| 1 | |
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| 2 | [def __R ['[*R]]] |
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| 3 | [def __C ['[*C]]] |
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| 4 | [def __H ['[*H]]] |
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| 5 | [def __O ['[*O]]] |
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| 6 | [def __R3 ['[*'''R<superscript>3</superscript>''']]] |
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| 7 | [def __R4 ['[*'''R<superscript>4</superscript>''']]] |
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| 8 | [def __quadrulple ('''α,β,γ,δ''')] |
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| 9 | [def __quat_formula ['[^q = '''α + βi + γj + δk''']]] |
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| 10 | [def __quat_complex_formula ['[^q = ('''α + βi) + (γ + δi)j''' ]]] |
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| 11 | [def __not_equal ['[^xy '''≠''' yx]]] |
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| 12 | |
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| 13 | |
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| 14 | [section Quaternions] |
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| 15 | |
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| 16 | [section Overview] |
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| 17 | |
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| 18 | Quaternions are a relative of complex numbers. |
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| 19 | |
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| 20 | Quaternions are in fact part of a small hierarchy of structures built |
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| 21 | upon the real numbers, which comprise only the set of real numbers |
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| 22 | (traditionally named __R), the set of complex numbers (traditionally named __C), |
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| 23 | the set of quaternions (traditionally named __H) and the set of octonions |
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| 24 | (traditionally named __O), which possess interesting mathematical properties |
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| 25 | (chief among which is the fact that they are ['division algebras], |
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| 26 | ['i.e.] where the following property is true: if ['[^y]] is an element of that |
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| 27 | algebra and is [*not equal to zero], then ['[^yx = yx']], where ['[^x]] and ['[^x']] |
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| 28 | denote elements of that algebra, implies that ['[^x = x']]). |
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| 29 | Each member of the hierarchy is a super-set of the former. |
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| 30 | |
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| 31 | One of the most important aspects of quaternions is that they provide an |
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| 32 | efficient way to parameterize rotations in __R3 (the usual three-dimensional space) |
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| 33 | and __R4. |
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| 34 | |
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| 35 | In practical terms, a quaternion is simply a quadruple of real numbers __quadrulple, |
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| 36 | which we can write in the form __quat_formula, where ['[^i]] is the same object as for complex numbers, |
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| 37 | and ['[^j]] and ['[^k]] are distinct objects which play essentially the same kind of role as ['[^i]]. |
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| 38 | |
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| 39 | An addition and a multiplication is defined on the set of quaternions, |
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| 40 | which generalize their real and complex counterparts. The main novelty |
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| 41 | here is that [*the multiplication is not commutative] (i.e. there are |
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| 42 | quaternions ['[^x]] and ['[^y]] such that __not_equal). A good mnemotechnical way of remembering |
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| 43 | things is by using the formula ['[^i*i = j*j = k*k = -1]]. |
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| 44 | |
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| 45 | Quaternions (and their kin) are described in far more details in this |
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| 46 | other [@../../libs/math/quaternion/TQE.pdf document] |
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| 47 | (with [@../../libs/math/quaternion/TQE_EA.pdf errata and addenda]). |
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| 48 | |
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| 49 | Some traditional constructs, such as the exponential, carry over without |
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| 50 | too much change into the realms of quaternions, but other, such as taking |
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| 51 | a square root, do not. |
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| 52 | |
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| 53 | [endsect] |
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| 54 | |
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| 55 | [section Header File] |
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| 56 | |
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| 57 | The interface and implementation are both supplied by the header file |
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| 58 | [@../../boost/math/quaternion.hpp quaternion.hpp]. |
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| 59 | |
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| 60 | [endsect] |
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| 61 | |
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| 62 | [section Synopsis] |
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| 63 | |
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| 64 | namespace boost{ namespace math{ |
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| 65 | |
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| 66 | template<typename T> class ``[link boost_math.quaternions.template_class_quaternion quaternion]``; |
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| 67 | template<> class ``[link boost_math.quaternions.quaternion_specializations quaternion<float>]``; |
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| 68 | template<> class ``[link boost_math.quaternion_double quaternion<double>]``; |
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| 69 | template<> class ``[link boost_math.quaternion_long_double quaternion<long double>]``; |
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| 70 | |
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| 71 | // operators |
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| 72 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (T const & lhs, quaternion<T> const & rhs); |
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| 73 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (quaternion<T> const & lhs, T const & rhs); |
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| 74 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
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| 75 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
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| 76 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_addition_operators operator +]`` (quaternion<T> const & lhs, quaternion<T> const & rhs); |
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| 77 | |
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| 78 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (T const & lhs, quaternion<T> const & rhs); |
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| 79 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (quaternion<T> const & lhs, T const & rhs); |
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| 80 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
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| 81 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
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| 82 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_subtraction_operators operator -]`` (quaternion<T> const & lhs, quaternion<T> const & rhs); |
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| 83 | |
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| 84 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (T const & lhs, quaternion<T> const & rhs); |
