| [article Boost.Octonions |
| [quickbook 1.3] |
| [copyright 2001-2003 Hubert Holin] |
| [purpose Octonions] |
| [license |
| 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 http://www.boost.org/LICENSE_1_0.txt]) |
| ] |
| [authors [Holin, Hubert]] |
| [category math] |
| [last-revision $Date: 2006-12-29 11:08:32 +0000 (Fri, 29 Dec 2006) $] |
| ] |
| |
| [def __R ['[*R]]] |
| [def __C ['[*C]]] |
| [def __H ['[*H]]] |
| [def __O ['[*O]]] |
| [def __R3 ['[*'''R<superscript>3</superscript>''']]] |
| [def __R4 ['[*'''R<superscript>4</superscript>''']]] |
| [def __octulple ('''α,β,γ,δ,ε,ζ,η,θ''')] |
| [def __oct_formula ['[^o = '''α + βi + γj + δk + εe' + ζi' + ηj' + θk' ''']]] |
| [def __oct_complex_formula ['[^o = ('''α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j' ''']]] |
| [def __oct_quat_formula ['[^o = ('''α + βi + γj + δk) + (ε + ζi + ηj - θj)e' ''']]] |
| [def __oct_not_equal ['[^x(yz) '''≠''' (xy)z]]] |
| |
| [def __asinh [link boost_quaternions.math_special_functions.asinh asinh]] |
| [def __acosh [link boost_quaternions.math_special_functions.acosh acosh]] |
| [def __atanh [link boost_quaternions.math_special_functions.atanh atanh]] |
| [def __sinc_pi [link boost_quaternions.math_special_functions.sinc_pi sinc_pi]] |
| [def __sinhc_pi [link boost_quaternions.math_special_functions.sinhc_pi sinhc_pi]] |
| |
| [def __log1p [link boost_quaternions.math_special_functions.log1p log1p]] |
| [def __expm1 [link boost_quaternions.math_special_functions.expm1 expm1]] |
| [def __hypot [link boost_quaternions.math_special_functions.hypot hypot]] |
| |
| This manual is also available in |
| [@http://sourceforge.net/projects/boost/files/boost-docs/ |
| printer friendly PDF format]. |
| |
| [section Octonions] |
| |
| [section Overview] |
| |
| Octonions, like [@../../quaternion/html/index.html quaternions], are a relative of complex numbers. |
| |
| Octonions see some use in theoretical physics. |
| |
| In practical terms, an octonion is simply an octuple of real numbers __octulple, |
| which we can write in the form __oct_formula, where ['[^i]], ['[^j]] and ['[^k]] |
| are the same objects as for quaternions, and ['[^e']], ['[^i']], ['[^j']] and ['[^k']] |
| are distinct objects which play essentially the same kind of role as ['[^i]] (or ['[^j]] or ['[^k]]). |
| |
| Addition and a multiplication is defined on the set of octonions, |
| which generalize their quaternionic counterparts. The main novelty this time |
| is that [*the multiplication is not only not commutative, is now not even |
| associative] (i.e. there are octonions ['[^x]], ['[^y]] and ['[^z]] such that __oct_not_equal). |
| A way of remembering things is by using the following multiplication table: |
| |
| [$../../../octonion/graphics/octonion_blurb17.jpeg] |
| |
| Octonions (and their kin) are described in far more details in this other |
| [@../../../quaternion/TQE.pdf document] |
| (with [@../../../quaternion/TQE_EA.pdf errata and addenda]). |
| |
| Some traditional constructs, such as the exponential, carry over without too |
| much change into the realms of octonions, but other, such as taking a square root, |
| do not (the fact that the exponential has a closed form is a result of the |
| author, but the fact that the exponential exists at all for octonions is known |
| since quite a long time ago). |
| |
| [endsect] |
| |
| [section Header File] |
| |
| The interface and implementation are both supplied by the header file |
| [@../../../../../boost/math/octonion.hpp octonion.hpp]. |
| |
| [endsect] |
| |
| [section Synopsis] |
| |
| namespace boost{ namespace math{ |
| |
| template<typename T> class ``[link boost_octonions.octonions.template_class_octonion octonion]``; |
| template<> class ``[link boost_octonions.octonions.octonion_specializations octonion<float>]``; |
| template<> class ``[link boost_octonions.octonion_double octonion<double>]``; |
| template<> class ``[link boost_octonions.octonion_long_double octonion<long double>]``; |
| |
| // operators |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.unary_plus_and_minus_operators operator +]`` (octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.unary_plus_and_minus_operators operator -]`` (octonion<T> const & o); |
| |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (T const & lhs, octonion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, T const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (T const & lhs, octonion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, T const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| template<typename T, typename charT, class traits> |
| ::std::basic_istream<charT,traits> & ``[link boost_octonions.octonions.non_mem.stream_extractor operator >>]`` (::std::basic_istream<charT,traits> & is, octonion<T> & o); |
| |
| template<typename T, typename charT, class traits> |
| ::std::basic_ostream<charT,traits> & ``[link boost_octonions.octonions.non_mem.stream_inserter operator <<]`` (::std::basic_ostream<charT,traits> & os, octonion<T> const & o); |
| |
| // values |
| |
| template<typename T> T ``[link boost_octonions.octonions.octonion_value_operations.real_and_unreal real]``(octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonion_value_operations.real_and_unreal unreal]``(octonion<T> const & o); |
| |
| template<typename T> T ``[link boost_octonions.octonions.octonion_value_operations.sup sup]``(octonion<T> const & o); |
