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 <boost/config.hpp> // for BOOST_NO_STD_LOCALE
#include <boost/math_fwd.hpp>
#include <boost/detail/workaround.hpp>
#include <boost/type_traits/is_convertible.hpp>
#include <boost/utility/enable_if.hpp>
#ifndef BOOST_NO_STD_LOCALE
# include <locale> // for the "<<" operator
#endif /* BOOST_NO_STD_LOCALE */
#include <complex>
#include <iosfwd> // for the "<<" and ">>" operators
#include <sstream> // for the "<<" operator
#include <boost/math/special_functions/sinc.hpp> // for the Sinus cardinal
#include <boost/math/special_functions/sinhc.hpp> // for the Hyperbolic Sinus cardinal
#if defined(BOOST_NO_CXX11_NOEXCEPT) || defined(BOOST_NO_CXX11_RVALUE_REFERENCES) || defined(BOOST_NO_SFINAE_EXPR)
#include <boost/type_traits/is_pod.hpp>
#endif
namespace boost
{
namespace math
{
namespace detail {
#if !defined(BOOST_NO_CXX11_NOEXCEPT) && !defined(BOOST_NO_CXX11_RVALUE_REFERENCES) && !defined(BOOST_NO_SFINAE_EXPR)
template <class T>
struct is_trivial_arithmetic_type_imp
{
typedef mpl::bool_<
noexcept(std::declval<T&>() += std::declval<T>())
&& noexcept(std::declval<T&>() -= std::declval<T>())
&& noexcept(std::declval<T&>() *= std::declval<T>())
&& noexcept(std::declval<T&>() /= std::declval<T>())
> type;
};
template <class T>
struct is_trivial_arithmetic_type : public is_trivial_arithmetic_type_imp<T>::type {};
#else
template <class T>
struct is_trivial_arithmetic_type : public boost::is_pod<T> {};
#endif
}
#ifndef BOOST_NO_CXX14_CONSTEXPR
namespace constexpr_detail
{
template <class T>
constexpr void swap(T& a, T& b)
{
T t(a);
a = b;
b = t;
}
}
#endif
template<typename T>
class quaternion
{
public:
typedef T value_type;
// constructor for H seen as R^4
// (also default constructor)
BOOST_CONSTEXPR 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
BOOST_CONSTEXPR 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
BOOST_CONSTEXPR quaternion(quaternion const & a_recopier)
: a(a_recopier.R_component_1()),
b(a_recopier.R_component_2()),
c(a_recopier.R_component_3()),
d(a_recopier.R_component_4()) {}
#ifndef BOOST_NO_CXX11_RVALUE_REFERENCES
BOOST_CONSTEXPR quaternion(quaternion && a_recopier)
: a(std::move(a_recopier.R_component_1())),
b(std::move(a_recopier.R_component_2())),
c(std::move(a_recopier.R_component_3())),
d(std::move(a_recopier.R_component_4())) {}
#endif
// templated copy constructor
template<typename X>
BOOST_CONSTEXPR 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_CONSTEXPR T real() const
{
return(a);
}
BOOST_CONSTEXPR quaternion<T> unreal() const
{
return(quaternion<T>(static_cast<T>(0), b, c, d));
}
BOOST_CONSTEXPR T R_component_1() const
{
return(a);
}
BOOST_CONSTEXPR T R_component_2() const
{
return(b);
}
BOOST_CONSTEXPR T R_component_3() const
{
return(c);
}
BOOST_CONSTEXPR T R_component_4() const
{
return(d);
}
BOOST_CONSTEXPR ::std::complex<T> C_component_1() const
{
return(::std::complex<T>(a, b));
}
BOOST_CONSTEXPR ::std::complex<T> C_component_2() const
{
return(::std::complex<T>(c, d));
}
BOOST_CXX14_CONSTEXPR void swap(quaternion& o)
{
#ifndef BOOST_NO_CXX14_CONSTEXPR
using constexpr_detail::swap;
#else
using std::swap;
#endif
swap(a, o.a);
swap(b, o.b);
swap(c, o.c);
swap(d, o.d);
}
// assignment operators
template<typename X>
BOOST_CXX14_CONSTEXPR quaternion<T> & operator = (quaternion<X> const & a_affecter)
{
a = static_cast<T>(a_affecter.R_component_1());
b = static_cast<T>(a_affecter.R_component_2());
c = static_cast<T>(a_affecter.R_component_3());
d = static_cast<T>(a_affecter.R_component_4());
return(*this);
}
BOOST_CXX14_CONSTEXPR quaternion<T> & operator = (quaternion<T> const & a_affecter)
{
a = a_affecter.a;
b = a_affecter.b;
c = a_affecter.c;
d = a_affecter.d;
return(*this);
}
#ifndef BOOST_NO_CXX11_RVALUE_REFERENCES
BOOST_CXX14_CONSTEXPR quaternion<T> & operator = (quaternion<T> && a_affecter)
{
a = std::move(a_affecter.a);
b = std::move(a_affecter.b);
c = std::move(a_affecter.c);
d = std::move(a_affecter.d);
return(*this);
}
#endif
BOOST_CXX14_CONSTEXPR quaternion<T> & operator = (T const & a_affecter)
{
a = a_affecter;
b = c = d = static_cast<T>(0);
return(*this);
}
BOOST_CXX14_CONSTEXPR quaternion<T> & operator = (::std::complex<T> const & a_affecter)
{
a = a_affecter.real();
b = a_affecter.imag();
c = d = static_cast<T>(0);
return(*this);
}
// 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);"
//
// Note2: Each operator comes in 2 forms - one for the simple case where
// type T throws no exceptions, and one exception-safe version
// for the case where it might.