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| 85 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (quaternion<T> const & lhs, T const & rhs); |
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| 86 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
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| 87 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
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| 88 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_multiplication_operators operator *]`` (quaternion<T> const & lhs, quaternion<T> const & rhs); |
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| 89 | |
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| 90 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (T const & lhs, quaternion<T> const & rhs); |
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| 91 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (quaternion<T> const & lhs, T const & rhs); |
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| 92 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
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| 93 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
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| 94 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.binary_division_operators operator /]`` (quaternion<T> const & lhs, quaternion<T> const & rhs); |
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| 95 | |
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| 96 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.unary_plus operator +]`` (quaternion<T> const & q); |
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| 97 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_non_member_operators.unary_minus operator -]`` (quaternion<T> const & q); |
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| 98 | |
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| 99 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (T const & lhs, quaternion<T> const & rhs); |
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| 100 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (quaternion<T> const & lhs, T const & rhs); |
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| 101 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
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| 102 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
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| 103 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.equality_operators operator ==]`` (quaternion<T> const & lhs, quaternion<T> const & rhs); |
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| 104 | |
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| 105 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (T const & lhs, quaternion<T> const & rhs); |
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| 106 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (quaternion<T> const & lhs, T const & rhs); |
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| 107 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
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| 108 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
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| 109 | template<typename T> bool ``[link boost_math.quaternions.quaternion_non_member_operators.inequality_operators operator !=]`` (quaternion<T> const & lhs, quaternion<T> const & rhs); |
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| 110 | |
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| 111 | template<typename T, typename charT, class traits> |
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| 112 | ::std::basic_istream<charT,traits>& ``[link boost_math.quaternions.quaternion_non_member_operators.stream_extractor operator >>]`` (::std::basic_istream<charT,traits> & is, quaternion<T> & q); |
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| 113 | |
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| 114 | template<typename T, typename charT, class traits> |
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| 115 | ::std::basic_ostream<charT,traits>& operator ``[link boost_math.quaternions.quaternion_non_member_operators.stream_inserter operator <<]`` (::std::basic_ostream<charT,traits> & os, quaternion<T> const & q); |
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| 116 | |
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| 117 | // values |
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| 118 | template<typename T> T ``[link boost_math.quaternions.quaternion_value_operations.real_and_unreal real]``(quaternion<T> const & q); |
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| 119 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_value_operations.real_and_unreal unreal]``(quaternion<T> const & q); |
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| 120 | |
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| 121 | template<typename T> T ``[link boost_math.quaternions.quaternion_value_operations.sup sup]``(quaternion<T> const & q); |
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| 122 | template<typename T> T ``[link boost_math.quaternions.quaternion_value_operations.l1 l1]``(quaternion<T> const & q); |
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| 123 | template<typename T> T ``[link boost_math.quaternions.quaternion_value_operations.abs abs]``(quaternion<T> const & q); |
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| 124 | template<typename T> T ``[link boost_math.quaternions.quaternion_value_operations.norm norm]``(quaternion<T>const & q); |
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| 125 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_value_operations.conj conj]``(quaternion<T> const & q); |
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| 126 | |
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| 127 | template<typename T> quaternion<T> ``[link boost_math.quaternions.creation_spherical spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2); |
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| 128 | template<typename T> quaternion<T> ``[link boost_math.quaternions.creation_semipolar semipolar]``(T const & rho, T const & alpha, T const & theta1, T const & theta2); |
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| 129 | template<typename T> quaternion<T> ``[link boost_math.quaternions.creation_multipolar multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2); |
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| 130 | template<typename T> quaternion<T> ``[link boost_math.quaternions.creation_cylindrospherical cylindrospherical]``(T const & t, T const & radius, T const & longitude, T const & latitude); |
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| 131 | template<typename T> quaternion<T> ``[link boost_math.quaternions.creation_cylindrical cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2); |
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| 132 | |
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| 133 | // transcendentals |
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| 134 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.exp exp]``(quaternion<T> const & q); |
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| 135 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.cos cos]``(quaternion<T> const & q); |
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| 136 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.sin sin]``(quaternion<T> const & q); |
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| 137 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.tan tan]``(quaternion<T> const & q); |
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| 138 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.cosh cosh]``(quaternion<T> const & q); |
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| 139 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.sinh sinh]``(quaternion<T> const & q); |
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| 140 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.tanh tanh]``(quaternion<T> const & q); |
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| 141 | template<typename T> quaternion<T> ``[link boost_math.quaternions.quaternion_transcendentals.pow pow]``(quaternion<T> const & q, int n); |
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| 142 | |
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| 143 | } // namespace math |
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| 144 | } // namespace boost |
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| 145 | |
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| 146 | [endsect] |
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| 147 | |
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| 148 | [section Template Class quaternion] |
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| 149 | |
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| 150 | namespace boost{ namespace math{ |
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| 151 | |
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| 152 | template<typename T> |