| template<typename T> T ``[link boost_octonions.octonions.octonion_value_operations.l1 l1]``(octonion<T>const & o); |
| template<typename T> T ``[link boost_octonions.octonions.octonion_value_operations.abs abs]``(octonion<T> const & o); |
| template<typename T> T ``[link boost_octonions.octonions.octonion_value_operations.norm norm]``(octonion<T>const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonion_value_operations.conj conj]``(octonion<T> const & o); |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.oct_create spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.oct_create multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.oct_create cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6); |
| |
| // transcendentals |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.exp exp]``(octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.cos cos]``(octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.sin sin]``(octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.tan tan]``(octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.cosh cosh]``(octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.sinh sinh]``(octonion<T> const & o); |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.tanh tanh]``(octonion<T> const & o); |
| |
| template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.pow pow]``(octonion<T> const & o, int n); |
| |
| } } // namespaces |
| |
| [endsect] |
| |
| [section Template Class octonion] |
| |
| namespace boost{ namespace math { |
| |
| template<typename T> |
| class octonion |
| { |
| public: |
| typedef T value_type; |
| |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>(), ::std::complex<T> const & z2 = ::std::complex<T>(), ::std::complex<T> const & z3 = ::std::complex<T>()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>()); |
| template<typename X> |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<X> const & a_recopier); |
| |
| T ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; |
| octonion<T> ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; |
| |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; |
| T ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; |
| |
| ::std::complex<T> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; |
| ::std::complex<T> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; |
| ::std::complex<T> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; |
| ::std::complex<T> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; |
| |
| ::boost::math::quaternion<T> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; |
| ::boost::math::quaternion<T> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; |
| |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<T> const & a_affecter); |
| template<typename X> |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const & a_affecter); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (T const & a_affecter); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<T> const & a_affecter); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<T> const & a_affecter); |
| |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (T const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<T> const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs); |
| |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (T const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<T> const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs); |
| |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (T const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<T> const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs); |
| |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (T const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<T> const & rhs); |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs); |
| }; |
| |
| } } // namespaces |
| |
| [endsect] |
| |
| [section Octonion Specializations] |
| |
| namespace boost{ namespace math{ |
| |
| template<> |
| class octonion<float> |
| { |
| public: |
| typedef float value_type; |
| |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>(), ::std::complex<float> const & z2 = ::std::complex<float>(), ::std::complex<float> const & z3 = ::std::complex<float>()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion<float> const & q0, ::boost::math::quaternion<float> const & q1 = ::boost::math::quaternion<float>()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<double> const & a_recopier); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<long double> const & a_recopier); |
| |
| float ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; |
| octonion<float> ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; |
| |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; |
| float ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; |
| |
| ::std::complex<float> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; |
| ::std::complex<float> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; |
| ::std::complex<float> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; |
| ::std::complex<float> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; |
| |
| ::boost::math::quaternion<float> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; |
| ::boost::math::quaternion<float> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; |
