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Quaternions

Overview
Header File
Synopsis
Template Class quaternion
Quaternion Specializations
Quaternion Member Typedefs
Quaternion Member Functions
Quaternion Non-Member Operators
Quaternion Value Operations
Quaternion Creation Functions
Quaternion Transcendentals
Test Program
Acknowledgements
History
To Do

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 ( α,β,γ,δ ), which we can write in the form q = α + βi + γj + δk , 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 xy ≠ yx). 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 document (with 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.

Header File

The interface and implementation are both supplied by the header file quaternion.hpp.

Synopsis

namespace boost{ namespace math{

template<typename T> class quaternion;
template<>           class quaternion<float>;
template<>           class quaternion<double>; 
template<>           class quaternion<long double>; 

// operators
template<typename T> quaternion<T> operator + (T const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, T const & rhs);
template<typename T> quaternion<T> operator + (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> quaternion<T> operator + (quaternion<T> const & lhs, quaternion<T> const & rhs);

template<typename T> quaternion<T> operator - (T const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, T const & rhs);
template<typename T> quaternion<T> operator - (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> quaternion<T> operator - (quaternion<T> const & lhs, quaternion<T> const & rhs);

template<typename T> quaternion<T> operator * (T const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, T const & rhs);
template<typename T> quaternion<T> operator * (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> quaternion<T> operator * (quaternion<T> const & lhs, quaternion<T> const & rhs);

template<typename T> quaternion<T> operator / (T const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, T const & rhs);
template<typename T> quaternion<T> operator / (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> quaternion<T> operator / (quaternion<T> const & lhs, quaternion<T> const & rhs); 

template<typename T> quaternion<T> operator + (quaternion<T> const & q);
template<typename T> quaternion<T> operator - (quaternion<T> const & q); 

template<typename T> bool operator == (T const & lhs, quaternion<T> const & rhs);
template<typename T> bool operator == (quaternion<T> const & lhs, T const & rhs);
template<typename T> bool operator == (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T> bool operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> bool operator == (quaternion<T> const & lhs, quaternion<T> const & rhs);

template<typename T> bool operator != (T const & lhs, quaternion<T> const & rhs);
template<typename T> bool operator != (quaternion<T> const & lhs, T const & rhs);
template<typename T> bool operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T> bool operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> bool operator != (quaternion<T> const & lhs, quaternion<T> const & rhs); 

template<typename T, typename charT, class traits>
::std::basic_istream<charT,traits>& operator >> (::std::basic_istream<charT,traits> & is, quaternion<T> & q);

template<typename T, typename charT, class traits>
::std::basic_ostream<charT,traits>& operator operator << (::std::basic_ostream<charT,traits> & os, quaternion<T> const & q);

// values
template<typename T>	T              real(quaternion<T> const & q);
template<typename T>	quaternion<T>  unreal(quaternion<T> const & q);

template<typename T>	T              sup(quaternion<T> const & q);
template<typename T>	T              l1(quaternion<T> const & q);
template<typename T>	T              abs(quaternion<T> const & q);
template<typename T>	T              norm(quaternion<T>const  & q);
template<typename T>	quaternion<T>  conj(quaternion<T> const & q);

template<typename T>	quaternion<T>  spherical(T const & rho, T const & theta, T const & phi1, T const & phi2);
template<typename T>	quaternion<T>  semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2);
template<typename T>	quaternion<T>  multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2);
template<typename T>	quaternion<T>  cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude);
template<typename T>	quaternion<T>  cylindrical(T const & r, T const & angle, T const & h1, T const & h2);

// transcendentals
template<typename T>	quaternion<T>  exp(quaternion<T> const & q);
template<typename T>	quaternion<T>  cos(quaternion<T> const & q);
template<typename T>	quaternion<T>  sin(quaternion<T> const & q);
template<typename T>	quaternion<T>  tan(quaternion<T> const & q);
template<typename T>	quaternion<T>  cosh(quaternion<T> const & q);
template<typename T>	quaternion<T>  sinh(quaternion<T> const & q);
template<typename T>	quaternion<T>  tanh(quaternion<T> const & q);
template<typename T>	quaternion<T>  pow(quaternion<T> const & q, int n);

} // namespace math
} // namespace boost

Template Class quaternion

namespace boost{ namespace math{

template<typename T>
class quaternion
{
public:

   typedef T value_type;

   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<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>());
   template<typename X> 
   explicit quaternion(quaternion<X> const & a_recopier);