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| 153 | class quaternion |
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| 154 | { |
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| 155 | public: |
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| 156 | |
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| 157 | typedef T ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; |
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| 158 | |
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| 159 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); |
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| 160 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>()); |
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| 161 | template<typename X> |
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| 162 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion<X> const & a_recopier); |
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| 163 | |
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| 164 | T ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; |
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| 165 | quaternion<T> ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; |
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| 166 | T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; |
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| 167 | T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; |
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| 168 | T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; |
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| 169 | T ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; |
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| 170 | ::std::complex<T> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; |
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| 171 | ::std::complex<T> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; |
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| 172 | |
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| 173 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<T> const & a_affecter); |
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| 174 | template<typename X> |
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| 175 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<X> const & a_affecter); |
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| 176 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(T const & a_affecter); |
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| 177 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex<T> const & a_affecter); |
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| 178 | |
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| 179 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(T const & rhs); |
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| 180 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex<T> const & rhs); |
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| 181 | template<typename X> |
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| 182 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion<X> const & rhs); |
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| 183 | |
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| 184 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(T const & rhs); |
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| 185 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex<T> const & rhs); |
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| 186 | template<typename X> |
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| 187 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion<X> const & rhs); |
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| 188 | |
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| 189 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(T const & rhs); |
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| 190 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex<T> const & rhs); |
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| 191 | template<typename X> |
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| 192 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion<X> const & rhs); |
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| 193 | |
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| 194 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(T const & rhs); |
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| 195 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex<T> const & rhs); |
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| 196 | template<typename X> |
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| 197 | quaternion<T>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion<X> const & rhs); |
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| 198 | }; |
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| 199 | |
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| 200 | } // namespace math |
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| 201 | } // namespace boost |
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| 202 | |
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| 203 | [endsect] |
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| 204 | |
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| 205 | [section Quaternion Specializations] |
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| 206 | |
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| 207 | namespace boost{ namespace math{ |
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| 208 | |
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| 209 | template<> |
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| 210 | class quaternion<float> |
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| 211 | { |
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| 212 | public: |
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| 213 | typedef float ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; |
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| 214 | |
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| 215 | explicit ``[link boost_math.quaternions.quaternion_member_functions.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); |
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| 216 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>()); |
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| 217 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion<double> const & a_recopier); |
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| 218 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion<long double> const & a_recopier); |
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| 219 | |
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| 220 | float ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; |
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| 221 | quaternion<float> ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; |
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| 222 | float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; |
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| 223 | float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; |
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| 224 | float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; |
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| 225 | float ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; |
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| 226 | ::std::complex<float> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; |
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| 227 | ::std::complex<float> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; |
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| 228 | |
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| 229 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<float> const & a_affecter); |
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| 230 | template<typename X> |
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| 231 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<X> const & a_affecter); |
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| 232 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(float const & a_affecter); |
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| 233 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex<float> const & a_affecter); |
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| 234 | |
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| 235 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(float const & rhs); |
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| 236 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex<float> const & rhs); |
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| 237 | template<typename X> |
