[def __R ['[*R]]] [def __C ['[*C]]] [def __H ['[*H]]] [def __O ['[*O]]] [def __R3 ['[*'''R3''']]] [def __R4 ['[*'''R4''']]] [def __quadrulple ('''α,β,γ,δ''')] [def __quat_formula ['[^q = '''α + βi + γj + δk''']]] [def __quat_complex_formula ['[^q = ('''α + βi) + (γ + δi)j''' ]]] [def __not_equal ['[^xy '''≠''' yx]]] [mathpart quaternions Quaternions] [section:quat_overview Overview] Quaternions are a relative of complex numbers. Quaternions are in fact part of a small hierarchy of structures built upon the real numbers, which comprise only the set of real numbers (traditionally named __R), the set of complex numbers (traditionally named __C), the set of quaternions (traditionally named __H) and the set of octonions (traditionally named __O), which possess interesting mathematical properties (chief among which is the fact that they are ['division algebras], ['i.e.] where the following property is true: if ['[^y]] is an element of that algebra and is [*not equal to zero], then ['[^yx = yx']], where ['[^x]] and ['[^x']] denote elements of that algebra, implies that ['[^x = x']]). Each member of the hierarchy is a super-set of the former. One of the most important aspects of quaternions is that they provide an efficient way to parameterize rotations in __R3 (the usual three-dimensional space) and __R4. In practical terms, a quaternion is simply a quadruple of real numbers __quadrulple, which we can write in the form __quat_formula, where ['[^i]] is the same object as for complex numbers, and ['[^j]] and ['[^k]] are distinct objects which play essentially the same kind of role as ['[^i]]. An addition and a multiplication is defined on the set of quaternions, which generalize their real and complex counterparts. The main novelty here is that [*the multiplication is not commutative] (i.e. there are quaternions ['[^x]] and ['[^y]] such that __not_equal). A good mnemotechnical way of remembering things is by using the formula ['[^i*i = j*j = k*k = -1]]. Quaternions (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 quaternions, but other, such as taking a square root, do not. [endsect] [/section:quat_overview Overview] [section:quat_header Header File] The interface and implementation are both supplied by the header file [@../../../../boost/math/quaternion.hpp quaternion.hpp]. [endsect] [section:quat_synopsis Synopsis] namespace boost{ namespace math{ template class ``[link math_toolkit.quat quaternion]``; template<> class ``[link math_toolkit.spec quaternion]``; template<> class ``[link math_quaternion_double quaternion]``; template<> class ``[link math_quaternion_long_double quaternion]``; // operators template quaternion ``[link math_toolkit.quat_non_mem.binary_addition_operators operator +]`` (T const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, T const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_addition_operators operator +]`` (::std::complex const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, ::std::complex const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_addition_operators operator +]`` (quaternion const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_subtraction_operators operator -]`` (T const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, T const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_subtraction_operators operator -]`` (::std::complex const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, ::std::complex const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_subtraction_operators operator -]`` (quaternion const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_multiplication_operators operator *]`` (T const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, T const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_multiplication_operators operator *]`` (::std::complex const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, ::std::complex const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_multiplication_operators operator *]`` (quaternion const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_division_operators operator /]`` (T const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_division_operators operator /]`` (quaternion const & lhs, T const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_division_operators operator /]`` (::std::complex const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_division_operators operator /]`` (quaternion const & lhs, ::std::complex const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.binary_division_operators operator /]`` (quaternion const & lhs, quaternion const & rhs); template quaternion ``[link math_toolkit.quat_non_mem.unary_plus operator +]`` (quaternion const & q); template quaternion ``[link math_toolkit.quat_non_mem.unary_minus operator -]`` (quaternion const & q); template bool ``[link math_toolkit.quat_non_mem.equality_operators operator ==]`` (T const & lhs, quaternion const & rhs); template bool ``[link math_toolkit.quat_non_mem.equality_operators operator ==]`` (quaternion const & lhs, T const & rhs); template bool ``[link math_toolkit.quat_non_mem.equality_operators operator ==]`` (::std::complex const & lhs, quaternion const & rhs); template bool ``[link math_toolkit.quat_non_mem.equality_operators operator ==]`` (quaternion const & lhs, ::std::complex const & rhs); template bool ``[link math_toolkit.quat_non_mem.equality_operators operator ==]`` (quaternion const & lhs, quaternion const & rhs); template bool ``[link math_toolkit.quat_non_mem.inequality_operators operator !