math-octonion.qbk

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[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 ('''&#x03B1;,&#x03B2;,&#x03B3;,&#x03B4;,&#x03B5;,&#x03B6;,&#x03B7;,&#x03B8;''')]
[def __oct_formula ['[^o = '''&#x03B1; + &#x03B2;i + &#x03B3;j + &#x03B4;k + &#x03B5;e' + &#x03B6;i' + &#x03B7;j' + &#x03B8;k' ''']]]
[def __oct_complex_formula ['[^o = ('''&#x03B1; + &#x03B2;i) + (&#x03B3; + &#x03B4;i)j + (&#x03B5; + &#x03B6;i)e' + (&#x03B7; - &#x03B8;i)j' ''']]]
[def __oct_quat_formula ['[^o = ('''&#x03B1; + &#x03B2;i + &#x03B3;j + &#x03B4;k) + (&#x03B5; + &#x03B6;i + &#x03B7;j - &#x03B8;j)e' ''']]]
[def __oct_not_equal ['[^x(yz) '''&#x2260;''' (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://svn.boost.org/svn/boost/sandbox/pdf/math/release/octonion.pdf 
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]