private:
BOOST_CXX14_CONSTEXPR quaternion<T> & do_add(T const & rhs, const mpl::true_&)
{
a += rhs;
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_add(T const & rhs, const mpl::false_&)
{
quaternion<T> result(a + rhs, b, c, d); // exception guard
swap(result);
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_add(std::complex<T> const & rhs, const mpl::true_&)
{
a += std::real(rhs);
b += std::imag(rhs);
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_add(std::complex<T> const & rhs, const mpl::false_&)
{
quaternion<T> result(a + std::real(rhs), b + std::imag(rhs), c, d); // exception guard
swap(result);
return *this;
}
template <class X>
BOOST_CXX14_CONSTEXPR quaternion<T> & do_add(quaternion<X> const & rhs, const mpl::true_&)
{
a += rhs.R_component_1();
b += rhs.R_component_2();
c += rhs.R_component_3();
d += rhs.R_component_4();
return *this;
}
template <class X>
BOOST_CXX14_CONSTEXPR quaternion<T> & do_add(quaternion<X> const & rhs, const mpl::false_&)
{
quaternion<T> result(a + rhs.R_component_1(), b + rhs.R_component_2(), c + rhs.R_component_3(), d + rhs.R_component_4()); // exception guard
swap(result);
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_subtract(T const & rhs, const mpl::true_&)
{
a -= rhs;
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_subtract(T const & rhs, const mpl::false_&)
{
quaternion<T> result(a - rhs, b, c, d); // exception guard
swap(result);
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_subtract(std::complex<T> const & rhs, const mpl::true_&)
{
a -= std::real(rhs);
b -= std::imag(rhs);
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_subtract(std::complex<T> const & rhs, const mpl::false_&)
{
quaternion<T> result(a - std::real(rhs), b - std::imag(rhs), c, d); // exception guard
swap(result);
return *this;
}
template <class X>
BOOST_CXX14_CONSTEXPR quaternion<T> & do_subtract(quaternion<X> const & rhs, const mpl::true_&)
{
a -= rhs.R_component_1();
b -= rhs.R_component_2();
c -= rhs.R_component_3();
d -= rhs.R_component_4();
return *this;
}
template <class X>
BOOST_CXX14_CONSTEXPR quaternion<T> & do_subtract(quaternion<X> const & rhs, const mpl::false_&)
{
quaternion<T> result(a - rhs.R_component_1(), b - rhs.R_component_2(), c - rhs.R_component_3(), d - rhs.R_component_4()); // exception guard
swap(result);
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_multiply(T const & rhs, const mpl::true_&)
{
a *= rhs;
b *= rhs;
c *= rhs;
d *= rhs;
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_multiply(T const & rhs, const mpl::false_&)
{
quaternion<T> result(a * rhs, b * rhs, c * rhs, d * rhs); // exception guard
swap(result);
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_divide(T const & rhs, const mpl::true_&)
{
a /= rhs;
b /= rhs;
c /= rhs;
d /= rhs;
return *this;
}
BOOST_CXX14_CONSTEXPR quaternion<T> & do_divide(T const & rhs, const mpl::false_&)
{
quaternion<T> result(a / rhs, b / rhs, c / rhs, d / rhs); // exception guard
swap(result);
return *this;
}
public:
BOOST_CXX14_CONSTEXPR quaternion<T> & operator += (T const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
BOOST_CXX14_CONSTEXPR quaternion<T> & operator += (::std::complex<T> const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
template<typename X> BOOST_CXX14_CONSTEXPR quaternion<T> & operator += (quaternion<X> const & rhs) { return do_add(rhs, detail::is_trivial_arithmetic_type<T>()); }
BOOST_CXX14_CONSTEXPR quaternion<T> & operator -= (T const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
BOOST_CXX14_CONSTEXPR quaternion<T> & operator -= (::std::complex<T> const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
template<typename X> BOOST_CXX14_CONSTEXPR quaternion<T> & operator -= (quaternion<X> const & rhs) { return do_subtract(rhs, detail::is_trivial_arithmetic_type<T>()); }
BOOST_CXX14_CONSTEXPR quaternion<T> & operator *= (T const & rhs) { return do_multiply(rhs, detail::is_trivial_arithmetic_type<T>()); }
BOOST_CXX14_CONSTEXPR quaternion<T> & operator *= (::std::complex<T> const & rhs)
{
T ar = rhs.real();
T br = rhs.imag();
quaternion<T> result(a*ar - b*br, a*br + b*ar, c*ar + d*br, -c*br+d*ar);
swap(result);
return(*this);
}
template<typename X>