=]`` (T const & lhs, quaternion const & rhs); template bool ``[link math_toolkit.quat_non_mem.inequality_operators operator !=]`` (quaternion const & lhs, T const & rhs); template bool ``[link math_toolkit.quat_non_mem.inequality_operators operator !=]`` (::std::complex const & lhs, quaternion const & rhs); template bool ``[link math_toolkit.quat_non_mem.inequality_operators operator !=]`` (quaternion const & lhs, ::std::complex const & rhs); template bool ``[link math_toolkit.quat_non_mem.inequality_operators operator !=]`` (quaternion const & lhs, quaternion const & rhs); template ::std::basic_istream& ``[link math_toolkit.quat_non_mem.stream_extractor operator >>]`` (::std::basic_istream & is, quaternion & q); template ::std::basic_ostream& operator ``[link math_toolkit.quat_non_mem.stream_inserter operator <<]`` (::std::basic_ostream & os, quaternion const & q); // values template T ``[link math_toolkit.value_op.real_and_unreal real]``(quaternion const & q); template quaternion ``[link math_toolkit.value_op.real_and_unreal unreal]``(quaternion const & q); template T ``[link math_toolkit.value_op.sup sup]``(quaternion const & q); template T ``[link math_toolkit.value_op.l1 l1]``(quaternion const & q); template T ``[link math_toolkit.value_op.abs abs]``(quaternion const & q); template T ``[link math_toolkit.value_op.norm norm]``(quaternionconst & q); template quaternion ``[link math_toolkit.value_op.conj conj]``(quaternion const & q); template quaternion ``[link math_quaternions.creation_spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2); template quaternion ``[link math_quaternions.creation_semipolar semipolar]``(T const & rho, T const & alpha, T const & theta1, T const & theta2); template quaternion ``[link math_quaternions.creation_multipolar multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2); template quaternion ``[link math_quaternions.creation_cylindrospherical cylindrospherical]``(T const & t, T const & radius, T const & longitude, T const & latitude); template quaternion ``[link math_quaternions.creation_cylindrical cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2); // transcendentals template quaternion ``[link math_toolkit.trans.exp exp]``(quaternion const & q); template quaternion ``[link math_toolkit.trans.cos cos]``(quaternion const & q); template quaternion ``[link math_toolkit.trans.sin sin]``(quaternion const & q); template quaternion ``[link math_toolkit.trans.tan tan]``(quaternion const & q); template quaternion ``[link math_toolkit.trans.cosh cosh]``(quaternion const & q); template quaternion ``[link math_toolkit.trans.sinh sinh]``(quaternion const & q); template quaternion ``[link math_toolkit.trans.tanh tanh]``(quaternion const & q); template quaternion ``[link math_toolkit.trans.pow pow]``(quaternion const & q, int n); } // namespace math } // namespace boost [endsect] [/section:quat_header Header File] [section:quat Template Class quaternion] namespace boost{ namespace math{ template class quaternion { public: typedef T ``[link math_toolkit.mem_typedef value_type]``; explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); template explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(quaternion const & a_recopier); T ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts real]``() const; quaternion ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts unreal]``() const; T ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_1]``() const; T ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_2]``() const; T ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_3]``() const; T ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_4]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_2]``() const; quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(T const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(T const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(T const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(T const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(T const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; } // namespace math } // namespace boost [endsect] [/section:quat Template Class quaternion] [section:spec Quaternion Specializations] namespace boost{ namespace math{ template<> class quaternion { public: typedef float ``[link math_toolkit.mem_typedef value_type]``; explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(quaternion const & a_recopier); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(quaternion const & a_recopier); float ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts real]``() const; quaternion ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts unreal]``() const; float ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_1]``() const; float ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_2]``() const; float ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_3]``() const; float ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_4]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_2]``() const; quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(float const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(float const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(float const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(float const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(float const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; [#math_quaternion_double] template<> class quaternion { public: typedef double ``[link math_toolkit.mem_typedef value_type]``; explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(quaternion const & a_recopier); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(quaternion const & a_recopier); double ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts real]``() const; quaternion ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts unreal]``() const; double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_1]``() const; double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_2]``() const; double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_3]``() const; double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_4]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_2]``() const; quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(double const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; [#math_quaternion_long_double] template<> class quaternion { public: typedef long double ``[link math_toolkit.mem_typedef value_type]``; explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(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); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(quaternion const & a_recopier); explicit ``[link math_toolkit.quat_mem_fun.constructors quaternion]``(quaternion const & a_recopier); long double ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts real]``() const; quaternion ``[link math_toolkit.quat_mem_fun.real_and_unreal_parts unreal]``() const; long double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_1]``() const; long double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_2]``() const; long double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_3]``() const; long double ``[link math_toolkit.quat_mem_fun.individual_real_components R_component_4]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.quat_mem_fun.individual_complex_components C_component_2]``() const; quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); template quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(quaternion const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(long double const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.assignment_operators operator = ]``(::std::complex const & a_affecter); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(long double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.addition_operators operator += ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(long double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.subtraction_operators operator -= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(long double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.multiplication_operators operator *= ]``(quaternion const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(long double const & rhs); quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(::std::complex const & rhs); template quaternion& ``[link math_toolkit.quat_mem_fun.division_operators operator /= ]``(quaternion const & rhs); }; } // namespace math } // namespace boost [endsect] [/section:spec Quaternion Specializations] [section:mem_typedef Quaternion 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 access to the type the template is built upon. [endsect] [section:quat_mem_fun Quaternion Member Functions] [h3 Constructors] Template version: explicit quaternion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); explicit quaternion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); template explicit quaternion(quaternion const & a_recopier); Float specialization version: explicit quaternion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f); explicit quaternion(::std::complex const & z0,::std::complex const & z1 = ::std::complex()); explicit quaternion(quaternion const & a_recopier); explicit quaternion(quaternion const & a_recopier); Double specialization version: explicit quaternion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0); explicit quaternion(::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); explicit quaternion(quaternion const & a_recopier); explicit quaternion(quaternion const & a_recopier); Long double specialization version: explicit quaternion(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); explicit quaternion( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex()); explicit quaternion(quaternion const & a_recopier); explicit quaternion(quaternion 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 four base type arguments. A constructor is also provided to build quaternions from one or two complex numbers 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; quaternion unreal() const; Like complex number, quaternions do have a meaningful notion of "real part", but unlike them there is no meaningful notion of "imaginary part". Instead there is an "unreal part" which itself is a quaternion, and usually nothing simpler (as opposed to the complex number case). 