/******************************************
 * C++ Quaternions
 * Version: 1.0.9
 * Author:  Douglas Wilhelm Harder
 * Date:    2008/03/03
 *
 * Copyright (c) 2006-2008 by Douglas Wilhelm Harder.
 * All rights reserved.
 ******************************************/

#ifndef CA_UWATERLOO_ALUMNI_DWHARDER_QUATERNION
#define CA_UWATERLOO_ALUMNI_DWHARDER_QUATERNION

#include <iostream>
#include <cmath>
#include <string>

template <typename T> class Quaternion;

template <typename T> std::ostream & operator << ( std::ostream &, const Quaternion<T> & );

template <typename T> Quaternion<T> operator + ( T, const Quaternion<T> & );
template <typename T> Quaternion<T> operator + ( long, const Quaternion<T> & );
template <typename T> Quaternion<T> operator - ( T, const Quaternion<T> & );
template <typename T> Quaternion<T> operator - ( long, const Quaternion<T> & );
template <typename T> Quaternion<T> operator * ( T, const Quaternion<T> & );
template <typename T> Quaternion<T> operator * ( long, const Quaternion<T> & );
template <typename T> Quaternion<T> operator / ( T, const Quaternion<T> & );
template <typename T> Quaternion<T> operator / ( long, const Quaternion<T> & );

template <typename T> bool operator == ( T, const Quaternion<T> & );
template <typename T> bool operator == ( long, const Quaternion<T> & );
template <typename T> bool operator != ( T, const Quaternion<T> & );
template <typename T> bool operator != ( long, const Quaternion<T> & );

template <typename T> T real( const Quaternion<T> & );
template <typename T> T imag_i( const Quaternion<T> & );
template <typename T> T imag_j( const Quaternion<T> & );
template <typename T> T imag_k( const Quaternion<T> & );
template <typename T> T csgn( const Quaternion<T> & );
template <typename T> T abs( const Quaternion<T> & );
template <typename T> T norm( const Quaternion<T> & );
template <typename T> T abs_imag( const Quaternion<T> & );
template <typename T> T norm_imag( const Quaternion<T> & );
template <typename T> T arg( const Quaternion<T> & );
template <typename T> Quaternion<T> imag( const Quaternion<T> & );
template <typename T> Quaternion<T> conj( const Quaternion<T> & );
template <typename T> Quaternion<T> signum( const Quaternion<T> & );
template <typename T> Quaternion<T> sqr( const Quaternion<T> & );
template <typename T> Quaternion<T> sqrt( const Quaternion<T> & );
template <typename T> Quaternion<T> rotate( const Quaternion<T> &, const Quaternion<T> & );
template <typename T> Quaternion<T> exp( const Quaternion<T> & );
template <typename T> Quaternion<T> log( const Quaternion<T> & );
template <typename T> Quaternion<T> log10( const Quaternion<T> & );
template <typename T> Quaternion<T> pow( const Quaternion<T> &, const Quaternion<T> & );
template <typename T> Quaternion<T> pow( const Quaternion<T> &, T );
template <typename T> Quaternion<T> inverse( const Quaternion<T> & );
template <typename T> Quaternion<T> sin( const Quaternion<T> & );
template <typename T> Quaternion<T> cos( const Quaternion<T> & );
template <typename T> Quaternion<T> tan( const Quaternion<T> & );
template <typename T> Quaternion<T> sec( const Quaternion<T> & );
template <typename T> Quaternion<T> csc( const Quaternion<T> & );
template <typename T> Quaternion<T> cot( const Quaternion<T> & );
template <typename T> Quaternion<T> sinh( const Quaternion<T> & );
template <typename T> Quaternion<T> cosh( const Quaternion<T> & );
template <typename T> Quaternion<T> tanh( const Quaternion<T> & );
template <typename T> Quaternion<T> sech( const Quaternion<T> & );
template <typename T> Quaternion<T> csch( const Quaternion<T> & );
template <typename T> Quaternion<T> coth( const Quaternion<T> & );
template <typename T> Quaternion<T> asin( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> acos( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> atan( const Quaternion<T> & );
template <typename T> Quaternion<T> asec( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> acsc( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> acot( const Quaternion<T> & );
template <typename T> Quaternion<T> asinh( const Quaternion<T> & );
template <typename T> Quaternion<T> acosh( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> atanh( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> asech( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> acsch( const Quaternion<T> & );
template <typename T> Quaternion<T> acoth( const Quaternion<T> &, const Quaternion<T> & = Quaternion<T>::I );
template <typename T> Quaternion<T> bessel_J( int, const Quaternion<T> & );
template <typename T> Quaternion<T> floor( const Quaternion<T> & );
template <typename T> Quaternion<T> ceil( const Quaternion<T> & );
template <typename T> Quaternion<T> horner( const Quaternion<T> &, T *, unsigned int );
template <typename T> Quaternion<T> horner( const Quaternion<T> &, T *, T *, unsigned int );

template <typename T = double> class Quaternion {
	private:
		T r, i, j, k;

		inline static Quaternion multiplier( T, T, const Quaternion & );
		inline static Quaternion make_inf( T, T );
		inline static Quaternion make_i( T, T );

	public:
		const static Quaternion ZERO;
		const static Quaternion ONE;
		const static Quaternion I;
		const static Quaternion J;
		const static Quaternion K;

		const static Quaternion UNITS[4];