| |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<float> const & a_affecter); |
| template<typename X> |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const & a_affecter); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (float const & a_affecter); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<float> const & a_affecter); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<float> const & a_affecter); |
| |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (float const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<float> const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<float> const & rhs); |
| template<typename X> |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs); |
| |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (float const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<float> const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<float> const & rhs); |
| template<typename X> |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs); |
| |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (float const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<float> const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<float> const & rhs); |
| template<typename X> |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs); |
| |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (float const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<float> const & rhs); |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<float> const & rhs); |
| template<typename X> |
| octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs); |
| }; |
| |
| [#boost_octonions.octonion_double] |
| |
| template<> |
| class octonion<double> |
| { |
| public: |
| typedef double value_type; |
| |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>(), ::std::complex<double> const & z2 = ::std::complex<double>(), ::std::complex<double> const & z3 = ::std::complex<double>()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion<double> const & q0, ::boost::math::quaternion<double> const & q1 = ::boost::math::quaternion<double>()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<float> const & a_recopier); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<long double> const & a_recopier); |
| |
| double ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; |
| octonion<double> ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; |
| |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; |
| double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; |
| |
| ::std::complex<double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; |
| ::std::complex<double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; |
| ::std::complex<double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; |
| ::std::complex<double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; |
| |
| ::boost::math::quaternion<double> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; |
| ::boost::math::quaternion<double> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; |
| |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<double> const & a_affecter); |
| template<typename X> |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const & a_affecter); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (double const & a_affecter); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<double> const & a_affecter); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<double> const & a_affecter); |
| |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (double const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<double> const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<double> const & rhs); |
| template<typename X> |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs); |
| |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (double const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<double> const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<double> const & rhs); |
| template<typename X> |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs); |
| |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (double const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<double> const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<double> const & rhs); |
| template<typename X> |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs); |
| |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (double const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<double> const & rhs); |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<double> const & rhs); |
| template<typename X> |
| octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs); |
| }; |
| |
| [#boost_octonions.octonion_long_double] |
| |
| template<> |
| class octonion<long double> |
| { |
| public: |
| typedef long double value_type; |
| |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``( ::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>(), ::std::complex<long double> const & z2 = ::std::complex<long double>(), ::std::complex<long double> const & z3 = ::std::complex<long double>()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``( ::boost::math::quaternion<long double> const & q0, ::boost::math::quaternion<long double> const & z1 = ::boost::math::quaternion<long double>()); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<float> const & a_recopier); |