BOOST_CXX14_CONSTEXPR 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());
quaternion<T> result(a*ar - b*br - c*cr - d*dr, a*br + b*ar + c*dr - d*cr, a*cr - b*dr + c*ar + d*br, a*dr + b*cr - c*br + d*ar);
swap(result);
return(*this);
}
BOOST_CXX14_CONSTEXPR quaternion<T> & operator /= (T const & rhs) { return do_divide(rhs, detail::is_trivial_arithmetic_type<T>()); }
BOOST_CXX14_CONSTEXPR quaternion<T> & operator /= (::std::complex<T> const & rhs)
{
T ar = rhs.real();
T br = rhs.imag();
T denominator = ar*ar+br*br;
quaternion<T> result((+a*ar + b*br) / denominator, (-a*br + b*ar) / denominator, (+c*ar - d*br) / denominator, (+c*br + d*ar) / denominator);
swap(result);
return(*this);
}
template<typename X>
BOOST_CXX14_CONSTEXPR 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;
quaternion<T> result((+a*ar+b*br+c*cr+d*dr)/denominator, (-a*br+b*ar-c*dr+d*cr)/denominator, (-a*cr+b*dr+c*ar-d*br)/denominator, (-a*dr-b*cr+c*br+d*ar)/denominator);
swap(result);
return(*this);
}
private:
T a, b, c, d;
};
// swap:
template <class T>
BOOST_CXX14_CONSTEXPR void swap(quaternion<T>& a, quaternion<T>& b) { a.swap(b); }
// operator+
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator + (const quaternion<T1>& a, const T2& b)
{
return quaternion<T1>(static_cast<T1>(a.R_component_1() + b), a.R_component_2(), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator + (const T1& a, const quaternion<T2>& b)
{
return quaternion<T2>(static_cast<T2>(b.R_component_1() + a), b.R_component_2(), b.R_component_3(), b.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator + (const quaternion<T1>& a, const std::complex<T2>& b)
{
return quaternion<T1>(a.R_component_1() + std::real(b), a.R_component_2() + std::imag(b), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator + (const std::complex<T1>& a, const quaternion<T2>& b)
{
return quaternion<T1>(b.R_component_1() + real(a), b.R_component_2() + imag(a), b.R_component_3(), b.R_component_4());
}
template <class T>
inline BOOST_CONSTEXPR quaternion<T> operator + (const quaternion<T>& a, const quaternion<T>& b)
{
return quaternion<T>(a.R_component_1() + b.R_component_1(), a.R_component_2() + b.R_component_2(), a.R_component_3() + b.R_component_3(), a.R_component_4() + b.R_component_4());
}
// operator-
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator - (const quaternion<T1>& a, const T2& b)
{
return quaternion<T1>(static_cast<T1>(a.R_component_1() - b), a.R_component_2(), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator - (const T1& a, const quaternion<T2>& b)
{
return quaternion<T2>(static_cast<T2>(a - b.R_component_1()), -b.R_component_2(), -b.R_component_3(), -b.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator - (const quaternion<T1>& a, const std::complex<T2>& b)
{
return quaternion<T1>(a.R_component_1() - std::real(b), a.R_component_2() - std::imag(b), a.R_component_3(), a.R_component_4());
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator - (const std::complex<T1>& a, const quaternion<T2>& b)
{
return quaternion<T1>(real(a) - b.R_component_1(), imag(a) - b.R_component_2(), -b.R_component_3(), -b.R_component_4());
}
template <class T>
inline BOOST_CONSTEXPR quaternion<T> operator - (const quaternion<T>& a, const quaternion<T>& b)
{
return quaternion<T>(a.R_component_1() - b.R_component_1(), a.R_component_2() - b.R_component_2(), a.R_component_3() - b.R_component_3(), a.R_component_4() - b.R_component_4());
}
// operator*
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator * (const quaternion<T1>& a, const T2& b)
{
return quaternion<T1>(static_cast<T1>(a.R_component_1() * b), a.R_component_2() * b, a.R_component_3() * b, a.R_component_4() * b);
}
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator * (const T1& a, const quaternion<T2>& b)
{
return quaternion<T2>(static_cast<T2>(a * b.R_component_1()), a * b.R_component_2(), a * b.R_component_3(), a * b.R_component_4());
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator * (const quaternion<T1>& a, const std::complex<T2>& b)
{
quaternion<T1> result(a);
result *= b;
return result;
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator * (const std::complex<T1>& a, const quaternion<T2>& b)
{
quaternion<T1> result(a);
result *= b;
return result;
}