		/******************************************
		 * Constructor and Copy Constructor
		 ******************************************/

		Quaternion( T, T, T, T );
		Quaternion( T = 0.0 );

		/******************************************
		 * Assignment Operator
		 ******************************************/

		const Quaternion & operator = ( const T & );

		/******************************************
		 * Mutating Arithmetic Operators
		 ******************************************/

		Quaternion & operator += ( const Quaternion & );
		Quaternion & operator -= ( const Quaternion & );
		Quaternion & operator *= ( const Quaternion & );
		Quaternion & operator /= ( const Quaternion & );

		Quaternion & operator += ( T );
		Quaternion & operator -= ( T );
		Quaternion & operator *= ( T );
		Quaternion & operator /= ( T );

		Quaternion & operator ++();
		Quaternion operator ++( int );
		Quaternion & operator --();
		Quaternion operator --( int );

		/******************************************
		 * Real-valued Functions
		 ******************************************/

		T real() const;
		T operator []( int ) const;
		T &operator []( int );
		T imag_i() const;
		T imag_j() const;
		T imag_k() const;
		T csgn() const;
		T abs() const;
		T norm() const;
		T abs_imag() const;
		T norm_imag() const;
		T arg() const;

		/******************************************
		 * Quaternion-valued Functions
		 ******************************************/

		Quaternion imag() const;
		Quaternion conj() const;
		Quaternion operator * () const;
		Quaternion signum() const;
		Quaternion sqr() const;
		Quaternion sqrt() const;
		Quaternion rotate( const Quaternion & ) const;

		/******************************************
		 * Boolean-valued Functions
		 ******************************************/

		bool is_imaginary() const;
		bool is_inf() const;
		bool is_nan() const;
		bool is_neg_inf() const;
		bool is_pos_inf() const;
		bool is_real() const;
		bool is_real_inf() const;
		bool is_zero() const;

		/******************************************
		 * Exponential and Logarithmic Functions
		 ******************************************/

		Quaternion exp() const;
		Quaternion log() const;
		Quaternion log10() const;
		Quaternion pow(const Quaternion & w) const;
		Quaternion pow(T x) const;
		Quaternion inverse() const;

		/******************************************
		 * Trigonometric and Hyperbolic Functions
		 ******************************************/

		Quaternion   sin() const;   Quaternion   cos() const;   Quaternion   tan() const;
		Quaternion   sec() const;   Quaternion   csc() const;   Quaternion   cot() const;
		Quaternion  sinh() const;   Quaternion  cosh() const;   Quaternion  tanh() const;
		Quaternion  sech() const;   Quaternion  csch() const;   Quaternion  coth() const;
		Quaternion  asin( const Quaternion & = I ) const;
		Quaternion  acos( const Quaternion & = I ) const;       Quaternion  atan() const;
		Quaternion  asec( const Quaternion & = I ) const;
		Quaternion  acsc( const Quaternion & = I ) const;       Quaternion  acot() const;

		Quaternion asinh() const;
		Quaternion acosh( const Quaternion & = I ) const;
		Quaternion atanh( const Quaternion & = I ) const;

		Quaternion asech( const Quaternion & = I ) const;
		Quaternion acsch() const;
		Quaternion acoth( const Quaternion & = I ) const;

		Quaternion bessel_J( int ) const;

		/******************************************
		 * Integer Functions
		 ******************************************/

		Quaternion ceil() const;
		Quaternion floor() const;

		/******************************************
		 * Horner's Rule
		 ******************************************/

		Quaternion horner( T *, unsigned int ) const;
		Quaternion horner( T *, T *, unsigned int ) const;

		/******************************************
		 * Random Factories
		 ******************************************/

		static Quaternion random();
		static Quaternion random_imag();
		static Quaternion random_real();

		/******************************************
		 * Binary Arithmetic Operators
		 ******************************************/

		Quaternion operator + ( const Quaternion & ) const;
		Quaternion operator + ( T ) const;

		Quaternion operator - ( const Quaternion & ) const;
		Quaternion operator - ( T ) const;

		Quaternion operator * ( const Quaternion & ) const;
		Quaternion operator * ( T ) const;

		Quaternion operator / ( const Quaternion & ) const;
		Quaternion operator / ( T ) const;

		/******************************************
		 * Unary Arithmetic Operators
		 ******************************************/

		Quaternion operator - () const;

		/******************************************
		 * Binary Boolean Operators
		 ******************************************/

		bool operator == ( const Quaternion & ) const;
		bool operator == ( T ) const;

		bool operator != ( const Quaternion & ) const;
		bool operator != ( T ) const;
};

#endif