| explicit ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<double> const & a_recopier); |
| |
| long double ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const; |
| octonion<long double> ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const; |
| |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const; |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const; |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const; |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const; |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const; |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const; |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const; |
| long double ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const; |
| |
| ::std::complex<long double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const; |
| ::std::complex<long double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const; |
| ::std::complex<long double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const; |
| ::std::complex<long double> ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const; |
| |
| ::boost::math::quaternion<long double> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const; |
| ::boost::math::quaternion<long double> ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const; |
| |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<long double> const & a_affecter); |
| template<typename X> |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const & a_affecter); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (long double const & a_affecter); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<long double> const & a_affecter); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<long double> const & a_affecter); |
| |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (long double const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<long double> const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<long double> const & rhs); |
| template<typename X> |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs); |
| |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (long double const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<long double> const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<long double> const & rhs); |
| template<typename X> |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs); |
| |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (long double const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<long double> const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<long double> const & rhs); |
| template<typename X> |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs); |
| |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (long double const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<long double> const & rhs); |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<long double> const & rhs); |
| template<typename X> |
| octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs); |
| }; |
| |
| } } // namespaces |
| |
| [endsect] |
| |
| [section Octonion Member Typedefs] |
| |
| [*value_type] |
| |
| Template version: |
| |
| typedef T value_type; |
| |
| Float specialization version: |
| |
| typedef float value_type; |
| |
| Double specialization version: |
| |
| typedef double value_type; |
| |
| Long double specialization version: |
| |
| typedef long double value_type; |
| |
| These provide easy acces to the type the template is built upon. |
| |
| [endsect] |
| |
| [section Octonion Member Functions] |
| |
| [h3 Constructors] |
| |
| Template version: |
| |
| explicit octonion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T()); |
| explicit octonion(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>(), ::std::complex<T> const & z2 = ::std::complex<T>(), ::std::complex<T> const & z3 = ::std::complex<T>()); |
| explicit octonion(::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>()); |
| template<typename X> |
| explicit octonion(octonion<X> const & a_recopier); |
| |
| Float specialization version: |
| |
| explicit octonion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f); |
| explicit octonion(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>(), ::std::complex<float> const & z2 = ::std::complex<float>(), ::std::complex<float> const & z3 = ::std::complex<float>()); |
| explicit octonion(::boost::math::quaternion<float> const & q0, ::boost::math::quaternion<float> const & q1 = ::boost::math::quaternion<float>()); |
| explicit octonion(octonion<double> const & a_recopier); |
| explicit octonion(octonion<long double> const & a_recopier); |
| |
| Double specialization version: |
| |
| explicit octonion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0); |
| explicit octonion(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>(), ::std::complex<double> const & z2 = ::std::complex<double>(), ::std::complex<double> const & z3 = ::std::complex<double>()); |
| explicit octonion(::boost::math::quaternion<double> const & q0, ::boost::math::quaternion<double> const & q1 = ::boost::math::quaternion<double>()); |
| explicit octonion(octonion<float> const & a_recopier); |
| explicit octonion(octonion<long double> const & a_recopier); |
| |
| Long double specialization version: |