   T                  real() const;
   quaternion<T>      unreal() const;
   T                  R_component_1() const;
   T                  R_component_2() const;
   T                  R_component_3() const;
   T                  R_component_4() const;
   ::std::complex<T>  C_component_1() const;
   ::std::complex<T>  C_component_2() const;

   quaternion<T>&     operator = (quaternion<T> const  & a_affecter);
   template<typename X>	
   quaternion<T>&     operator = (quaternion<X> const  & a_affecter);
   quaternion<T>&     operator = (T const  & a_affecter);
   quaternion<T>&     operator = (::std::complex<T> const & a_affecter);

   quaternion<T>&     operator += (T const & rhs);
   quaternion<T>&     operator += (::std::complex<T> const & rhs);
   template<typename X>
   quaternion<T>&     operator += (quaternion<X> const & rhs);

   quaternion<T>&     operator -= (T const & rhs);
   quaternion<T>&     operator -= (::std::complex<T> const & rhs);
   template<typename X>
   quaternion<T>&     operator -= (quaternion<X> const & rhs);

   quaternion<T>&     operator *= (T const & rhs);
   quaternion<T>&     operator *= (::std::complex<T> const & rhs);
   template<typename X>
   quaternion<T>&     operator *= (quaternion<X> const & rhs);

   quaternion<T>&     operator /= (T const & rhs);
   quaternion<T>&     operator /= (::std::complex<T> const & rhs);
   template<typename X>
   quaternion<T>&     operator /= (quaternion<X> const & rhs);
};

} // namespace math
} // namespace boost

Quaternion Specializations

namespace boost{ namespace math{

template<>
class quaternion<float>
{
public:
   typedef float value_type;
	
   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<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>());
   explicit quaternion(quaternion<double> const & a_recopier);
   explicit quaternion(quaternion<long double> const & a_recopier);
	
   float                  real() const;
   quaternion<float>      unreal() const;
   float                  R_component_1() const;
   float                  R_component_2() const;
   float                  R_component_3() const;
   float                  R_component_4() const;
   ::std::complex<float>  C_component_1() const;
   ::std::complex<float>  C_component_2() const;

   quaternion<float>&     operator = (quaternion<float> const  & a_affecter);
   template<typename X>	
   quaternion<float>&     operator = (quaternion<X> const  & a_affecter);
   quaternion<float>&     operator = (float const  & a_affecter);
   quaternion<float>&     operator = (::std::complex<float> const & a_affecter);

   quaternion<float>&     operator += (float const & rhs);
   quaternion<float>&     operator += (::std::complex<float> const & rhs);
   template<typename X>
   quaternion<float>&     operator += (quaternion<X> const & rhs);

   quaternion<float>&     operator -= (float const & rhs);
   quaternion<float>&     operator -= (::std::complex<float> const & rhs);
   template<typename X>
   quaternion<float>&     operator -= (quaternion<X> const & rhs);

   quaternion<float>&     operator *= (float const & rhs);
   quaternion<float>&     operator *= (::std::complex<float> const & rhs);
   template<typename X>
   quaternion<float>&     operator *= (quaternion<X> const & rhs);

   quaternion<float>&     operator /= (float const & rhs);
   quaternion<float>&     operator /= (::std::complex<float> const & rhs);
   template<typename X>
   quaternion<float>&     operator /= (quaternion<X> const & rhs);
};

template<>
class quaternion<double>
{
public:
   typedef double value_type;
	
   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<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>());
   explicit quaternion(quaternion<float> const & a_recopier);
   explicit quaternion(quaternion<long double> const & a_recopier);
	
   double                  real() const;
   quaternion<double>      unreal() const;
   double                  R_component_1() const;
   double                  R_component_2() const;
   double                  R_component_3() const;
   double                  R_component_4() const;
   ::std::complex<double>  C_component_1() const;
   ::std::complex<double>  C_component_2() const;

   quaternion<double>&     operator = (quaternion<double> const  & a_affecter);
   template<typename X>	
   quaternion<double>&     operator = (quaternion<X> const  & a_affecter);
   quaternion<double>&     operator = (double const  & a_affecter);
   quaternion<double>&     operator = (::std::complex<double> const & a_affecter);

   quaternion<double>&     operator += (double const & rhs);
   quaternion<double>&     operator += (::std::complex<double> const & rhs);
   template<typename X>
   quaternion<double>&     operator += (quaternion<X> const & rhs);

   quaternion<double>&     operator -= (double const & rhs);
   quaternion<double>&     operator -= (::std::complex<double> const & rhs);
   template<typename X>
   quaternion<double>&     operator -= (quaternion<X> const & rhs);

   quaternion<double>&     operator *= (double const & rhs);
   quaternion<double>&     operator *= (::std::complex<double> const & rhs);
   template<typename X>
   quaternion<double>&     operator *= (quaternion<X> const & rhs);

   quaternion<double>&     operator /= (double const & rhs);
   quaternion<double>&     operator /= (::std::complex<double> const & rhs);
   template<typename X>
   quaternion<double>&     operator /= (quaternion<X> const & rhs);
};

template<>
class quaternion<long double>
{
public:
   typedef long double value_type;
	