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; A quaternion having four real components, these are returned by these four functions. Hence real and R_component_1 return the same value. [h4 Individual Complex Components] ::std::complex C_component_1() const; ::std::complex C_component_2() const; A quaternion likewise has two complex components, and as we have seen above, for any quaternion __quat_formula we also have __quat_complex_formula. These functions return them. The real part of `q.C_component_1()` is the same as `q.real()`. [h3 Quaternion Member Operators] [h4 Assignment Operators] quaternion& operator = (quaternion const & a_affecter); template quaternion& operator = (quaternion const& a_affecter); quaternion& operator = (T const& a_affecter); quaternion& operator = (::std::complex 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 Addition Operators] quaternion& operator += (T const & rhs) quaternion& operator += (::std::complex const & rhs); template quaternion& operator += (quaternion 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. [h4 Subtraction Operators] quaternion& operator -= (T const & rhs) quaternion& operator -= (::std::complex const & rhs); template quaternion& operator -= (quaternion 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. [h4 Multiplication Operators] quaternion& operator *= (T const & rhs) quaternion& operator *= (::std::complex const & rhs); template quaternion& operator *= (quaternion const & rhs); These perform the mathematical operation `(*this)*rhs` [*in this order] (order is important as multiplication is not commutative for quaternions) 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. [h4 Division Operators] quaternion& operator /= (T const & rhs) quaternion& operator /= (::std::complex const & rhs); template quaternion& operator /= (quaternion const & rhs); These perform the mathematical operation `(*this)*inverse_of(rhs)` [*in this order] (order is important as multiplication is not commutative for quaternions) 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. [endsect] [/section:mem_typedef Quaternion Member Typedefs] [section:quat_non_mem Quaternion Non-Member Operators] [h4 Unary Plus] template quaternion operator + (quaternion const & q); This unary operator simply returns q. [h4 Unary Minus] template quaternion operator - (quaternion const & q); This unary operator returns the opposite of q. [h4 Binary Addition Operators] template quaternion operator + (T const & lhs, quaternion const & rhs); template quaternion operator + (quaternion const & lhs, T const & rhs); template quaternion operator + (::std::complex const & lhs, quaternion const & rhs); template quaternion operator + (quaternion const & lhs, ::std::complex const & rhs); template quaternion operator + (quaternion const & lhs, quaternion const & rhs); These operators return `quaternion(lhs) += rhs`. [h4 Binary Subtraction Operators] template quaternion operator - (T const & lhs, quaternion const & rhs); template quaternion operator - (quaternion const & lhs, T const & rhs); template quaternion operator - (::std::complex const & lhs, quaternion const & rhs); template quaternion operator - (quaternion const & lhs, ::std::complex const & rhs); template quaternion operator - (quaternion const & lhs, quaternion const & rhs); These operators return `quaternion(lhs) -= rhs`. [h4 Binary Multiplication Operators] template quaternion operator * (T const & lhs, quaternion const & rhs); template quaternion operator * (quaternion const & lhs, T const & rhs); template quaternion operator * (::std::complex const & lhs, quaternion const & rhs); template quaternion operator * (quaternion const & lhs, ::std::complex const & rhs); template quaternion operator * (quaternion const & lhs, quaternion const & rhs); These operators return `quaternion(lhs) *= rhs`. [h4 Binary Division Operators] template quaternion operator / (T const & lhs, quaternion const & rhs); template quaternion operator / (quaternion const & lhs, T const & rhs); template quaternion operator / (::std::complex const & lhs, quaternion const & rhs); template quaternion operator / (quaternion const & lhs, ::std::complex const & rhs); template quaternion operator / (quaternion const & lhs, quaternion const & rhs); These operators return `quaternion(lhs) /= rhs`. It is of course still an error to divide by zero... [h4 Equality Operators] template bool operator == (T const & lhs, quaternion const & rhs); template bool operator == (quaternion const & lhs, T const & rhs); template bool operator == (::std::complex