| |
| explicit octonion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L); |
| explicit octonion( ::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>(), ::std::complex<long double> const & z2 = ::std::complex<long double>(), ::std::complex<long double> const & z3 = ::std::complex<long double>()); |
| explicit octonion(::boost::math::quaternion<long double> const & q0, ::boost::math::quaternion<long double> const & q1 = ::boost::math::quaternion<long double>()); |
| explicit octonion(octonion<float> const & a_recopier); |
| explicit octonion(octonion<double> const & a_recopier); |
| |
| A default constructor is provided for each form, which initializes each component |
| to the default values for their type (i.e. zero for floating numbers). |
| This constructor can also accept one to eight base type arguments. |
| A constructor is also provided to build octonions from one to four complex numbers |
| sharing the same base type, and another taking one or two quaternions |
| sharing the same base type. The unspecialized template also sports a |
| templarized copy constructor, while the specialized forms have copy |
| constructors from the other two specializations, which are explicit |
| when a risk of precision loss exists. For the unspecialized form, |
| the base type's constructors must not throw. |
| |
| Destructors and untemplated copy constructors (from the same type) |
| are provided by the compiler. Converting copy constructors make use |
| of a templated helper function in a "detail" subnamespace. |
| |
| [h3 Other member functions] |
| |
| [h4 Real and Unreal Parts] |
| |
| T real() const; |
| octonion<T> unreal() const; |
| |
| Like complex number, octonions 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 octonion, |
| and usually nothing simpler (as opposed to the complex number case). |
| These are returned by the first two functions. |
| |
| [h4 Individual Real Components] |
| |
| T R_component_1() const; |
| T R_component_2() const; |
| T R_component_3() const; |
| T R_component_4() const; |
| T R_component_5() const; |
| T R_component_6() const; |
| T R_component_7() const; |
| T R_component_8() const; |
| |
| A octonion having eight real components, these are returned by |
| these eight functions. Hence real and R_component_1 return the same value. |
| |
| [h4 Individual Complex Components] |
| |
| ::std::complex<T> C_component_1() const; |
| ::std::complex<T> C_component_2() const; |
| ::std::complex<T> C_component_3() const; |
| ::std::complex<T> C_component_4() const; |
| |
| A octonion likewise has four complex components. Actually, octonions |
| are indeed a (left) vector field over the complexes, but beware, as |
| for any octonion __oct_formula we also have __oct_complex_formula |
| (note the [*minus] sign in the last factor). |
| What the C_component_n functions return, however, are the complexes |
| which could be used to build the octonion using the constructor, and |
| [*not] the components of the octonion on the basis ['[^(1, j, e', j')]]. |
| |
| [h4 Individual Quaternion Components] |
| |
| ::boost::math::quaternion<T> H_component_1() const; |
| ::boost::math::quaternion<T> H_component_2() const; |
| |
| Likewise, for any octonion __oct_formula we also have __oct_quat_formula, though there |
| is no meaningful vector-space-like structure based on the quaternions. |
| What the H_component_n functions return are the quaternions which |
| could be used to build the octonion using the constructor. |
| |
| [h3 Octonion Member Operators] |
| [h4 Assignment Operators] |
| |
| octonion<T> & operator = (octonion<T> const & a_affecter); |
| template<typename X> |
| octonion<T> & operator = (octonion<X> const & a_affecter); |
| octonion<T> & operator = (T const & a_affecter); |
| octonion<T> & operator = (::std::complex<T> const & a_affecter); |
| octonion<T> & operator = (::boost::math::quaternion<T> const & a_affecter); |
| |
| These perform the expected assignment, with type modification if |
| necessary (for instance, assigning from a base type will set the |
| real part to that value, and all other components to zero). |
| For the unspecialized form, the base type's assignment operators must not throw. |
| |
| [h4 Other Member Operators] |
| |
| octonion<T> & operator += (T const & rhs) |
| octonion<T> & operator += (::std::complex<T> const & rhs); |
| octonion<T> & operator += (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & operator += (octonion<X> const & rhs); |
| |
| These perform the mathematical operation `(*this)+rhs` and store the result in |
| `*this`. The unspecialized form has exception guards, which the specialized |
| forms do not, so as to insure exception safety. For the unspecialized form, |
| the base type's assignment operators must not throw. |
| |
| octonion<T> & operator -= (T const & rhs) |
| octonion<T> & operator -= (::std::complex<T> const & rhs); |
| octonion<T> & operator -= (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & operator -= (octonion<X> const & rhs); |
| |
| These perform the mathematical operation `(*this)-rhs` and store the result |
| in `*this`. The unspecialized form has exception guards, which the |
| specialized forms do not, so as to insure exception safety. |
| For the unspecialized form, the base type's assignment operators must not throw. |
| |