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| 238 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion<X> const & rhs); |
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| 239 | |
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| 240 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(float const & rhs); |
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| 241 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex<float> const & rhs); |
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| 242 | template<typename X> |
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| 243 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion<X> const & rhs); |
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| 244 | |
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| 245 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(float const & rhs); |
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| 246 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex<float> const & rhs); |
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| 247 | template<typename X> |
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| 248 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion<X> const & rhs); |
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| 249 | |
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| 250 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(float const & rhs); |
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| 251 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex<float> const & rhs); |
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| 252 | template<typename X> |
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| 253 | quaternion<float>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion<X> const & rhs); |
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| 254 | }; |
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| 255 | |
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| 256 | [#boost_math.quaternion_double] |
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| 257 | |
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| 258 | template<> |
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| 259 | class quaternion<double> |
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| 260 | { |
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| 261 | public: |
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| 262 | typedef double ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; |
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| 263 | |
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| 264 | explicit ``[link boost_math.quaternions.quaternion_member_functions.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); |
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| 265 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>()); |
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| 266 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion<float> const & a_recopier); |
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| 267 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion<long double> const & a_recopier); |
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| 268 | |
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| 269 | double ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; |
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| 270 | quaternion<double> ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; |
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| 271 | double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; |
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| 272 | double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; |
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| 273 | double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; |
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| 274 | double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; |
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| 275 | ::std::complex<double> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; |
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| 276 | ::std::complex<double> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; |
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| 277 | |
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| 278 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<double> const & a_affecter); |
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| 279 | template<typename X> |
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| 280 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<X> const & a_affecter); |
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| 281 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(double const & a_affecter); |
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| 282 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex<double> const & a_affecter); |
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| 283 | |
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| 284 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(double const & rhs); |
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| 285 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex<double> const & rhs); |
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| 286 | template<typename X> |
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| 287 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion<X> const & rhs); |
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| 288 | |
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| 289 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(double const & rhs); |
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| 290 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex<double> const & rhs); |
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| 291 | template<typename X> |
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| 292 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion<X> const & rhs); |
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| 293 | |
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| 294 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(double const & rhs); |
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| 295 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex<double> const & rhs); |
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| 296 | template<typename X> |
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| 297 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion<X> const & rhs); |
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| 298 | |
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| 299 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(double const & rhs); |
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| 300 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex<double> const & rhs); |
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| 301 | template<typename X> |
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| 302 | quaternion<double>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion<X> const & rhs); |
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| 303 | }; |
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| 304 | |
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| 305 | [#boost_math.quaternion_long_double] |
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| 306 | |
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| 307 | template<> |
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| 308 | class quaternion<long double> |
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| 309 | { |
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| 310 | public: |
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| 311 | typedef long double ``[link boost_math.quaternions.quaternion_member_typedefs value_type]``; |
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| 312 | |
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| 313 | explicit ``[link boost_math.quaternions.quaternion_member_functions.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); |
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| 314 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>()); |
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| 315 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion<float> const & a_recopier); |
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| 316 | explicit ``[link boost_math.quaternions.quaternion_member_functions.constructors quaternion]``(quaternion<double> const & a_recopier); |
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| 317 | |
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| 318 | long double ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts real]``() const; |
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| 319 | quaternion<long double> ``[link boost_math.quaternions.quaternion_member_functions.real_and_unreal_parts unreal]``() const; |
