R_component_3() == static_cast<T>(0))&&
(lhs.R_component_4() == static_cast<T>(0))
);
}
template<typename T>
inline bool operator == (quaternion<T> const & lhs, quaternion<T> const & rhs)
{
return (
(rhs.R_component_1() == lhs.R_component_1())&&
(rhs.R_component_2() == lhs.R_component_2())&&
(rhs.R_component_3() == lhs.R_component_3())&&
(rhs.R_component_4() == lhs.R_component_4())
);
}
#define BOOST_QUATERNION_NOT_EQUAL_GENERATOR \
{ \
return(!(lhs == rhs)); \
}
template<typename T>
inline bool operator != (T const & lhs, quaternion<T> const & rhs)
BOOST_QUATERNION_NOT_EQUAL_GENERATOR
template<typename T>
inline bool operator != (quaternion<T> const & lhs, T const & rhs)
BOOST_QUATERNION_NOT_EQUAL_GENERATOR
template<typename T>
inline bool operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs)
BOOST_QUATERNION_NOT_EQUAL_GENERATOR
template<typename T>
inline bool operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs)
BOOST_QUATERNION_NOT_EQUAL_GENERATOR
template<typename T>
inline bool operator != (quaternion<T> const & lhs, quaternion<T> const & rhs)
BOOST_QUATERNION_NOT_EQUAL_GENERATOR
#undef BOOST_QUATERNION_NOT_EQUAL_GENERATOR
// Note: we allow the following formats, whith a, b, c, and d reals
// a
// (a), (a,b), (a,b,c), (a,b,c,d)
// (a,(c)), (a,(c,d)), ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b),(c,d))
template<typename T, typename charT, class traits>
::std::basic_istream<charT,traits> & operator >> ( ::std::basic_istream<charT,traits> & is,
quaternion<T> & q)
{
#ifdef BOOST_NO_STD_LOCALE
#else
const ::std::ctype<charT> & ct = ::std::use_facet< ::std::ctype<charT> >(is.getloc());
#endif /* BOOST_NO_STD_LOCALE */
T a = T();
T b = T();
T c = T();
T d = T();
::std::complex<T> u = ::std::complex<T>();
::std::complex<T> v = ::std::complex<T>();
charT ch = charT();
char cc;
is >> ch; // get the first lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == '(') // read "(", possible: (a), (a,b), (a,b,c), (a,b,c,d), (a,(c)), (a,(c,d)), ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b,),(c,d,))
{
is >> ch; // get the second lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == '(') // read "((", possible: ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b,),(c,d,))
{
is.putback(ch);
is >> u; // we extract the first and second components
a = u.real();
b = u.imag();
if (!is.good()) goto finish;
is >> ch; // get the next lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == ')') // format: ((a)) or ((a,b))
{
q = quaternion<T>(a,b);
}
else if (cc == ',') // read "((a)," or "((a,b),", possible: ((a),c), ((a),(c)), ((a),(c,d)), ((a,b),c), ((a,b),(c)), ((a,b,),(c,d,))
{
is >> v; // we extract the third and fourth components
c = v.real();
d = v.imag();
if (!is.good()) goto finish;
is >> ch; // get the last lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == ')') // format: ((a),c), ((a),(c)), ((a),(c,d)), ((a,b),c), ((a,b),(c)) or ((a,b,),(c,d,))
{
q = quaternion<T>(a,b,c,d);
}
else // error
{
is.setstate(::std::ios_base::failbit);
}
}
else // error
{
is.setstate(::std::ios_base::failbit);
}
}
else // read "(a", possible: (a), (a,b), (a,b,c), (a,b,c,d), (a,(c)), (a,(c,d))
{
is.putback(ch);
is >> a; // we extract the first component
if (!is.good()) goto finish;
is >> ch; // get the third lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == ')') // format: (a)
{
q = quaternion<T>(a);
}
else if (cc == ',') // read "(a,", possible: (a,b), (a,b,c), (a,b,c,d), (a,(c)), (a,(c,d))
{
is >> ch; // get the fourth lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == '(') // read "(a,(", possible: (a,(c)), (a,(c,d))
{
is.putback(ch);
is >> v; // we extract the third and fourth component