   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<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>());
   explicit quaternion(quaternion<float> const & a_recopier);
   explicit quaternion(quaternion<double> const & a_recopier);
	
   long double                  real() const;
   quaternion<long double>      unreal() const;
   long double                  R_component_1() const;
   long double                  R_component_2() const;
   long double                  R_component_3() const;
   long double                  R_component_4() const;
   ::std::complex<long double>  C_component_1() const;
   ::std::complex<long double>  C_component_2() const;

   quaternion<long double>&     operator = (quaternion<long double> const  & a_affecter);
   template<typename X>	
   quaternion<long double>&     operator = (quaternion<X> const  & a_affecter);
   quaternion<long double>&     operator = (long double const  & a_affecter);
   quaternion<long double>&     operator = (::std::complex<long double> const & a_affecter);

   quaternion<long double>&     operator += (long double const & rhs);
   quaternion<long double>&     operator += (::std::complex<long double> const & rhs);
   template<typename X>
   quaternion<long double>&     operator += (quaternion<X> const & rhs);

   quaternion<long double>&     operator -= (long double const & rhs);
   quaternion<long double>&     operator -= (::std::complex<long double> const & rhs);
   template<typename X>
   quaternion<long double>&     operator -= (quaternion<X> const & rhs);

   quaternion<long double>&     operator *= (long double const & rhs);
   quaternion<long double>&     operator *= (::std::complex<long double> const & rhs);
   template<typename X>
   quaternion<long double>&     operator *= (quaternion<X> const & rhs);

   quaternion<long double>&     operator /= (long double const & rhs);
   quaternion<long double>&     operator /= (::std::complex<long double> const & rhs);
   template<typename X>
   quaternion<long double>&     operator /= (quaternion<X> const & rhs);
};

} // namespace math
} // namespace boost

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 acces to the type the template is built upon.

Quaternion Member Functions

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<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>());
template<typename X>
explicit quaternion(quaternion<X> 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<float> const & z0,::std::complex<float> const & z1 = ::std::complex<float>());
explicit quaternion(quaternion<double> const & a_recopier); 
explicit quaternion(quaternion<long double> 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<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>());
explicit quaternion(quaternion<float> const & a_recopier);
explicit quaternion(quaternion<long double> 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<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>());
explicit quaternion(quaternion<float> const & a_recopier);
explicit quaternion(quaternion<double> const & a_recopier);

A default constructor is provided for each form, which initializes each component to the default values for their type (i.e. zero for floating numbers). This constructor can also accept one to 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.

Other member functions

Real and Unreal Parts

T             real() const;
quaternion<T> 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.

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.

Individual Complex Components

::std::complex<T>	C_component_1() const;
::std::complex<T>	C_component_2() const;

A quaternion likewise has two complex components, and as we have seen above, for any quaternion q = α + βi + γj + δk we also have q = ( α + βi) + (γ + δi)j . These functions return them. The real part of q.C_component_1() is the same as q.real().

Quaternion Member Operators

Assignment Operators

quaternion<T>& operator = (quaternion<T> const & a_affecter);
template<typename X> 
quaternion<T>& operator = (quaternion<X> const& a_affecter);
quaternion<T>& operator = (T const& a_affecter);
quaternion<T>& operator = (::std::complex<T> const& a_affecter);

These perform the expected assignment, with type modification if necessary (for instance, assigning from a base type will set the real part to that value, and all other components to zero). For the unspecialized form, the base type's assignment operators must not throw.

Addition Operators

quaternion<T>& operator += (T const & rhs)
quaternion<T>& operator += (::std::complex<T> const & rhs);
template<typename X>	
quaternion<T>& operator += (quaternion<X> const & rhs);

These perform the mathematical operation (*this)+rhs and store the result in *this. The unspecialized form has exception guards, which the specialized forms do not, so as to insure exception safety. For the unspecialized form, the base type's assignment operators must not throw.

Subtraction Operators

quaternion<T>& operator -= (T const & rhs)
quaternion<T>& operator -= (::std::complex<T> const & rhs);
template<typename X>	
quaternion<T>& operator -= (quaternion<X> const & rhs);

These perform the mathematical operation (*this)-rhs and store the result in *this. The unspecialized form has exception guards, which the specialized forms do not, so as to insure exception safety. For the unspecialized form, the base type's assignment operators must not throw.