const & lhs, quaternion const & rhs); template bool operator == (quaternion const & lhs, ::std::complex const & rhs); template bool operator == (quaternion const & lhs, quaternion const & rhs); These return true if and only if the four components of `quaternion(lhs)` are equal to their counterparts in `quaternion(rhs)`. As with any floating-type entity, this is essentially meaningless. [h4 Inequality Operators] template bool operator != (T const & lhs, quaternion const & rhs); template bool operator != (quaternion const & lhs, T const & rhs); template bool operator != (::std::complex const & lhs, quaternion const & rhs); template bool operator != (quaternion const & lhs, ::std::complex const & rhs); template bool operator != (quaternion const & lhs, quaternion const & rhs); These return true if and only if `quaternion(lhs) == quaternion(rhs)` is false. As with any floating-type entity, this is essentially meaningless. [h4 Stream Extractor] template ::std::basic_istream& operator >> (::std::basic_istream & is, quaternion & q); Extracts a quaternion q of one of the following forms (with a, b, c and d of type `T`): [^a (a), (a,b), (a,b,c), (a,b,c,d) (a,(c)), (a,(c,d)), ((a)), ((a),c), ((a),(c)), ((a),(c,d)), ((a,b)), ((a,b),c), ((a,b),(c)), ((a,b),(c,d))] 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`. The rationale for the list of accepted formats is that either we have a list of up to four reals, or else we have a couple of complex numbers, and in that case if it formatted as a proper complex number, then it should be accepted. Thus potential ambiguities are lifted (for instance (a,b) is (a,b,0,0) and not (a,0,b,0), i.e. it is parsed as a list of two real numbers and not two complex numbers which happen to have imaginary parts equal to zero). [h4 Stream Inserter] template ::std::basic_ostream& operator << (::std::basic_ostream & os, quaternion const & q); Inserts the quaternion q onto the stream `os` as if it were implemented as follows: template ::std::basic_ostream& operator << ( ::std::basic_ostream & os, quaternion const & q) { ::std::basic_ostringstream s; s.flags(os.flags()); s.imbue(os.getloc()); s.precision(os.precision()); s << '(' << q.R_component_1() << ',' << q.R_component_2() << ',' << q.R_component_3() << ',' << q.R_component_4() << ')'; return os << s.str(); } [endsect] [/section:quat_non_mem Quaternion Non-Member Operators] [section:value_op Quaternion Value Operations] [h4 real and unreal] template T real(quaternion const & q); template quaternion unreal(quaternion const & q); These return `q.real()` and `q.unreal()` respectively. [h4 conj] template quaternion conj(quaternion const & q); This returns the conjugate of the quaternion. [h4 sup] template T sup(quaternion const & q); This return the sup norm (the greatest among `abs(q.R_component_1())...abs(q.R_component_4()))` of the quaternion. [h4 l1] template T l1(quaternion const & q); This return the l1 norm `(abs(q.R_component_1())+...+abs(q.R_component_4()))` of the quaternion. [h4 abs] template T abs(quaternion const & q); This return the magnitude (Euclidean norm) of the quaternion. [h4 norm] template T norm(quaternionconst & q); This return the (Cayley) norm of the quaternion. 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) quaternions, the Euclidean norm (also known as magnitude) is the square root of the Cayley norm. [endsect] [/section:value_op Quaternion Value Operations] [section:create Quaternion Creation Functions] template quaternion spherical(T const & rho, T const & theta, T const & phi1, T const & phi2); template quaternion semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2); template quaternion multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2); template quaternion cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude); template quaternion cylindrical(T const & r, T const & angle, T const & h1, T const & h2); These build quaternions in a way similar to the way polar builds complex numbers, as there is no strict equivalent to polar coordinates for quaternions. [#math_quaternions.creation_spherical] `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. [#math_quaternions.creation_cylindrical] `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 quaternion. The third and fourth inputs are placed into the third and fourth __R components of the quaternion, respectively. [#math_quaternions.creation_multipolar] `multipolar` is yet another simple generalization of polar coordinates. This time, both __C components of the quaternion are given in polar coordinates. [#math_quaternions.creation_cylindrospherical] `cylindrospherical` is specific to quaternions. It is often interesting to consider __H as the cartesian product of __R by __R3 (the quaternionic multiplication as then a special form, as given here). This function therefore builds a quaternion from this representation, with the __R3 component given in usual __R3 spherical coordinates. [#math_quaternions.creation_semipolar] `semipolar` is another generator which is specific to quaternions. It takes as a first input the magnitude of the quaternion, as a second input an angle