c = v.real();
d = v.imag();
if (!is.good()) goto finish;
is >> ch; // get the ninth lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == ')') // format: (a,(c)) or (a,(c,d))
{
q = quaternion<T>(a,b,c,d);
}
else // error
{
is.setstate(::std::ios_base::failbit);
}
}
else // read "(a,b", possible: (a,b), (a,b,c), (a,b,c,d)
{
is.putback(ch);
is >> b; // we extract the second component
if (!is.good()) goto finish;
is >> ch; // get the fifth lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == ')') // format: (a,b)
{
q = quaternion<T>(a,b);
}
else if (cc == ',') // read "(a,b,", possible: (a,b,c), (a,b,c,d)
{
is >> c; // we extract the third component
if (!is.good()) goto finish;
is >> ch; // get the seventh lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == ')') // format: (a,b,c)
{
q = quaternion<T>(a,b,c);
}
else if (cc == ',') // read "(a,b,c,", possible: (a,b,c,d)
{
is >> d; // we extract the fourth component
if (!is.good()) goto finish;
is >> ch; // get the ninth lexeme
if (!is.good()) goto finish;
#ifdef BOOST_NO_STD_LOCALE
cc = ch;
#else
cc = ct.narrow(ch, char());
#endif /* BOOST_NO_STD_LOCALE */
if (cc == ')') // format: (a,b,c,d)
{
q = quaternion<T>(a,b,c,d);
}
else // error
{
is.setstate(::std::ios_base::failbit);
}
}
else // error
{
is.setstate(::std::ios_base::failbit);
}
}
else // error
{
is.setstate(::std::ios_base::failbit);
}
}
}
else // error
{
is.setstate(::std::ios_base::failbit);
}
}
}
else // format: a
{
is.putback(ch);
is >> a; // we extract the first component
if (!is.good()) goto finish;
q = quaternion<T>(a);
}
finish:
return(is);
}
template<typename T, typename charT, class traits>
::std::basic_ostream<charT,traits> & operator << ( ::std::basic_ostream<charT,traits> & os,
quaternion<T> const & q)
{
::std::basic_ostringstream<charT,traits> s;
s.flags(os.flags());
#ifdef BOOST_NO_STD_LOCALE
#else
s.imbue(os.getloc());
#endif /* BOOST_NO_STD_LOCALE */
s.precision(os.precision());
s << '(' << q.R_component_1() << ','
<< q.R_component_2() << ','
<< q.R_component_3() << ','
<< q.R_component_4() << ')';
return os << s.str();
}
// values
template<typename T>
inline T real(quaternion<T> const & q)
{
return(q.real());
}
template<typename T>
inline quaternion<T> unreal(quaternion<T> const & q)
{
return(q.unreal());
}
#define BOOST_QUATERNION_VALARRAY_LOADER \
using ::std::valarray; \
\
valarray<T> temp(4); \
\
temp[0] = q.R_component_1(); \
temp[1] = q.R_component_2(); \
temp[2] = q.R_component_3(); \
temp[3] = q.R_component_4();
template<typename T>
inline T sup(quaternion<T> const & q)
{
#ifdef BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP
using ::std::abs;
#endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */
BOOST_QUATERNION_VALARRAY_LOADER
return((abs(temp).max)());
}
template<typename T>
inline T l1(quaternion<T> const & q)
{
#ifdef BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP
using ::std::abs;
#endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */
BOOST_QUATERNION_VALARRAY_LOADER
return(abs(temp).sum());
}
template<typename T>
inline T abs(quaternion<T> const & q)
{
#ifdef BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP
using ::std::abs;
#endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */
using ::std::sqrt;
BOOST_QUATERNION_VALARRAY_LOADER
T maxim = (abs(temp).max)(); // overflow protection
if (maxim == static_cast<T>(0))
{
return(maxim);
}
else
{
T mixam = static_cast<T>(1)/maxim; // prefer multiplications over divisions
temp *= mixam;
temp *= temp;
return(maxim*sqrt(temp.sum()));
}
//return(sqrt(norm(q)));
}
#undef BOOST_QUATERNION_VALARRAY_LOADER
// Note: This is the Cayley norm, not the Euclidian norm...
