Class Quaternion

java.lang.Object

org.episteme.core.mathematics.numbers.complex.Quaternion

All Implemented Interfaces:

AbelianGroup<Quaternion>, AbelianMonoid<Quaternion>, Group<Quaternion>, Magma<Quaternion>, Monoid<Quaternion>, Field<Quaternion>, Ring<Quaternion>, Semiring<Quaternion>, Set<Quaternion>

public final class Quaternion
extends Object
implements Field<Quaternion>

Represents a Quaternion number (H).

q = a + bi + cj + dk Quaternions form a non-commutative division ring (skew field).

* @author Silvere Martin-Michiellot

Since:

1.0

Author:

Gemini AI (Google DeepMind)

Field Summary

Fields

Modifier and Type	Field	Description
static final Quaternion	I
static final Quaternion	J
static final Quaternion	K
static final Quaternion	ONE
static final Quaternion	ZERO

Constructor Summary

Constructors

Constructor	Description
Quaternion(double a, double b, double c, double d)
Quaternion(Real a, Real b, Real c, Real d)

Method Summary

Modifier and Type	Method	Description
Quaternion	add(Quaternion other)
Quaternion	add(Quaternion a, Quaternion b)	Returns the sum of two elements (additive notation).
int	characteristic()	Returns the characteristic of this field.
Quaternion	conjugate()
boolean	contains(Quaternion element)	Tests whether this set contains the specified element.
String	description()	Returns a human-readable description of this set.
boolean	equals(Object obj)
Real	getI()
Real	getJ()
Real	getK()
Real	getReal()
int	hashCode()
Quaternion	inverse()
Quaternion	inverse(Quaternion element)	Returns the multiplicative inverse of a non-zero element.
boolean	isEmpty()	Returns true if this set contains no elements.
boolean	isMultiplicationCommutative()	Fields always have commutative multiplication.
Quaternion	multiply(double scalar)
Quaternion	multiply(Quaternion other)
Quaternion	multiply(Quaternion a, Quaternion b)	Returns the product of two elements.
Quaternion	multiply(Real scalar)
Quaternion	negate()
Quaternion	negate(Quaternion element)	Returns the additive inverse (negation) of an element.
Real	norm()
Real	normSquared()
static Quaternion	of(double real)
static Quaternion	of(double a, double b, double c, double d)
static Quaternion	of(Real real)
static Quaternion	of(Real a, Real b, Real c, Real d)
Quaternion	one()	Returns the multiplicative identity (one element).
Quaternion	operate(Quaternion left, Quaternion right)	Performs the binary operation on two elements.
Quaternion	subtract(Quaternion other)
String	toString()
Quaternion	zero()	Returns the additive identity (zero element).

Methods inherited from class Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Methods inherited from interface AbelianGroup

isCommutative, subtract

Methods inherited from interface AbelianMonoid

identity, isCommutative

Methods inherited from interface Field

divide, hasUnity

Methods inherited from interface Monoid

isAssociative

Methods inherited from interface Semiring

pow

Field Details

ZERO

public static final Quaternion ZERO

ONE

public static final Quaternion ONE

I

public static final Quaternion I

J

public static final Quaternion J

K

public static final Quaternion K

Constructor Details

Quaternion

public Quaternion(Real a,
 Real b,
 Real c,
 Real d)

Quaternion

public Quaternion(double a,
 double b,
 double c,
 double d)

Method Details

of

public static Quaternion of(Real a,
 Real b,
 Real c,
 Real d)

of

public static Quaternion of(double a,
 double b,
 double c,
 double d)

of

public static Quaternion of(Real real)

of

public static Quaternion of(double real)

getReal

public Real getReal()

getI

public Real getI()

getJ

public Real getJ()

getK

public Real getK()

add

public Quaternion add(Quaternion other)

subtract

public Quaternion subtract(Quaternion other)

multiply

public Quaternion multiply(Quaternion other)

multiply

public Quaternion multiply(Real scalar)

multiply

public Quaternion multiply(double scalar)

conjugate

public Quaternion conjugate()

normSquared

public Real normSquared()

norm

public Real norm()

inverse

public Quaternion inverse()

negate

public Quaternion negate()

toString

public String toString()

Overrides:

toString in class Object

equals

public boolean equals(Object obj)

Overrides:

equals in class Object

hashCode

public int hashCode()

Overrides:

hashCode in class Object

operate

public Quaternion operate(Quaternion left,
 Quaternion right)

Description copied from interface: Magma

Performs the binary operation on two elements.