in the range `0` to `+pi/2` such that magnitudes of the first two __C components of the quaternion are the product of the first input and the sine and cosine of this angle, respectively, and finally as third and fourth inputs angles in the range `-pi/2` to `+pi/2` which represent the arguments of the first and second __C components of the quaternion, respectively. As usual, nothing untoward happens if what should be magnitudes are negative numbers or angles are out of their natural ranges, as symmetries and periodicities kick in. In this version of our implementation of quaternions, there is no analogue of the complex value operation `arg` as the situation is somewhat more complicated. Unit quaternions are linked both to rotations in __R3 and in __R4, and the correspondences are not too complicated, but there is currently a lack of standard (de facto or de jure) matrix library with which the conversions could work. This should be remedied in a further revision. In the mean time, an example of how this could be done is presented here for [@../../example/HSO3.hpp __R3], and here for [@../../example/HSO4.hpp __R4] ([@../../example/HSO3SO4.cpp example test file]). [endsect] [/section:create Quaternion Creation Functions] [section:trans Quaternion Transcendentals] There is no `log` or `sqrt` provided for quaternions 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 quaternions ("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. This author claims the complete proof of the "closed formula" as his own, as well as its independent invention (there are claims to prior invention of the formula, such as one by Professor Shoemake, and it is possible that the formula had been known a couple of centuries back, but in absence of bibliographical reference, the matter is pending, awaiting further investigation; on the other hand, the definition and existence of the exponential on the quaternions, is of course a fact known for a very long time). Basically, any converging power series with real coefficients which allows for a closed formula in __C can be transposed to __H. 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 quaternion exp(quaternion const & q); Computes the exponential of the quaternion. [h4 cos] template quaternion cos(quaternion const & q); Computes the cosine of the quaternion [h4 sin] template quaternion sin(quaternion const & q); Computes the sine of the quaternion. [h4 tan] template quaternion tan(quaternion const & q); Computes the tangent of the quaternion. [h4 cosh] template quaternion cosh(quaternion const & q); Computes the hyperbolic cosine of the quaternion. [h4 sinh] template quaternion sinh(quaternion const & q); Computes the hyperbolic sine of the quaternion. [h4 tanh] template quaternion tanh(quaternion const & q); Computes the hyperbolic tangent of the quaternion. [h4 pow] template quaternion pow(quaternion const & q, int n); Computes the n-th power of the quaternion q. [endsect] [/section:trans Quaternion Transcendentals] [section:quat_tests Test Program] The [@../../test/quaternion_test.cpp quaternion_test.cpp] test program tests quaternions specializations for float, double and long double ([@../quaternion/output.txt sample output], with message output enabled). If you define the symbol TEST_VERBOSE, you will get additional output ([@../quaternion/output_more.txt verbose output]); this will only be helpful 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:quat_tests Test Program] [section:exp The Quaternionic Exponential] Please refer to the following PDF's: *[@../quaternion/TQE.pdf The Quaternionic Exponential (and beyond)] *[@../quaternion/TQE_EA.pdf The Quaternionic Exponential (and beyond) ERRATA & ADDENDA] [endsect] [/section:exp The Quaternionic Exponential] [section:acknowledgement 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 discution about this library. [endsect] [/section:quat_tests Test Program] [section:quat_history History] * 1.5.9 - 13/5/2013: Incorporated into Boost.Math. * 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. * 1.5.7 - 24/02/2003: transitioned to the unit test framework; now included by the library header (rather than the test files). * 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 and instead of ; 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 - 26/03/2001: boostification, inlining of all operators except input, output and pow, fixed exception safety of some members (template version) and output operator, added spherical, semipolar, multipolar, cylindrospherical and cylindrical. * 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:quat_history History] [section:quat_todo To Do] * Improve testing. * Rewrite input operator using Spirit (creates a dependency). * Put in place an Expression Template mechanism (perhaps borrowing from uBlas). * Use uBlas for the link with rotations (and move from the [@../../example/HSO3SO4.cpp example] implementation to an efficient one). [endsect] [/section:quat_todo To Do] [endmathpart] [/mathpart quaternions Quaternions] [/ Copyright 1999, 2005, 2013 Hubert Holin. 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). ]