template<typename T>
inline T norm(quaternion<T>const & q)
{
return(real(q*conj(q)));
}
template<typename T>
inline quaternion<T> conj(quaternion<T> const & q)
{
return(quaternion<T>( +q.R_component_1(),
-q.R_component_2(),
-q.R_component_3(),
-q.R_component_4()));
}
template<typename T>
inline quaternion<T> spherical( T const & rho,
T const & theta,
T const & phi1,
T const & phi2)
{
using ::std::cos;
using ::std::sin;
//T a = cos(theta)*cos(phi1)*cos(phi2);
//T b = sin(theta)*cos(phi1)*cos(phi2);
//T c = sin(phi1)*cos(phi2);
//T d = sin(phi2);
T courrant = static_cast<T>(1);
T d = sin(phi2);
courrant *= cos(phi2);
T c = sin(phi1)*courrant;
courrant *= cos(phi1);
T b = sin(theta)*courrant;
T a = cos(theta)*courrant;
return(rho*quaternion<T>(a,b,c,d));
}
template<typename T>
inline quaternion<T> semipolar( T const & rho,
T const & alpha,
T const & theta1,
T const & theta2)
{
using ::std::cos;
using ::std::sin;
T a = cos(alpha)*cos(theta1);
T b = cos(alpha)*sin(theta1);
T c = sin(alpha)*cos(theta2);
T d = sin(alpha)*sin(theta2);
return(rho*quaternion<T>(a,b,c,d));
}
template<typename T>
inline quaternion<T> multipolar( T const & rho1,
T const & theta1,
T const & rho2,
T const & theta2)
{
using ::std::cos;
using ::std::sin;
T a = rho1*cos(theta1);
T b = rho1*sin(theta1);
T c = rho2*cos(theta2);
T d = rho2*sin(theta2);
return(quaternion<T>(a,b,c,d));
}
template<typename T>
inline quaternion<T> cylindrospherical( T const & t,
T const & radius,
T const & longitude,
T const & latitude)
{
using ::std::cos;
using ::std::sin;
T b = radius*cos(longitude)*cos(latitude);
T c = radius*sin(longitude)*cos(latitude);
T d = radius*sin(latitude);
return(quaternion<T>(t,b,c,d));
}
template<typename T>
inline quaternion<T> cylindrical(T const & r,
T const & angle,
T const & h1,
T const & h2)
{
using ::std::cos;
using ::std::sin;
T a = r*cos(angle);
T b = r*sin(angle);
return(quaternion<T>(a,b,h1,h2));
}
// transcendentals
// (please see the documentation)
template<typename T>
inline quaternion<T> exp(quaternion<T> const & q)
{
using ::std::exp;
using ::std::cos;
using ::boost::math::sinc_pi;
T u = exp(real(q));
T z = abs(unreal(q));
T w = sinc_pi(z);
return(u*quaternion<T>(cos(z),
w*q.R_component_2(), w*q.R_component_3(),
w*q.R_component_4()));
}
template<typename T>
inline quaternion<T> cos(quaternion<T> const & q)
{
using ::std::sin;
using ::std::cos;
using ::std::cosh;
using ::boost::math::sinhc_pi;
T z = abs(unreal(q));
T w = -sin(q.real())*sinhc_pi(z);
return(quaternion<T>(cos(q.real())*cosh(z),
w*q.R_component_2(), w*q.R_component_3(),
w*q.R_component_4()));
}
template<typename T>
inline quaternion<T> sin(quaternion<T> const & q)
{
using ::std::sin;
using ::std::cos;
using ::std::cosh;
using ::boost::math::sinhc_pi;
T z = abs(unreal(q));
T w = +cos(q.real())*sinhc_pi(z);
return(quaternion<T>(sin(q.real())*cosh(z),
w*q.R_component_2(), w*q.R_component_3(),
w*q.R_component_4()));
}
template<typename T>
inline quaternion<T> tan(quaternion<T> const & q)
{
return(sin(q)/cos(q));
}
template<typename T>
inline quaternion<T> cosh(quaternion<T> const & q)
{
return((exp(+q)+exp(-q))/static_cast<T>(2));
}
template<typename T>
inline quaternion<T> sinh(quaternion<T> const & q)
{
return((exp(+q)-exp(-q))/static_cast<T>(2));
}
template<typename T>
inline quaternion<T> tanh(quaternion<T> const & q)
{
return(sinh(q)/cosh(q));
}
template<typename T>
quaternion<T> pow(quaternion<T> const & q,
int n)
{
if (n > 1)
{
int m = n>>1;
quaternion<T> result = pow(q, m);
result *= result;
if (n != (m<<1))
{
result *= q; // n odd
}
return(result);
}
else if (n == 1)
{
return(q);
}
else if (n == 0)
{
return(quaternion<T>(static_cast<T>(1)));
}
else /* n < 0 */
{
return(pow(quaternion<T>(static_cast<T>(1))/q,-n));
}
}
// helper templates for converting copy constructors (definition)
namespace detail
{
template< typename T,
typename U
>
quaternion<T> quaternion_type_converter(quaternion<U> const & rhs)
{
return(quaternion<T>( static_cast<T>(rhs.R_component_1()),
static_cast<T>(rhs.R_component_2()),
static_cast<T>(rhs.R_component_3()),
static_cast<T>(rhs.R_component_4())));
}
}
}
}
#endif /* BOOST_QUATERNION_HPP */