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| 320 | long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_1]``() const; |
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| 321 | long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_2]``() const; |
|---|
| 322 | long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_3]``() const; |
|---|
| 323 | long double ``[link boost_math.quaternions.quaternion_member_functions.individual_real_components R_component_4]``() const; |
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| 324 | ::std::complex<long double> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_1]``() const; |
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| 325 | ::std::complex<long double> ``[link boost_math.quaternions.quaternion_member_functions.individual_complex__components C_component_2]``() const; |
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| 326 | |
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| 327 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<long double> const & a_affecter); |
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| 328 | template<typename X> |
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| 329 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(quaternion<X> const & a_affecter); |
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| 330 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(long double const & a_affecter); |
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| 331 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.assignment_operators operator = ]``(::std::complex<long double> const & a_affecter); |
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| 332 | |
|---|
| 333 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(long double const & rhs); |
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| 334 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(::std::complex<long double> const & rhs); |
|---|
| 335 | template<typename X> |
|---|
| 336 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.addition_operators operator += ]``(quaternion<X> const & rhs); |
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| 337 | |
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| 338 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(long double const & rhs); |
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| 339 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(::std::complex<long double> const & rhs); |
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| 340 | template<typename X> |
|---|
| 341 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.subtraction_operators operator -= ]``(quaternion<X> const & rhs); |
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| 342 | |
|---|
| 343 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(long double const & rhs); |
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| 344 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(::std::complex<long double> const & rhs); |
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| 345 | template<typename X> |
|---|
| 346 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.multiplication_operators operator *= ]``(quaternion<X> const & rhs); |
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| 347 | |
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| 348 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(long double const & rhs); |
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| 349 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(::std::complex<long double> const & rhs); |
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| 350 | template<typename X> |
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| 351 | quaternion<long double>& ``[link boost_math.quaternions.quaternion_member_functions.division_operators operator /= ]``(quaternion<X> const & rhs); |
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| 352 | }; |
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| 353 | |
|---|
| 354 | } // namespace math |
|---|
| 355 | } // namespace boost |
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| 356 | |
|---|
| 357 | [endsect] |
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| 358 | |
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| 359 | [section Quaternion Member Typedefs] |
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| 360 | |
|---|
| 361 | [*value_type] |
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| 362 | |
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| 363 | Template version: |
|---|
| 364 | |
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| 365 | typedef T value_type; |
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| 366 | |
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| 367 | Float specialization version: |
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| 368 | |
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| 369 | typedef float value_type; |
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| 370 | |
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| 371 | Double specialization version: |
|---|
| 372 | |
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| 373 | typedef double value_type; |
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| 374 | |
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| 375 | Long double specialization version: |
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| 376 | |
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| 377 | typedef long double value_type; |
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| 378 | |
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| 379 | These provide easy acces to the type the template is built upon. |
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| 380 | |
|---|
| 381 | [endsect] |
|---|
| 382 | |
|---|
| 383 | [section Quaternion Member Functions] |
|---|
| 384 | [h3 Constructors] |
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| 385 | |
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| 386 | Template version: |
|---|
| 387 | |
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| 388 | explicit quaternion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T()); |
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| 389 | explicit quaternion(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>()); |
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| 390 | template<typename X> |
|---|
| 391 | explicit quaternion(quaternion<X> const & a_recopier); |
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| 392 | |
|---|
| 393 | Float specialization version: |
|---|
| 394 | |
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| 395 | 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); |
|---|
| 396 | explicit quaternion(::std::complex<float> const & z0,::std::complex<float> const & z1 = ::std::complex<float>()); |
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| 397 | explicit quaternion(quaternion<double> const & a_recopier); |
|---|
| 398 | explicit quaternion(quaternion<long double> const & a_recopier); |
|---|
| 399 | |
|---|
| 400 | Double specialization version: |
|---|
| 401 | |
|---|
| 402 | 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); |
|---|
| 403 | explicit quaternion(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>()); |
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| 404 | explicit quaternion(quaternion<float> const & a_recopier); |
|---|
| 405 | explicit quaternion(quaternion<long double> const & a_recopier); |
|---|
| 406 | |
|---|
| 407 | Long double specialization version: |
|---|
| 408 | |
|---|
| 409 | 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); |
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| 410 | explicit quaternion( ::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>()); |
|---|
| 411 | explicit quaternion(quaternion<float> const & a_recopier); |
|---|
| 412 | explicit quaternion(quaternion<double> const & a_recopier); |
|---|
| 413 | |
|---|
| 414 | A default constructor is provided for each form, which initializes |
|---|
| 415 | each component to the default values for their type |
|---|
| 416 | (i.e. zero for floating numbers). This constructor can also accept |
|---|
| 417 | one to four base type arguments. A constructor is also provided to |
|---|
| 418 | build quaternions from one or two complex numbers sharing the same |
|---|
| 419 | base type. The unspecialized template also sports a templarized copy |
|---|
| 420 | constructor, while the specialized forms have copy constructors |
|---|
| 421 | from the other two specializations, which are explicit when a risk of |
|---|
| 422 | precision loss exists. For the unspecialized form, the base type's |
|---|
| 423 | constructors must not throw. |
|---|
| 424 | |
|---|
| 425 | Destructors and untemplated copy constructors (from the same type) are |
|---|
| 426 | provided by the compiler. Converting copy constructors make use of a |
|---|
| 427 | templated helper function in a "detail" subnamespace. |
|---|
| 428 | |
|---|
| 429 | [h3 Other member functions] |
|---|
| 430 | [h4 Real and Unreal Parts] |