Multiplication Operators

quaternion<T>& operator *= (T const & rhs)
quaternion<T>& operator *= (::std::complex<T> const & rhs);
template<typename X>	
quaternion<T>& operator *= (quaternion<X> const & rhs);

These perform the mathematical operation (*this)*rhsin 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.

Division Operators

quaternion<T>& operator /= (T const & rhs)
quaternion<T>& operator /= (::std::complex<T> const & rhs);
template<typename X>	
quaternion<T>& operator /= (quaternion<X> 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.

Quaternion Non-Member Operators

Unary Plus

template<typename T>	
quaternion<T> operator + (quaternion<T> const & q);

This unary operator simply returns q.

Unary Minus

template<typename T>
quaternion<T> operator - (quaternion<T> const & q);

This unary operator returns the opposite of q.

Binary Addition Operators

template<typename T>	quaternion<T>	operator + (T const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator + (quaternion<T> const & lhs, T const & rhs);
template<typename T>	quaternion<T>	operator + (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator + (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T>	quaternion<T>	operator + (quaternion<T> const & lhs, quaternion<T> const & rhs);

These operators return quaternion<T>(lhs)+=rhs.

Binary Subtraction Operators

template<typename T>	quaternion<T>	operator - (T const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator - (quaternion<T> const & lhs, T const & rhs);
template<typename T>	quaternion<T>	operator - (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator - (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T>	quaternion<T>	operator - (quaternion<T> const & lhs, quaternion<T> const & rhs);

These operators return quaternion<T>(lhs)-=rhs.

Binary Multiplication Operators

template<typename T>	quaternion<T>	operator * (T const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator * (quaternion<T> const & lhs, T const & rhs);
template<typename T>	quaternion<T>	operator * (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator * (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T>	quaternion<T>	operator * (quaternion<T> const & lhs, quaternion<T> const & rhs);

These operators return quaternion<T>(lhs)*=rhs.

Binary Division Operators

template<typename T>	quaternion<T>	operator / (T const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator / (quaternion<T> const & lhs, T const & rhs);
template<typename T>	quaternion<T>	operator / (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T>	quaternion<T>	operator / (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T>	quaternion<T>	operator / (quaternion<T> const & lhs, quaternion<T> const & rhs);

These operators return quaternion<T>(lhs)/=rhs. It is of course still an error to divide by zero...

Equality Operators

template<typename T>	bool	operator == (T const & lhs, quaternion<T> const & rhs);
template<typename T>	bool	operator == (quaternion<T> const & lhs, T const & rhs);
template<typename T>	bool	operator == (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T>	bool	operator == (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T>	bool	operator == (quaternion<T> const & lhs, quaternion<T> const & rhs);

These return true if and only if the four components of quaternion<T>(lhs) are equal to their counterparts in quaternion<T>(rhs). As with any floating-type entity, this is essentially meaningless.

Inequality Operators

template<typename T>	bool	operator != (T const & lhs, quaternion<T> const & rhs);
template<typename T>	bool	operator != (quaternion<T> const & lhs, T const & rhs);
template<typename T>	bool	operator != (::std::complex<T> const & lhs, quaternion<T> const & rhs);
template<typename T>	bool	operator != (quaternion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T>	bool	operator != (quaternion<T> const & lhs, quaternion<T> const & rhs);

These return true if and only if quaternion<T>(lhs)==quaternion<T>(rhs) is false. As with any floating-type entity, this is essentially meaningless.

Stream Extractor

template<typename T, typename charT, class traits>
::std::basic_istream<charT,traits>& operator >> (::std::basic_istream<charT,traits> & is, quaternion<T> & 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 formated 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).

Stream Inserter

template<typename T, typename charT, class traits>
::std::basic_ostream<charT,traits>& operator << (::std::basic_ostream<charT,traits> & os, quaternion<T> const & q);

Inserts the quaternion q onto the stream os as if it were implemented as follows:

template<typename T, typename charT, class traits>
::std::basic_ostream<charT,traits>& operator << (	
               ::std::basic_ostream<charT,traits> & os,
               quaternion<T> const & q)
{
   ::std::basic_ostringstream<charT,traits>	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();
}

Quaternion Value Operations

real and unreal

template<typename T>	T              real(quaternion<T> const & q);
template<typename T>	quaternion<T>  unreal(quaternion<T> const & q);

These return q.real() and q.unreal() respectively.