This is the fundamental operation of a magma. The result must be an element of this magma (closure property).

Properties: None required (not necessarily associative or commutative)

Specified by:

operate in interface Magma<Quaternion>

Parameters:

left - the first operand

right - the second operand

Returns:

the result of a ∗ b

See Also:

• Magma.isAssociative()
• Magma.isCommutative()

zero

public Quaternion zero()

Description copied from interface: AbelianMonoid

Returns the additive identity (zero element).

Delegates to AbelianMonoid.identity().

Specified by:

zero in interface AbelianMonoid<Quaternion>

Returns:

the zero element

one

public Quaternion one()

Description copied from interface: Semiring

Returns the multiplicative identity (one element).

Satisfies: 1 × a = a × 1 = a for all elements a.

Specified by:

one in interface Semiring<Quaternion>

Returns:

the multiplicative identity

negate

public Quaternion negate(Quaternion element)

Description copied from interface: AbelianGroup

Returns the additive inverse (negation) of an element.

Satisfies: a + (-a) = (-a) + a = 0

Specified by:

negate in interface AbelianGroup<Quaternion>

Parameters:

element - the element to negate

Returns:

-element

See Also:

• Group.inverse(Object)

inverse

public Quaternion inverse(Quaternion element)

Description copied from interface: Field

Returns the multiplicative inverse of a non-zero element.

For element a ≠0, returns a⁻¹ such that: a × a⁻¹ = a⁻¹ × a = 1

Examples:

• In â„š: inverse(2/3) = 3/2
• In ℝ: inverse(5.0) = 0.2
• In â„‚: inverse(3+4i) = (3-4i)/25

Specified by:

inverse in interface Field<Quaternion>

Specified by:

inverse in interface Group<Quaternion>

Parameters:

element - the element to invert (must be non-zero)

Returns:

the multiplicative inverse

See Also:

• Field.divide(Object, Object)
• Semiring.one()

isMultiplicationCommutative

public boolean isMultiplicationCommutative()

Description copied from interface: Field

Fields always have commutative multiplication.

Specified by:

isMultiplicationCommutative in interface Field<Quaternion>

Specified by:

isMultiplicationCommutative in interface Semiring<Quaternion>

Returns:

always true

characteristic

public int characteristic()

Description copied from interface: Field

Returns the characteristic of this field.

The characteristic is the smallest positive integer n such that:
1 + 1 + ... + 1 (n times) = 0
If no such n exists, the characteristic is 0.

Examples:

• char(â„š) = char(ℝ) = char(â„‚) = 0
• char(𝔽ₚ) = p (for prime p)

Specified by:

characteristic in interface Field<Quaternion>

Returns:

the characteristic (0 for infinite fields, p for finite fields)

description

public String description()

Description copied from interface: Set

Returns a human-readable description of this set.

Examples:

• "ℝ (Real Numbers)"
• "ℤ/12ℤ (Integers modulo 12)"
• "{1, 2, 3, 4, 5}"

Specified by:

description in interface Set<Quaternion>

Returns:

a description of this set

isEmpty

public boolean isEmpty()

Description copied from interface: Set

Returns true if this set contains no elements.

The empty set (∅) is a fundamental concept in set theory. It is the unique set containing no elements.

Specified by:

isEmpty in interface Set<Quaternion>

Returns:

true if this set is empty

contains

public boolean contains(Quaternion element)

Description copied from interface: Set

Tests whether this set contains the specified element.

This is the fundamental operation of a set - membership testing.

Specified by:

contains in interface Set<Quaternion>

Parameters:

element - the element to test for membership

Returns:

true if this set contains the element, false otherwise

See Also:

• Set.isEmpty()
• invalid reference

  #size()

add

public Quaternion add(Quaternion a,
 Quaternion b)

Description copied from interface: AbelianMonoid

Returns the sum of two elements (additive notation).

Delegates to Magma.operate(Object, Object).

Specified by:

add in interface AbelianMonoid<Quaternion>

Parameters:

a - the first addend

b - the second addend

Returns:

a + b

multiply

public Quaternion multiply(Quaternion a,
 Quaternion b)

Description copied from interface: Semiring

Returns the product of two elements.

Multiplication must be associative and distribute over addition.

Specified by:

multiply in interface Semiring<Quaternion>

Parameters:

a - the first factor

b - the second factor

Returns:

a × b