|---|
| 431 | |
|---|
| 432 | T real() const; |
|---|
| 433 | quaternion<T> unreal() const; |
|---|
| 434 | |
|---|
| 435 | Like complex number, quaternions do have a meaningful notion of "real part", |
|---|
| 436 | but unlike them there is no meaningful notion of "imaginary part". |
|---|
| 437 | Instead there is an "unreal part" which itself is a quaternion, |
|---|
| 438 | and usually nothing simpler (as opposed to the complex number case). |
|---|
| 439 | These are returned by the first two functions. |
|---|
| 440 | |
|---|
| 441 | [h4 Individual Real Components] |
|---|
| 442 | |
|---|
| 443 | T R_component_1() const; |
|---|
| 444 | T R_component_2() const; |
|---|
| 445 | T R_component_3() const; |
|---|
| 446 | T R_component_4() const; |
|---|
| 447 | |
|---|
| 448 | A quaternion having four real components, these are returned by these four |
|---|
| 449 | functions. Hence real and R_component_1 return the same value. |
|---|
| 450 | |
|---|
| 451 | [h4 Individual Complex Components] |
|---|
| 452 | |
|---|
| 453 | ::std::complex<T> C_component_1() const; |
|---|
| 454 | ::std::complex<T> C_component_2() const; |
|---|
| 455 | |
|---|
| 456 | A quaternion likewise has two complex components, and as we have seen above, |
|---|
| 457 | for any quaternion __quat_formula we also have __quat_complex_formula. These functions return them. |
|---|
| 458 | The real part of `q.C_component_1()` is the same as `q.real()`. |
|---|
| 459 | |
|---|
| 460 | [h3 Quaternion Member Operators] |
|---|
| 461 | [h4 Assignment Operators] |
|---|
| 462 | |
|---|
| 463 | quaternion<T>& operator = (quaternion<T> const & a_affecter); |
|---|
| 464 | template<typename X> |
|---|
| 465 | quaternion<T>& operator = (quaternion<X> const& a_affecter); |
|---|
| 466 | quaternion<T>& operator = (T const& a_affecter); |
|---|
| 467 | quaternion<T>& operator = (::std::complex<T> const& a_affecter); |
|---|
| 468 | |
|---|
| 469 | These perform the expected assignment, with type modification if necessary |
|---|
| 470 | (for instance, assigning from a base type will set the real part to that |
|---|
| 471 | value, and all other components to zero). For the unspecialized form, |
|---|
| 472 | the base type's assignment operators must not throw. |
|---|
| 473 | |
|---|
| 474 | [h4 Addition Operators] |
|---|
| 475 | |
|---|
| 476 | quaternion<T>& operator += (T const & rhs) |
|---|
| 477 | quaternion<T>& operator += (::std::complex<T> const & rhs); |
|---|
| 478 | template<typename X> |
|---|
| 479 | quaternion<T>& operator += (quaternion<X> const & rhs); |
|---|
| 480 | |
|---|
| 481 | These perform the mathematical operation `(*this)+rhs` and store the result in |
|---|
| 482 | `*this`. The unspecialized form has exception guards, which the specialized |
|---|
| 483 | forms do not, so as to insure exception safety. For the unspecialized form, |
|---|
| 484 | the base type's assignment operators must not throw. |
|---|
| 485 | |
|---|
| 486 | [h4 Subtraction Operators] |
|---|
| 487 | |
|---|
| 488 | quaternion<T>& operator -= (T const & rhs) |
|---|
| 489 | quaternion<T>& operator -= (::std::complex<T> const & rhs); |
|---|
| 490 | template<typename X> |
|---|
| 491 | quaternion<T>& operator -= (quaternion<X> const & rhs); |
|---|
| 492 | |
|---|
| 493 | These perform the mathematical operation `(*this)-rhs` and store the result |
|---|
| 494 | in `*this`. The unspecialized form has exception guards, which the |
|---|
| 495 | specialized forms do not, so as to insure exception safety. |
|---|
| 496 | For the unspecialized form, the base type's assignment operators |
|---|
| 497 | must not throw. |
|---|
| 498 | |
|---|
| 499 | [h4 Multiplication Operators] |
|---|
| 500 | |
|---|
| 501 | quaternion<T>& operator *= (T const & rhs) |
|---|
| 502 | quaternion<T>& operator *= (::std::complex<T> const & rhs); |
|---|
| 503 | template<typename X> |
|---|
| 504 | quaternion<T>& operator *= (quaternion<X> const & rhs); |
|---|
| 505 | |
|---|
| 506 | These perform the mathematical operation `(*this)*rhs` [*in this order] |
|---|
| 507 | (order is important as multiplication is not commutative for quaternions) |
|---|
| 508 | and store the result in `*this`. The unspecialized form has exception guards, |
|---|
| 509 | which the specialized forms do not, so as to insure exception safety. |
|---|
| 510 | For the unspecialized form, the base type's assignment operators must not throw. |
|---|
| 511 | |
|---|
| 512 | [h4 Division Operators] |
|---|
| 513 | |
|---|
| 514 | quaternion<T>& operator /= (T const & rhs) |
|---|
| 515 | quaternion<T>& operator /= (::std::complex<T> const & rhs); |
|---|
| 516 | template<typename X> |
|---|
| 517 | quaternion<T>& operator /= (quaternion<X> const & rhs); |
|---|
| 518 | |
|---|
| 519 | These perform the mathematical operation `(*this)*inverse_of(rhs)` [*in this |
|---|
| 520 | order] (order is important as multiplication is not commutative for quaternions) |
|---|
| 521 | and store the result in `*this`. The unspecialized form has exception guards, |
|---|
| 522 | which the specialized forms do not, so as to insure exception safety. |
|---|
| 523 | For the unspecialized form, the base type's assignment operators must not throw. |
|---|
| 524 | |
|---|
| 525 | [endsect] |
|---|
| 526 | [section Quaternion Non-Member Operators] |
|---|
| 527 | |
|---|
| 528 | [h4 Unary Plus] |
|---|
| 529 | |
|---|
| 530 | template<typename T> |
|---|
| 531 | quaternion<T> operator + (quaternion<T> const & q); |
|---|
| 532 | |
|---|
| 533 | This unary operator simply returns q. |
|---|
| 534 | |
|---|
| 535 | [h4 Unary Minus] |
|---|
| 536 | |
|---|
| 537 | template<typename T> |
|---|
| 538 | quaternion<T> operator - (quaternion<T> const & q); |
|---|
| 539 | |
|---|
| 540 | This unary operator returns the opposite of q. |
|---|
| 541 | |
|---|
| 542 | [h4 Binary Addition Operators] |
|---|
| 543 | |
|---|
| 544 | template<typename T> quaternion<T> operator + (T const & lhs, quaternion<T> const & rhs); |
|---|
| 545 | template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, T const & rhs); |
|---|
| 546 | template<typename T> quaternion<T> operator + (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 547 | template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
|---|
| 548 | template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 549 | |
|---|
| 550 | These operators return `quaternion<T>(lhs) += rhs`. |
|---|
| 551 | |
|---|
| 552 | [h4 Binary Subtraction Operators] |
|---|
| 553 | |
|---|
| 554 | template<typename T> quaternion<T> operator - (T const & lhs, quaternion<T> const & rhs); |
|---|
| 555 | template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, T const & rhs); |
|---|
| 556 | template<typename T> quaternion<T> operator - (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 557 | template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
|---|
| 558 | template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 559 | |
|---|
| 560 | These operators return `quaternion<T>(lhs) -= rhs`. |
|---|
| 561 | |
|---|
| 562 | [h4 Binary Multiplication Operators] |
|---|
| 563 | |
|---|
| 564 | template<typename T> quaternion<T> operator * (T const & lhs, quaternion<T> const & rhs); |
|---|
| 565 | template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, T const & rhs); |
|---|
| 566 | template<typename T> quaternion<T> operator * (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 567 | template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
|---|
| 568 | template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 569 | |
|---|
| 570 | These operators return `quaternion<T>(lhs) *= rhs`. |
|---|
| 571 | |
|---|
| 572 | [h4 Binary Division Operators] |
|---|
| 573 | |
|---|
| 574 | template<typename T> quaternion<T> operator / (T const & lhs, quaternion<T> const & rhs); |
|---|
| 575 | template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, T const & rhs); |
|---|
| 576 | template<typename T> quaternion<T> operator / (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 577 | template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
|---|
| 578 | template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 579 | |
|---|
| 580 | These operators return `quaternion<T>(lhs) /= rhs`. It is of course still an |
|---|
| 581 | error to divide by zero... |
|---|
| 582 | |
|---|
| 583 | [h4 Equality Operators] |
|---|
| 584 | |
|---|
| 585 | template<typename T> bool operator == (T const & lhs, quaternion<T> const & rhs); |
|---|
| 586 | template<typename T> bool operator == (quaternion<T> const & lhs, T const & rhs); |
|---|
| 587 | template<typename T> bool operator == (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 588 | template<typename T> bool operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
|---|
| 589 | template<typename T> bool operator == (quaternion<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 590 | |
|---|
| 591 | These return true if and only if the four components of `quaternion<T>(lhs)` |
|---|
| 592 | are equal to their counterparts in `quaternion<T>(rhs)`. As with any |
|---|