conj

template<typename T>	quaternion<T>  conj(quaternion<T> const & q);

This returns the conjugate of the quaternion.

sup

template<typename T> T sup(quaternion<T> const & q);

This return the sup norm (the greatest among abs(q.R_component_1())...abs(q.R_component_4())) of the quaternion.

l1

template<typename T>	T  l1(quaternion<T> const & q);

This return the l1 norm (abs(q.R_component_1())+...+abs(q.R_component_4())) of the quaternion.

abs

template<typename T>	T  abs(quaternion<T> const & q);

This return the magnitude (Euclidian norm) of the quaternion.

norm

template<typename T>	T  norm(quaternion<T>const  & 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 Euclidian norm (also known as magnitude) is the square root of the Cayley norm.

Quaternion Creation Functions

template<typename T>	quaternion<T>	spherical(T const & rho, T const & theta, T const & phi1, T const & phi2);
template<typename T>	quaternion<T>	semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2);
template<typename T>	quaternion<T>	multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2);
template<typename T>	quaternion<T>	cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude);
template<typename T>	quaternion<T>	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.

spherical is a simple transposition of polar, it takes as inputs a (positive) magnitude and a point on the hypersphere, given by three angles. The first of these, theta has a natural range of -pi to +pi, and the other two have natural ranges of -pi/2 to +pi/2 (as is the case with the usual spherical coordinates in R3 ). Due to the many symmetries and periodicities, nothing untoward happens if the magnitude is negative or the angles are outside their natural ranges. The expected degeneracies (a magnitude of zero ignores the angles settings...) do happen however.

cylindrical is likewise a simple transposition of the usual cylindrical coordinates in R3 , which in turn is another derivative of planar polar coordinates. The first two inputs are the polar coordinates of the first C component of the quaternion. The third and fourth inputs are placed into the third and fourth R components of the quaternion, respectively.

multipolar is yet another simple generalization of polar coordinates. This time, both C components of the quaternion are given in polar coordinates.

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.

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 R3 , and here for R4 (example test file).

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 independant 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 and the boost/math/special_functions/sinhc.hpp headers.

exp

template<typename T>	quaternion<T> exp(quaternion<T> const & q);

Computes the exponential of the quaternion.

cos

template<typename T>	quaternion<T>  cos(quaternion<T> const & q);

Computes the cosine of the quaternion

sin

template<typename T>	quaternion<T>  sin(quaternion<T> const & q);

Computes the sine of the quaternion.

tan

template<typename T>	quaternion<T>  tan(quaternion<T> const & q);

Computes the tangent of the quaternion.

cosh

template<typename T>	quaternion<T>  cosh(quaternion<T> const & q);

Computes the hyperbolic cosine of the quaternion.

sinh

template<typename T>	quaternion<T>  sinh(quaternion<T> const & q);

Computes the hyperbolic sine of the quaternion.

tanh

template<typename T>	quaternion<T>  tanh(quaternion<T> const & q);

Computes the hyperbolic tangent of the quaternion.

pow

template<typename T>	quaternion<T>  pow(quaternion<T> const & q, int n);

Computes the n-th power of the quaternion q.

Test Program

The quaternion_test.cpp test program tests quaternions specializations for float, double and long double (sample output, with message output enabled).

If you define the symbol BOOST_QUATERNION_TEST_VERBOSE, you will get additional output (verbose output); this will only be helpfull if you enable message output at the same time, of course (by uncommenting the relevant line in the test or by adding --log_level=messages to your command line,...). In that case, and if you are running interactively, you may in addition define the symbol BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to interactively test the input operator with input of your choice from the standard input (instead of hard-coding it in the test).

Acknowledgements

The mathematical text has been typeset with Nisus Writer. Jens Maurer has helped with portability and standard adherence, and was the Review Manager for this library. More acknowledgements in the History section. Thank you to all who contributed to the discution about this library.

History

  • 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format.
  • 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).
  • 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com).
  • 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option.
  • 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different translation units); attempt at an improved compatibility with Microsoft compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; other compatibility fixes.
  • 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu).
  • 1.5.2 - 07/07/2001: introduced namespace math.
  • 1.5.1 - 07/06/2001: (end of Boost review) now includes <boost/math/special_functions/sinc.hpp> and <boost/math/special_functions/sinhc.hpp> instead of <boost/special_functions.hpp>; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code.
  • 1.5.0 - 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.

To Do

  • Improve testing.
  • Rewrite input operatore 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 implementation to an efficient one).
Copyright 2001 -2002 Daryle Walker, 2001-2003 Hubert Holin, 2005 John Maddock

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