...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
Octonions, like 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 (
α,β,γ,δ,ε,ζ,η,θ
), which we can write in the form o =
α + βi + γj + δk + εe' + ζi' + ηj' + θk'
, 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 quaternions x
,
y
and z
such that x(yz)
≠
(xy)z
). A way of remembering things is by using the
following multiplication table:
Octonions (and their kin) are described in far more details in this other document (with 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).
The interface and implementation are both supplied by the header file octonion.hpp.
namespace boost{ namespace math{ template<typename T> class octonion; template<> class octonion<float>; template<> class octonion<double>; template<> class octonion<long double>; // 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); 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); 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); 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); template<typename T> octonion<T> operator + (octonion<T> const & o); template<typename T> octonion<T> operator - (octonion<T> const & o); 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); 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); template<typename T, typename charT, class traits> ::std::basic_istream<charT,traits> & operator >> (::std::basic_istream<charT,traits> & is, octonion<T> & o); template<typename T, typename charT, class traits> ::std::basic_ostream<charT,traits> & operator << (::std::basic_ostream<charT,traits> & os, octonion<T> const & o); // values template<typename T> T real(octonion<T> const & o); template<typename T> octonion<T> unreal(octonion<T> const & o); template<typename T> T sup(octonion<T> const & o); template<typename T> T l1(octonion<T>const & o); template<typename T> T abs(octonion<T> const & o); template<typename T> T norm(octonion<T>const & o); template<typename T> octonion<T> conj(octonion<T> const & o); 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); // transcendentals template<typename T> octonion<T> exp(octonion<T> const & o); template<typename T> octonion<T> cos(octonion<T> const & o); template<typename T> octonion<T> sin(octonion<T> const & o); template<typename T> octonion<T> tan(octonion<T> const & o); template<typename T> octonion<T> cosh(octonion<T> const & o); template<typename T> octonion<T> sinh(octonion<T> const & o); template<typename T> octonion<T> tanh(octonion<T> const & o); template<typename T> octonion<T> pow(octonion<T> const & o, int n); } } // namespaces
namespace boost{ namespace math { template<typename T> class octonion { public: typedef T value_type; 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); T real() const; octonion<T> unreal() const; 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; ::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; ::boost::math::quaternion<T> H_component_1() const; ::boost::math::quaternion<T> H_component_2() const; 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); 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); 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); 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); 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); }; } } // namespaces
namespace boost{ namespace math{ template<> class octonion<float> { public: typedef float value_type; 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); float real() const; octonion<float> unreal() const; float R_component_1() const; float R_component_2() const; float R_component_3() const; float R_component_4() const; float R_component_5() const; float R_component_6() const; float R_component_7() const; float R_component_8() const; ::std::complex<float> C_component_1() const; ::std::complex<float> C_component_2() const; ::std::complex<float> C_component_3() const; ::std::complex<float> C_component_4() const; ::boost::math::quaternion<float> H_component_1() const; ::boost::math::quaternion<float> H_component_2() const; octonion<float> & operator = (octonion<float> const & a_affecter); template<typename X> octonion<float> & operator = (octonion<X> const & a_affecter); octonion<float> & operator = (float const & a_affecter); octonion<float> & operator = (::std::complex<float> const & a_affecter); octonion<float> & operator = (::boost::math::quaternion<float> const & a_affecter); octonion<float> & operator += (float const & rhs); octonion<float> & operator += (::std::complex<float> const & rhs); octonion<float> & operator += (::boost::math::quaternion<float> const & rhs); template<typename X> octonion<float> & operator += (octonion<X> const & rhs); octonion<float> & operator -= (float const & rhs); octonion<float> & operator -= (::std::complex<float> const & rhs); octonion<float> & operator -= (::boost::math::quaternion<float> const & rhs); template<typename X> octonion<float> & operator -= (octonion<X> const & rhs); octonion<float> & operator *= (float const & rhs); octonion<float> & operator *= (::std::complex<float> const & rhs); octonion<float> & operator *= (::boost::math::quaternion<float> const & rhs); template<typename X> octonion<float> & operator *= (octonion<X> const & rhs); octonion<float> & operator /= (float const & rhs); octonion<float> & operator /= (::std::complex<float> const & rhs); octonion<float> & operator /= (::boost::math::quaternion<float> const & rhs); template<typename X> octonion<float> & operator /= (octonion<X> const & rhs); };
template<> class octonion<double> { public: typedef double value_type; 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); double real() const; octonion<double> unreal() const; double R_component_1() const; double R_component_2() const; double R_component_3() const; double R_component_4() const; double R_component_5() const; double R_component_6() const; double R_component_7() const; double R_component_8() const; ::std::complex<double> C_component_1() const; ::std::complex<double> C_component_2() const; ::std::complex<double> C_component_3() const; ::std::complex<double> C_component_4() const; ::boost::math::quaternion<double> H_component_1() const; ::boost::math::quaternion<double> H_component_2() const; octonion<double> & operator = (octonion<double> const & a_affecter); template<typename X> octonion<double> & operator = (octonion<X> const & a_affecter); octonion<double> & operator = (double const & a_affecter); octonion<double> & operator = (::std::complex<double> const & a_affecter); octonion<double> & operator = (::boost::math::quaternion<double> const & a_affecter); octonion<double> & operator += (double const & rhs); octonion<double> & operator += (::std::complex<double> const & rhs); octonion<double> & operator += (::boost::math::quaternion<double> const & rhs); template<typename X> octonion<double> & operator += (octonion<X> const & rhs); octonion<double> & operator -= (double const & rhs); octonion<double> & operator -= (::std::complex<double> const & rhs); octonion<double> & operator -= (::boost::math::quaternion<double> const & rhs); template<typename X> octonion<double> & operator -= (octonion<X> const & rhs); octonion<double> & operator *= (double const & rhs); octonion<double> & operator *= (::std::complex<double> const & rhs); octonion<double> & operator *= (::boost::math::quaternion<double> const & rhs); template<typename X> octonion<double> & operator *= (octonion<X> const & rhs); octonion<double> & operator /= (double const & rhs); octonion<double> & operator /= (::std::complex<double> const & rhs); octonion<double> & operator /= (::boost::math::quaternion<double> const & rhs); template<typename X> octonion<double> & operator /= (octonion<X> const & rhs); };