| 593 | floating-type entity, this is essentially meaningless. |
|---|
| 594 | |
|---|
| 595 | [h4 Inequality Operators] |
|---|
| 596 | |
|---|
| 597 | template<typename T> bool operator != (T const & lhs, quaternion<T> const & rhs); |
|---|
| 598 | template<typename T> bool operator != (quaternion<T> const & lhs, T const & rhs); |
|---|
| 599 | template<typename T> bool operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 600 | template<typename T> bool operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs); |
|---|
| 601 | template<typename T> bool operator != (quaternion<T> const & lhs, quaternion<T> const & rhs); |
|---|
| 602 | |
|---|
| 603 | These return true if and only if `quaternion<T>(lhs) == quaternion<T>(rhs)` is |
|---|
| 604 | false. As with any floating-type entity, this is essentially meaningless. |
|---|
| 605 | |
|---|
| 606 | [h4 Stream Extractor] |
|---|
| 607 | |
|---|
| 608 | template<typename T, typename charT, class traits> |
|---|
| 609 | ::std::basic_istream<charT,traits>& operator >> (::std::basic_istream<charT,traits> & is, quaternion<T> & q); |
|---|
| 610 | |
|---|
| 611 | Extracts a quaternion q of one of the following forms |
|---|
| 612 | (with a, b, c and d of type `T`): |
|---|
| 613 | |
|---|
| 614 | [^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))] |
|---|
| 615 | |
|---|
| 616 | The input values must be convertible to `T`. If bad input is encountered, |
|---|
| 617 | calls `is.setstate(ios::failbit)` (which may throw ios::failure (27.4.5.3)). |
|---|
| 618 | |
|---|
| 619 | [*Returns:] `is`. |
|---|
| 620 | |
|---|
| 621 | The rationale for the list of accepted formats is that either we have a |
|---|
| 622 | list of up to four reals, or else we have a couple of complex numbers, |
|---|
| 623 | and in that case if it formated as a proper complex number, then it should |
|---|
| 624 | be accepted. Thus potential ambiguities are lifted (for instance (a,b) is |
|---|
| 625 | (a,b,0,0) and not (a,0,b,0), i.e. it is parsed as a list of two real numbers |
|---|
| 626 | and not two complex numbers which happen to have imaginary parts equal to zero). |
|---|
| 627 | |
|---|
| 628 | [h4 Stream Inserter] |
|---|
| 629 | |
|---|
| 630 | template<typename T, typename charT, class traits> |
|---|
| 631 | ::std::basic_ostream<charT,traits>& operator << (::std::basic_ostream<charT,traits> & os, quaternion<T> const & q); |
|---|
| 632 | |
|---|
| 633 | Inserts the quaternion q onto the stream `os` as if it were implemented as follows: |
|---|
| 634 | |
|---|
| 635 | template<typename T, typename charT, class traits> |
|---|
| 636 | ::std::basic_ostream<charT,traits>& operator << ( |
|---|
| 637 | ::std::basic_ostream<charT,traits> & os, |
|---|
| 638 | quaternion<T> const & q) |
|---|
| 639 | { |
|---|
| 640 | ::std::basic_ostringstream<charT,traits> s; |
|---|
| 641 | |
|---|
| 642 | s.flags(os.flags()); |
|---|
| 643 | s.imbue(os.getloc()); |
|---|
| 644 | s.precision(os.precision()); |
|---|
| 645 | |
|---|
| 646 | s << '(' << q.R_component_1() << ',' |
|---|
| 647 | << q.R_component_2() << ',' |
|---|
| 648 | << q.R_component_3() << ',' |
|---|
| 649 | << q.R_component_4() << ')'; |
|---|
| 650 | |
|---|
| 651 | return os << s.str(); |
|---|
| 652 | } |
|---|
| 653 | |
|---|
| 654 | [endsect] |
|---|
| 655 | |
|---|
| 656 | [section Quaternion Value Operations] |
|---|
| 657 | |
|---|
| 658 | [h4 real and unreal] |
|---|
| 659 | |
|---|
| 660 | template<typename T> T real(quaternion<T> const & q); |
|---|
| 661 | template<typename T> quaternion<T> unreal(quaternion<T> const & q); |
|---|
| 662 | |
|---|
| 663 | These return `q.real()` and `q.unreal()` respectively. |
|---|
| 664 | |
|---|
| 665 | [h4 conj] |
|---|
| 666 | |
|---|
| 667 | template<typename T> quaternion<T> conj(quaternion<T> const & q); |
|---|
| 668 | |
|---|
| 669 | This returns the conjugate of the quaternion. |
|---|
| 670 | |
|---|
| 671 | [h4 sup] |
|---|
| 672 | |
|---|
| 673 | template<typename T> T sup(quaternion<T> const & q); |
|---|
| 674 | |
|---|
| 675 | This return the sup norm (the greatest among |
|---|
| 676 | `abs(q.R_component_1())...abs(q.R_component_4()))` of the quaternion. |
|---|
| 677 | |
|---|
| 678 | [h4 l1] |
|---|
| 679 | |
|---|
| 680 | template<typename T> T l1(quaternion<T> const & q); |
|---|
| 681 | |
|---|
| 682 | This return the l1 norm `(abs(q.R_component_1())+...+abs(q.R_component_4()))` |
|---|
| 683 | of the quaternion. |
|---|
| 684 | |
|---|
| 685 | [h4 abs] |
|---|
| 686 | |
|---|
| 687 | template<typename T> T abs(quaternion<T> const & q); |
|---|
| 688 | |
|---|
| 689 | This return the magnitude (Euclidian norm) of the quaternion. |
|---|
| 690 | |
|---|
| 691 | [h4 norm] |
|---|
| 692 | |
|---|
| 693 | template<typename T> T norm(quaternion<T>const & q); |
|---|
| 694 | |
|---|
| 695 | This return the (Cayley) norm of the quaternion. |
|---|
| 696 | The term "norm" might be confusing, as most people associate it with the |
|---|
| 697 | Euclidian norm (and quadratic functionals). For this version of |
|---|
| 698 | (the mathematical objects known as) quaternions, the Euclidian norm |
|---|
| 699 | (also known as magnitude) is the square root of the Cayley norm. |
|---|
| 700 | |
|---|
| 701 | [endsect] |
|---|
| 702 | |
|---|
| 703 | [section Quaternion Creation Functions] |
|---|
| 704 | |
|---|
| 705 | template<typename T> quaternion<T> spherical(T const & rho, T const & theta, T const & phi1, T const & phi2); |
|---|
| 706 | template<typename T> quaternion<T> semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2); |
|---|
| 707 | template<typename T> quaternion<T> multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2); |
|---|
| 708 | template<typename T> quaternion<T> cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude); |
|---|
| 709 | template<typename T> quaternion<T> cylindrical(T const & r, T const & angle, T const & h1, T const & h2); |
|---|
| 710 | |
|---|
| 711 | These build quaternions in a way similar to the way polar builds complex |
|---|
| 712 | numbers, as there is no strict equivalent to polar coordinates for quaternions. |
|---|
| 713 | |
|---|
| 714 | [#boost_math.quaternions.creation_spherical] `spherical` is a simple transposition of `polar`, it takes as inputs |
|---|
| 715 | a (positive) magnitude and a point on the hypersphere, given by three angles. |
|---|
| 716 | The first of these, `theta` has a natural range of `-pi` to `+pi`, and the other |
|---|
| 717 | two have natural ranges of `-pi/2` to `+pi/2` (as is the case with the usual |
|---|
| 718 | spherical coordinates in __R3). Due to the many symmetries and periodicities, |
|---|
| 719 | nothing untoward happens if the magnitude is negative or the angles are |
|---|
| 720 | outside their natural ranges. The expected degeneracies (a magnitude of |
|---|
| 721 | zero ignores the angles settings...) do happen however. |
|---|
| 722 | |
|---|
| 723 | [#boost_math.quaternions.creation_cylindrical] `cylindrical` is likewise a simple transposition of the usual |
|---|
| 724 | cylindrical coordinates in __R3, which in turn is another derivative of |
|---|
| 725 | planar polar coordinates. The first two inputs are the polar coordinates of |
|---|
| 726 | the first __C component of the quaternion. The third and fourth inputs |
|---|
| 727 | are placed into the third and fourth __R components of the quaternion, |
|---|
| 728 | respectively. |
|---|
| 729 | |
|---|
| 730 | [#boost_math.quaternions.creation_multipolar] `multipolar` is yet another simple generalization of polar coordinates. |
|---|
| 731 | This time, both __C components of the quaternion are given in polar coordinates. |
|---|
| 732 | |
|---|
| 733 | [#boost_math.quaternions.creation_cylindrospherical] `cylindrospherical` is specific to quaternions. It is often interesting to |
|---|
| 734 | consider __H as the cartesian product of __R by __R3 (the quaternionic |
|---|
| 735 | multiplication as then a special form, as given here). This function |
|---|
| 736 | therefore builds a quaternion from this representation, with the __R3 |
|---|
| 737 | component given in usual __R3 spherical coordinates. |
|---|
| 738 | |
|---|
| 739 | [#boost_math.quaternions.creation_semipolar] `semipolar` is another generator which is specific to quaternions. |
|---|
| 740 | It takes as a first input the magnitude of the quaternion, as a |
|---|
| 741 | second input an angle in the range `0` to `+pi/2` such that magnitudes |
|---|
| 742 | of the first two __C components of the quaternion are the product of the |
|---|
| 743 | first input and the sine and cosine of this angle, respectively, and finally |
|---|
| 744 | as third and fourth inputs angles in the range `-pi/2` to `+pi/2` which |
|---|
| 745 | represent the arguments of the first and second __C components of |
|---|
| 746 | the quaternion, respectively. As usual, nothing untoward happens if |
|---|
| 747 | what should be magnitudes are negative numbers or angles are out of their |
|---|
| 748 | natural ranges, as symmetries and periodicities kick in. |
|---|
| 749 | |
|---|
| 750 | In this version of our implementation of quaternions, there is no |