| octonion<T> & operator *= (T const & rhs) |
| octonion<T> & operator *= (::std::complex<T> const & rhs); |
| octonion<T> & operator *= (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & operator *= (octonion<X> const & rhs); |
| |
| These perform the mathematical operation `(*this)*rhs` in this order |
| (order is important as multiplication is not commutative for octonions) |
| and store the result in `*this`. The unspecialized form has exception guards, |
| which the specialized forms do not, so as to insure exception safety. |
| For the unspecialized form, the base type's assignment operators must |
| not throw. Also, for clarity's sake, you should always group the |
| factors in a multiplication by groups of two, as the multiplication is |
| not even associative on the octonions (though there are of course cases |
| where this does not matter, it usually does). |
| |
| octonion<T> & operator /= (T const & rhs) |
| octonion<T> & operator /= (::std::complex<T> const & rhs); |
| octonion<T> & operator /= (::boost::math::quaternion<T> const & rhs); |
| template<typename X> |
| octonion<T> & operator /= (octonion<X> const & rhs); |
| |
| These perform the mathematical operation `(*this)*inverse_of(rhs)` |
| in this order (order is important as multiplication is not commutative |
| for octonions) and store the result in `*this`. The unspecialized form |
| has exception guards, which the specialized forms do not, so as to |
| insure exception safety. For the unspecialized form, the base |
| type's assignment operators must not throw. As for the multiplication, |
| remember to group any two factors using parenthesis. |
| |
| [endsect] |
| |
| [section:non_mem Octonion Non-Member Operators] |
| |
| [h4 Unary Plus and Minus Operators] |
| |
| template<typename T> octonion<T> operator + (octonion<T> const & o); |
| |
| This unary operator simply returns o. |
| |
| template<typename T> octonion<T> operator - (octonion<T> const & o); |
| |
| This unary operator returns the opposite of o. |
| |
| [h4 Binary Addition Operators] |
| |
| template<typename T> octonion<T> operator + (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator + (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> operator + (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> operator + (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> operator + (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| These operators return `octonion<T>(lhs) += rhs`. |
| |
| [h4 Binary Subtraction Operators] |
| |
| template<typename T> octonion<T> operator - (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator - (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> operator - (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> operator - (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> operator - (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| These operators return `octonion<T>(lhs) -= rhs`. |
| |
| [h4 Binary Multiplication Operators] |
| |
| template<typename T> octonion<T> operator * (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator * (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> operator * (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> operator * (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> operator * (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| These operators return `octonion<T>(lhs) *= rhs`. |
| |
| [h4 Binary Division Operators] |
| |
| template<typename T> octonion<T> operator / (T const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator / (octonion<T> const & lhs, T const & rhs); |
| template<typename T> octonion<T> operator / (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> octonion<T> operator / (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> octonion<T> operator / (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| These operators return `octonion<T>(lhs) /= rhs`. It is of course still an |
| error to divide by zero... |
| |
| [h4 Binary Equality Operators] |
| |
| template<typename T> bool operator == (T const & lhs, octonion<T> const & rhs); |
| template<typename T> bool operator == (octonion<T> const & lhs, T const & rhs); |
| template<typename T> bool operator == (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool operator == (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> bool operator == (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool operator == (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> bool operator == (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| These return true if and only if the four components of `octonion<T>(lhs)` |
| are equal to their counterparts in `octonion<T>(rhs)`. As with any |
| floating-type entity, this is essentially meaningless. |
| |
| [h4 Binary Inequality Operators] |
| |
| template<typename T> bool operator != (T const & lhs, octonion<T> const & rhs); |
| template<typename T> bool operator != (octonion<T> const & lhs, T const & rhs); |
| template<typename T> bool operator != (::std::complex<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool operator != (octonion<T> const & lhs, ::std::complex<T> const & rhs); |
| template<typename T> bool operator != (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs); |
| template<typename T> bool operator != (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs); |