template<> class octonion<long double> { public: typedef long double value_type; 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 & z1 = ::boost::math::quaternion<long double>()); explicit octonion(octonion<float> const & a_recopier); explicit octonion(octonion<double> const & a_recopier); long double real() const; octonion<long double> unreal() const; long double R_component_1() const; long double R_component_2() const; long double R_component_3() const; long double R_component_4() const; long double R_component_5() const; long double R_component_6() const; long double R_component_7() const; long double R_component_8() const; ::std::complex<long double> C_component_1() const; ::std::complex<long double> C_component_2() const; ::std::complex<long double> C_component_3() const; ::std::complex<long double> C_component_4() const; ::boost::math::quaternion<long double> H_component_1() const; ::boost::math::quaternion<long double> H_component_2() const; octonion<long double> & operator = (octonion<long double> const & a_affecter); template<typename X> octonion<long double> & operator = (octonion<X> const & a_affecter); octonion<long double> & operator = (long double const & a_affecter); octonion<long double> & operator = (::std::complex<long double> const & a_affecter); octonion<long double> & operator = (::boost::math::quaternion<long double> const & a_affecter); octonion<long double> & operator += (long double const & rhs); octonion<long double> & operator += (::std::complex<long double> const & rhs); octonion<long double> & operator += (::boost::math::quaternion<long double> const & rhs); template<typename X> octonion<long double> & operator += (octonion<X> const & rhs); octonion<long double> & operator -= (long double const & rhs); octonion<long double> & operator -= (::std::complex<long double> const & rhs); octonion<long double> & operator -= (::boost::math::quaternion<long double> const & rhs); template<typename X> octonion<long double> & operator -= (octonion<X> const & rhs); octonion<long double> & operator *= (long double const & rhs); octonion<long double> & operator *= (::std::complex<long double> const & rhs); octonion<long double> & operator *= (::boost::math::quaternion<long double> const & rhs); template<typename X> octonion<long double> & operator *= (octonion<X> const & rhs); octonion<long double> & operator /= (long double const & rhs); octonion<long double> & operator /= (::std::complex<long double> const & rhs); octonion<long double> & operator /= (::boost::math::quaternion<long double> const & rhs); template<typename X> octonion<long double> & operator /= (octonion<X> const & rhs); }; } } // namespaces
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.
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.
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.
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.
::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
o =
α + βi + γj + δk + εe' + ζi' + ηj' + θk'
we also have o = (
α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j'
(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')
.
::boost::math::quaternion<T> H_component_1() const; ::boost::math::quaternion<T> H_component_2() const;
Likewise, for any octonion o =
α + βi + γj + δk + εe' + ζi' + ηj' + θk'
we also have o = (
α + βi + γj + δk) + (ε + ζi + ηj - θj)e'
, 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.
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.
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.
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.
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
.
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
.
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
.
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...
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.
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.
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
.
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(); }
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.
template<typename T> octonion<T> conj(octonion<T> const & o);
This returns the conjugate of the octonion.
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.
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.
template<typename T> T abs(octonion<T> const & o);
This return the magnitude (Euclidian norm) of the octonion.
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.
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.
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 and the boost/math/special_functions/sinhc.hpp headers.
template<typename T> octonion<T> exp(octonion<T> const & o);
Computes the exponential of the octonion.
template<typename T> octonion<T> cos(octonion<T> const & o);
Computes the cosine of the octonion
template<typename T> octonion<T> sin(octonion<T> const & o);
Computes the sine of the octonion.
template<typename T> octonion<T> tan(octonion<T> const & o);
Computes the tangent of the octonion.
template<typename T> octonion<T> cosh(octonion<T> const & o);
Computes the hyperbolic cosine of the octonion.
template<typename T> octonion<T> sinh(octonion<T> const & o);
Computes the hyperbolic sine of the octonion.
template<typename T> octonion<T> tanh(octonion<T> const & o);
Computes the hyperbolic tangent of the octonion.
template<typename T> octonion<T> pow(octonion<T> const & o, int n);
Computes the n-th power of the octonion q.
The octonion_test.cpp test program tests octonions specialisations for float, double and long double (sample output).
If you define the symbol BOOST_OCTONION_TEST_VERBOSE, you will get additional output (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).
The mathematical text has been typeset with 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.
Copyright © 2001 -2002 Daryle Walker, 2001-2003 Hubert Holin, 2005 John Maddock |