template <class T>
inline BOOST_CXX14_CONSTEXPR quaternion<T> operator * (const quaternion<T>& a, const quaternion<T>& b)
{
quaternion<T> result(a);
result *= b;
return result;
}
// operator/
template <class T1, class T2>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator / (const quaternion<T1>& a, const T2& b)
{
return quaternion<T1>(a.R_component_1() / b, a.R_component_2() / b, a.R_component_3() / b, a.R_component_4() / b);
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator / (const T1& a, const quaternion<T2>& b)
{
quaternion<T2> result(a);
result /= b;
return result;
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T2, T1>::value, quaternion<T1> >::type
operator / (const quaternion<T1>& a, const std::complex<T2>& b)
{
quaternion<T1> result(a);
result /= b;
return result;
}
template <class T1, class T2>
inline BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<T1, T2>::value, quaternion<T2> >::type
operator / (const std::complex<T1>& a, const quaternion<T2>& b)
{
quaternion<T2> result(a);
result /= b;
return result;
}
template <class T>
inline BOOST_CXX14_CONSTEXPR quaternion<T> operator / (const quaternion<T>& a, const quaternion<T>& b)
{
quaternion<T> result(a);
result /= b;
return result;
}
template<typename T>
inline BOOST_CONSTEXPR const quaternion<T>& operator + (quaternion<T> const & q)
{
return q;
}
template<typename T>
inline BOOST_CONSTEXPR 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 R, typename T>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<R, T>::value, bool>::type operator == (R 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, typename R>
inline BOOST_CONSTEXPR typename boost::enable_if_c<boost::is_convertible<R, T>::value, bool>::type operator == (quaternion<T> const & lhs, R const & rhs)
{
return rhs == lhs;
}
template<typename T>
inline BOOST_CONSTEXPR 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 BOOST_CONSTEXPR bool operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs)
{
return rhs == lhs;
}
template<typename T>
inline BOOST_CONSTEXPR 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())
);
}
template<typename R, typename T> inline BOOST_CONSTEXPR bool operator != (R const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
template<typename T, typename R> inline BOOST_CONSTEXPR bool operator != (quaternion<T> const & lhs, R const & rhs) { return !(lhs == rhs); }
template<typename T> inline BOOST_CONSTEXPR bool operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
template<typename T> inline BOOST_CONSTEXPR bool operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs) { return !(lhs == rhs); }
template<typename T> inline BOOST_CONSTEXPR bool operator != (quaternion<T> const & lhs, quaternion<T> const & rhs) { return !(lhs == rhs); }
// 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 BOOST_CONSTEXPR T real(quaternion<T> const & q)
{
return(q.real());
}
template<typename T>
inline BOOST_CONSTEXPR quaternion<T> unreal(quaternion<T> const & q)
{
return(q.unreal());
}
template<typename T>
inline T sup(quaternion<T> const & q)
{
using ::std::abs;
return (std::max)((std::max)(abs(q.R_component_1()), abs(q.R_component_2())), (std::max)(abs(q.R_component_3()), abs(q.R_component_4())));
}
template<typename T>
inline T l1(quaternion<T> const & q)
{
using ::std::abs;
return abs(q.R_component_1()) + abs(q.R_component_2()) + abs(q.R_component_3()) + abs(q.R_component_4());
}
template<typename T>
inline T abs(quaternion<T> const & q)
{
using ::std::abs;
using ::std::sqrt;
T maxim = sup(q); // overflow protection
if (maxim == static_cast<T>(0))
{
return(maxim);
}
else
{
T mixam = static_cast<T>(1)/maxim; // prefer multiplications over divisions
T a = q.R_component_1() * mixam;
T b = q.R_component_2() * mixam;
T c = q.R_component_3() * mixam;
T d = q.R_component_4() * mixam;
a *= a;
b *= b;
c *= c;
d *= d;
return(maxim * sqrt(a + b + c + d));
}
//return(sqrt(norm(q)));
}
// Note: This is the Cayley norm, not the Euclidian norm...
template<typename T>
inline BOOST_CXX14_CONSTEXPR T norm(quaternion<T>const & q)
{
return(real(q*conj(q)));
}
template<typename T>
inline BOOST_CONSTEXPR 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));
}
}
}
}
#endif /* BOOST_QUATERNION_HPP */