| template<typename T> bool operator != (octonion<T> const & lhs, octonion<T> const & rhs); |
| |
| These return true if and only if `octonion<T>(lhs) == octonion<T>(rhs)` |
| is false. As with any floating-type entity, this is essentially meaningless. |
| |
| [h4 Stream Extractor] |
| |
| template<typename T, typename charT, class traits> |
| ::std::basic_istream<charT,traits> & operator >> (::std::basic_istream<charT,traits> & is, octonion<T> & o); |
| |
| Extracts an octonion `o`. We accept any format which seems reasonable. |
| However, since this leads to a great many ambiguities, decisions were made |
| to lift these. In case of doubt, stick to lists of reals. |
| |
| The input values must be convertible to T. If bad input is encountered, |
| calls `is.setstate(ios::failbit)` (which may throw `ios::failure` (27.4.5.3)). |
| |
| Returns `is`. |
| |
| [h4 Stream Inserter] |
| |
| template<typename T, typename charT, class traits> |
| ::std::basic_ostream<charT,traits> & operator << (::std::basic_ostream<charT,traits> & os, octonion<T> const & o); |
| |
| Inserts the octonion `o` onto the stream `os` as if it were implemented as follows: |
| |
| template<typename T, typename charT, class traits> |
| ::std::basic_ostream<charT,traits> & operator << ( ::std::basic_ostream<charT,traits> & os, |
| octonion<T> const & o) |
| { |
| ::std::basic_ostringstream<charT,traits> s; |
| |
| s.flags(os.flags()); |
| s.imbue(os.getloc()); |
| s.precision(os.precision()); |
| |
| s << '(' << o.R_component_1() << ',' |
| << o.R_component_2() << ',' |
| << o.R_component_3() << ',' |
| << o.R_component_4() << ',' |
| << o.R_component_5() << ',' |
| << o.R_component_6() << ',' |
| << o.R_component_7() << ',' |
| << o.R_component_8() << ')'; |
| |
| return os << s.str(); |
| } |
| |
| [endsect] |
| |
| [section Octonion Value Operations] |
| |
| [h4 Real and Unreal] |
| |
| template<typename T> T real(octonion<T> const & o); |
| template<typename T> octonion<T> unreal(octonion<T> const & o); |
| |
| These return `o.real()` and `o.unreal()` respectively. |
| |
| [h4 conj] |
| |
| template<typename T> octonion<T> conj(octonion<T> const & o); |
| |
| This returns the conjugate of the octonion. |
| |
| [h4 sup] |
| |
| template<typename T> T sup(octonion<T> const & o); |
| |
| This return the sup norm (the greatest among |
| `abs(o.R_component_1())...abs(o.R_component_8()))` of the octonion. |
| |
| [h4 l1] |
| |
| template<typename T> T l1(octonion<T> const & o); |
| |
| This return the l1 norm (`abs(o.R_component_1())+...+abs(o.R_component_8())`) |
| of the octonion. |
| |
| [h4 abs] |
| |
| template<typename T> T abs(octonion<T> const & o); |
| |
| This return the magnitude (Euclidian norm) of the octonion. |
| |
| [h4 norm] |
| |
| template<typename T> T norm(octonion<T>const & o); |
| |
| This return the (Cayley) norm of the octonion. The term "norm" might |
| be confusing, as most people associate it with the Euclidian norm |
| (and quadratic functionals). For this version of (the mathematical |
| objects known as) octonions, the Euclidian norm (also known as |
| magnitude) is the square root of the Cayley norm. |
| |
| [endsect] |
| |
| [section:oct_create Octonion Creation Functions] |
| |
| template<typename T> octonion<T> spherical(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6); |
| template<typename T> octonion<T> multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4); |
| template<typename T> octonion<T> cylindrical(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6); |
| |
| These build octonions in a way similar to the way polar builds |
| complex numbers, as there is no strict equivalent to |
| polar coordinates for octonions. |
| |
| `spherical` is a simple transposition of `polar`, it takes as inputs a |
| (positive) magnitude and a point on the hypersphere, given |
| by three angles. The first of these, ['theta] has a natural range of |
| -pi to +pi, and the other two have natural ranges of |
| -pi/2 to +pi/2 (as is the case with the usual spherical |
| coordinates in __R3). Due to the many symmetries and periodicities, |
| nothing untoward happens if the magnitude is negative or the angles are |
| outside their natural ranges. The expected degeneracies (a magnitude of |
| zero ignores the angles settings...) do happen however. |
| |
| `cylindrical` is likewise a simple transposition of the usual |
| cylindrical coordinates in __R3, which in turn is another derivative of |
| planar polar coordinates. The first two inputs are the polar |
| coordinates of the first __C component of the octonion. The third and |
| fourth inputs are placed into the third and fourth __R components of the |
| octonion, respectively. |
| |
| `multipolar` is yet another simple generalization of polar coordinates. |
| This time, both __C components of the octonion are given in polar coordinates. |
| |
| In this version of our implementation of octonions, there is no |
| analogue of the complex value operation arg as the situation is |
| somewhat more complicated. |
| |
| [endsect] |
| |
| [section Octonions Transcendentals] |
| |
| There is no `log` or `sqrt` provided for octonions in this implementation, |
| and `pow` is likewise restricted to integral powers of the exponent. |
| There are several reasons to this: on the one hand, the equivalent of |
| analytic continuation for octonions ("branch cuts") remains to be |