|---|
| 751 | analogue of the complex value operation `arg` as the situation is |
|---|
| 752 | somewhat more complicated. Unit quaternions are linked both to |
|---|
| 753 | rotations in __R3 and in __R4, and the correspondences are not too complicated, |
|---|
| 754 | but there is currently a lack of standard (de facto or de jure) matrix |
|---|
| 755 | library with which the conversions could work. This should be remedied in |
|---|
| 756 | a further revision. In the mean time, an example of how this could be |
|---|
| 757 | done is presented here for |
|---|
| 758 | [@../../libs/math/quaternion/HSO3.hpp __R3], and here for |
|---|
| 759 | [@../../libs/math/quaternion/HSO4.hpp __R4] |
|---|
| 760 | ([@../../libs/math/quaternion/HSO3SO4.cpp example test file]). |
|---|
| 761 | |
|---|
| 762 | [endsect] |
|---|
| 763 | |
|---|
| 764 | [section Quaternion Transcendentals] |
|---|
| 765 | |
|---|
| 766 | There is no `log` or `sqrt` provided for quaternions in this implementation, |
|---|
| 767 | and `pow` is likewise restricted to integral powers of the exponent. |
|---|
| 768 | There are several reasons to this: on the one hand, the equivalent of |
|---|
| 769 | analytic continuation for quaternions ("branch cuts") remains to be |
|---|
| 770 | investigated thoroughly (by me, at any rate...), and we wish to avoid the |
|---|
| 771 | nonsense introduced in the standard by exponentiations of complexes by |
|---|
| 772 | complexes (which is well defined, but not in the standard...). |
|---|
| 773 | Talking of nonsense, saying that `pow(0,0)` is "implementation defined" is just |
|---|
| 774 | plain brain-dead... |
|---|
| 775 | |
|---|
| 776 | We do, however provide several transcendentals, chief among which is the |
|---|
| 777 | exponential. This author claims the complete proof of the "closed formula" |
|---|
| 778 | as his own, as well as its independant invention (there are claims to prior |
|---|
| 779 | invention of the formula, such as one by Professor Shoemake, and it is |
|---|
| 780 | possible that the formula had been known a couple of centuries back, but in |
|---|
| 781 | absence of bibliographical reference, the matter is pending, awaiting further |
|---|
| 782 | investigation; on the other hand, the definition and existence of the |
|---|
| 783 | exponential on the quaternions, is of course a fact known for a very long time). |
|---|
| 784 | Basically, any converging power series with real coefficients which allows for a |
|---|
| 785 | closed formula in __C can be transposed to __H. More transcendentals of this |
|---|
| 786 | type could be added in a further revision upon request. It should be |
|---|
| 787 | noted that it is these functions which force the dependency upon the |
|---|
| 788 | [@../../boost/math/special_functions/sinc.hpp boost/math/special_functions/sinc.hpp] and the |
|---|
| 789 | [@../../boost/math/special_functions/sinhc.hpp boost/math/special_functions/sinhc.hpp] headers. |
|---|
| 790 | |
|---|
| 791 | [h4 exp] |
|---|
| 792 | |
|---|
| 793 | template<typename T> quaternion<T> exp(quaternion<T> const & q); |
|---|
| 794 | |
|---|
| 795 | Computes the exponential of the quaternion. |
|---|
| 796 | |
|---|
| 797 | [h4 cos] |
|---|
| 798 | |
|---|
| 799 | template<typename T> quaternion<T> cos(quaternion<T> const & q); |
|---|
| 800 | |
|---|
| 801 | Computes the cosine of the quaternion |
|---|
| 802 | |
|---|
| 803 | [h4 sin] |
|---|
| 804 | |
|---|
| 805 | template<typename T> quaternion<T> sin(quaternion<T> const & q); |
|---|
| 806 | |
|---|
| 807 | Computes the sine of the quaternion. |
|---|
| 808 | |
|---|
| 809 | [h4 tan] |
|---|
| 810 | |
|---|
| 811 | template<typename T> quaternion<T> tan(quaternion<T> const & q); |
|---|
| 812 | |
|---|
| 813 | Computes the tangent of the quaternion. |
|---|
| 814 | |
|---|
| 815 | [h4 cosh] |
|---|
| 816 | |
|---|
| 817 | template<typename T> quaternion<T> cosh(quaternion<T> const & q); |
|---|
| 818 | |
|---|
| 819 | Computes the hyperbolic cosine of the quaternion. |
|---|
| 820 | |
|---|
| 821 | [h4 sinh] |
|---|
| 822 | |
|---|
| 823 | template<typename T> quaternion<T> sinh(quaternion<T> const & q); |
|---|
| 824 | |
|---|
| 825 | Computes the hyperbolic sine of the quaternion. |
|---|
| 826 | |
|---|
| 827 | [h4 tanh] |
|---|
| 828 | |
|---|
| 829 | template<typename T> quaternion<T> tanh(quaternion<T> const & q); |
|---|
| 830 | |
|---|
| 831 | Computes the hyperbolic tangent of the quaternion. |
|---|
| 832 | |
|---|
| 833 | [h4 pow] |
|---|
| 834 | |
|---|
| 835 | template<typename T> quaternion<T> pow(quaternion<T> const & q, int n); |
|---|
| 836 | |
|---|
| 837 | Computes the n-th power of the quaternion q. |
|---|
| 838 | |
|---|
| 839 | [endsect] |
|---|
| 840 | |
|---|
| 841 | [section Test Program] |
|---|
| 842 | |
|---|
| 843 | The [@../../libs/math/quaternion/quaternion_test.cpp quaternion_test.cpp] |
|---|
| 844 | test program tests quaternions specializations for float, double and long double |
|---|
| 845 | ([@../../libs/math/quaternion/output.txt sample output], with message output |
|---|
| 846 | enabled). |
|---|
| 847 | |
|---|
| 848 | If you define the symbol BOOST_QUATERNION_TEST_VERBOSE, you will get |
|---|
| 849 | additional output ([@../../libs/math/quaternion/output_more.txt verbose output]); |
|---|
| 850 | this will only be helpfull if you enable message output at the same time, |
|---|
| 851 | of course (by uncommenting the relevant line in the test or by adding |
|---|
| 852 | [^--log_level=messages] to your command line,...). In that case, and if you |
|---|
| 853 | are running interactively, you may in addition define the symbol |
|---|
| 854 | BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to interactively test the input |
|---|
| 855 | operator with input of your choice from the standard input |
|---|
| 856 | (instead of hard-coding it in the test). |
|---|
| 857 | |
|---|
| 858 | [endsect] |
|---|
| 859 | |
|---|
| 860 | [section Acknowledgements] |
|---|
| 861 | |
|---|
| 862 | The mathematical text has been typeset with |
|---|
| 863 | [@http://www.nisus-soft.com/ Nisus Writer]. Jens Maurer has helped with |
|---|
| 864 | portability and standard adherence, and was the Review Manager |
|---|
| 865 | for this library. More acknowledgements in the History section. |
|---|
| 866 | Thank you to all who contributed to the discution about this library. |
|---|
| 867 | |
|---|
| 868 | [endsect] |
|---|
| 869 | |
|---|
| 870 | [section History] |
|---|
| 871 | |
|---|
| 872 | * 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. |
|---|
| 873 | * 1.5.7 - 24/02/2003: transitionned to the unit test framework; <boost/config.hpp> now included by the library header (rather than the test files). |
|---|
| 874 | * 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com). |
|---|
| 875 | * 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option. |
|---|
| 876 | * 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. |
|---|
| 877 | * 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu). |
|---|
| 878 | * 1.5.2 - 07/07/2001: introduced namespace math. |
|---|
| 879 | * 1.5.1 - 07/06/2001: (end of Boost review) now includes <boost/math/special_functions/sinc.hpp> and <boost/math/special_functions/sinhc.hpp> instead of <boost/special_functions.hpp>; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code. |
|---|
| 880 | * 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. |
|---|
| 881 | * 1.4.0 - 09/01/2001: added tan and tanh. |
|---|
| 882 | * 1.3.1 - 08/01/2001: cosmetic fixes. |
|---|
| 883 | * 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) algorithm. |
|---|
| 884 | * 1.2.0 - 25/05/2000: fixed the division operators and output; changed many signatures. |
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| 885 | * 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh. |
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| 886 | * 1.0.0 - 10/08/1999: first public version. |
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| 887 | |
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| 888 | [endsect] |
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| 889 | [section To Do] |
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| 890 | |
|---|
| 891 | * Improve testing. |
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| 892 | * Rewrite input operatore using Spirit (creates a dependency). |
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| 893 | * Put in place an Expression Template mechanism (perhaps borrowing from uBlas). |
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| 894 | * Use uBlas for the link with rotations (and move from the |
|---|
| 895 | [@../../libs/math/quaternion/HSO3SO4.cpp example] |
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| 896 | implementation to an efficient one). |
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| 897 | |
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| 898 | [endsect] |
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| 899 | [endsect] |
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