| investigated thoroughly (by me, at any rate...), and we wish to avoid |
| the nonsense introduced in the standard by exponentiations of |
| complexes by complexes (which is well defined, but not in the standard...). |
| Talking of nonsense, saying that `pow(0,0)` is "implementation defined" is |
| just plain brain-dead... |
| |
| We do, however provide several transcendentals, chief among which is |
| the exponential. That it allows for a "closed formula" is a result |
| of the author (the existence and definition of the exponential, on the |
| octonions among others, on the other hand, is a few centuries old). |
| Basically, any converging power series with real coefficients which |
| allows for a closed formula in __C can be transposed to __O. More |
| transcendentals of this type could be added in a further revision upon |
| request. It should be noted that it is these functions which force the |
| dependency upon the |
| [@../../../../../boost/math/special_functions/sinc.hpp boost/math/special_functions/sinc.hpp] |
| and the |
| [@../../../../../boost/math/special_functions/sinhc.hpp boost/math/special_functions/sinhc.hpp] |
| headers. |
| |
| [h4 exp] |
| |
| template<typename T> |
| octonion<T> exp(octonion<T> const & o); |
| |
| Computes the exponential of the octonion. |
| |
| [h4 cos] |
| |
| template<typename T> |
| octonion<T> cos(octonion<T> const & o); |
| |
| Computes the cosine of the octonion |
| |
| [h4 sin] |
| |
| template<typename T> |
| octonion<T> sin(octonion<T> const & o); |
| |
| Computes the sine of the octonion. |
| |
| [h4 tan] |
| |
| template<typename T> |
| octonion<T> tan(octonion<T> const & o); |
| |
| Computes the tangent of the octonion. |
| |
| [h4 cosh] |
| |
| template<typename T> |
| octonion<T> cosh(octonion<T> const & o); |
| |
| Computes the hyperbolic cosine of the octonion. |
| |
| [h4 sinh] |
| |
| template<typename T> |
| octonion<T> sinh(octonion<T> const & o); |
| |
| Computes the hyperbolic sine of the octonion. |
| |
| [h4 tanh] |
| |
| template<typename T> |
| octonion<T> tanh(octonion<T> const & o); |
| |
| Computes the hyperbolic tangent of the octonion. |
| |
| [h4 pow] |
| |
| template<typename T> |
| octonion<T> pow(octonion<T> const & o, int n); |
| |
| Computes the n-th power of the octonion q. |
| |
| [endsect] |
| |
| [section Test Program] |
| |
| The [@../../../octonion/octonion_test.cpp octonion_test.cpp] |
| test program tests octonions specialisations for float, double and long double |
| ([@../../../octonion/output.txt sample output]). |
| |
| If you define the symbol BOOST_OCTONION_TEST_VERBOSE, you will get additional |
| output ([@../../../octonion/output_more.txt verbose output]); this will |
| only be helpfull if you enable message output at the same time, of course |
| (by uncommenting the relevant line in the test or by adding --log_level=messages |
| to your command line,...). In that case, and if you are running interactively, |
| you may in addition define the symbol BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to |
| interactively test the input operator with input of your choice from the |
| standard input (instead of hard-coding it in the test). |
| |
| [endsect] |
| |
| [section Acknowledgements] |
| |
| The mathematical text has been typeset with |
| [@http://www.nisus-soft.com/ Nisus Writer]. |
| Jens Maurer has helped with portability and standard adherence, and was the |
| Review Manager for this library. More acknowledgements in the |
| History section. Thank you to all who contributed to the discussion about this library. |
| |
| [endsect] |
| |
| [section History] |
| |
| * 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. |
| * 1.5.7 - 25/02/2003: transitionned to the unit test framework; <boost/config.hpp> now included by the library header (rather than the test files), via <boost/math/quaternion.hpp>. |
| * 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com). |
| * 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option. |
| * 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different translation units); attempt at an improved compatibility with Microsoft compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; other compatibility fixes. |
| * 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu). |
| * 1.5.2 - 07/07/2001: introduced namespace math. |
| * 1.5.1 - 07/06/2001: (end of Boost review) now includes <boost/math/special_functions/sinc.hpp> and <boost/math/special_functions/sinhc.hpp> instead of <boost/special_functions.hpp>; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. |
| * 1.5.0 - 23/03/2001: boostification, inlining of all operators except input, output and pow, fixed exception safety of some members (template version). |
| * 1.4.0 - 09/01/2001: added tan and tanh. |
| * 1.3.1 - 08/01/2001: cosmetic fixes. |
| * 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) algorithm. |
| * 1.2.0 - 25/05/2000: fixed the division operators and output; changed many signatures. |
| * 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. |
| * 1.0.0 - 10/08/1999: first public version. |
| |
| [endsect] |
| |
| [section To Do] |
| |
| * Improve testing. |
| * Rewrite input operatore using Spirit (creates a dependency). |
| * Put in place an Expression Template mechanism (perhaps borrowing from uBlas). |
| |
| [endsect] |
| |
| [endsect] |
