org.apache.commons.math3.complex

## Class Quaternion

- java.lang.Object
- - org.apache.commons.math3.complex.Quaternion

- All Implemented Interfaces:
    Serializable

      public final class Quaternion
      extends Object
      implements Serializable

  This class implements quaternions (Hamilton's hypercomplex numbers).
  Instance of this class are guaranteed to be immutable.
  Since:
    3.1
  See Also:
    Serialized Form

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static Quaternion` | `I` i
    `static Quaternion` | `IDENTITY` Identity quaternion.
    `static Quaternion` | `J` j
    `static Quaternion` | `K` k
    `static Quaternion` | `ZERO` Zero quaternion.

  - ### Constructor Summary

    Constructors
    Constructor and Description
    `Quaternion(double[] v)` Builds a pure quaternion from a vector (assuming that the scalar part is zero).
    `Quaternion(double scalar, double[] v)` Builds a quaternion from scalar and vector parts.
    `Quaternion(double a, double b, double c, double d)` Builds a quaternion from its components.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `Quaternion` | `add(Quaternion q)` Computes the sum of the instance and another quaternion.
    `static Quaternion` | `add(Quaternion q1, Quaternion q2)` Computes the sum of two quaternions.
    `double` | `dotProduct(Quaternion q)` Computes the dot-product of the instance by a quaternion.
    `static double` | `dotProduct(Quaternion q1, Quaternion q2)` Computes the dot-product of two quaternions.
    `boolean` | `equals(Object other)`
    `boolean` | `equals(Quaternion q, double eps)` Checks whether this instance is equal to another quaternion within a given tolerance.
    `Quaternion` | `getConjugate()` Returns the conjugate quaternion of the instance.
    `Quaternion` | `getInverse()` Returns the inverse of this instance.
    `double` | `getNorm()` Computes the norm of the quaternion.
    `Quaternion` | `getPositivePolarForm()` Returns the polar form of the quaternion.
    `double` | `getQ0()` Gets the first component of the quaternion (scalar part).
    `double` | `getQ1()` Gets the second component of the quaternion (first component of the vector part).
    `double` | `getQ2()` Gets the third component of the quaternion (second component of the vector part).
    `double` | `getQ3()` Gets the fourth component of the quaternion (third component of the vector part).
    `double` | `getScalarPart()` Gets the scalar part of the quaternion.
    `double[]` | `getVectorPart()` Gets the three components of the vector part of the quaternion.
    `int` | `hashCode()`
    `boolean` | `isPureQuaternion(double eps)` Checks whether the instance is a pure quaternion within a given tolerance.
    `boolean` | `isUnitQuaternion(double eps)` Checks whether the instance is a unit quaternion within a given tolerance.
    `Quaternion` | `multiply(double alpha)` Multiplies the instance by a scalar.
    `Quaternion` | `multiply(Quaternion q)` Returns the Hamilton product of the instance by a quaternion.
    `static Quaternion` | `multiply(Quaternion q1, Quaternion q2)` Returns the Hamilton product of two quaternions.
    `Quaternion` | `normalize()` Computes the normalized quaternion (the versor of the instance).
    `Quaternion` | `subtract(Quaternion q)` Subtracts a quaternion from the instance.
    `static Quaternion` | `subtract(Quaternion q1, Quaternion q2)` Subtracts two quaternions.
    `String` | `toString()`

    - ### Methods inherited from class java.lang.Object

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

- - ### Field Detail

    - #### IDENTITY

          public static final Quaternion IDENTITY

      Identity quaternion.

    - #### ZERO

          public static final Quaternion ZERO

      Zero quaternion.

    - #### I

          public static final Quaternion I

      i

    - #### J

          public static final Quaternion J

      j

    - #### K

          public static final Quaternion K

      k

  - ### Constructor Detail

    - #### Quaternion

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

      Builds a quaternion from its components.
      Parameters:
        `a` - Scalar component.
        `b` - First vector component.
        `c` - Second vector component.
        `d` - Third vector component.

    - #### Quaternion

          public Quaternion(double scalar,
                    double[] v)
                     throws DimensionMismatchException

      Builds a quaternion from scalar and vector parts.
      Parameters:
        `scalar` - Scalar part of the quaternion.
        `v` - Components of the vector part of the quaternion.
      Throws:
        `DimensionMismatchException` - if the array length is not 3.

    - #### Quaternion

          public Quaternion(double[] v)

      Builds a pure quaternion from a vector (assuming that the scalar part is zero).
      Parameters:
        `v` - Components of the vector part of the pure quaternion.

  - ### Method Detail

    - #### getConjugate

          public Quaternion getConjugate()

      Returns the conjugate quaternion of the instance.
      Returns:
        the conjugate quaternion

    - #### multiply

          public static Quaternion multiply(Quaternion q1,
                            Quaternion q2)

      Returns the Hamilton product of two quaternions.
      Parameters:
        `q1` - First quaternion.
        `q2` - Second quaternion.
      Returns:
        the product `q1` and `q2`, in that order.

    - #### multiply

          public Quaternion multiply(Quaternion q)

      Returns the Hamilton product of the instance by a quaternion.
      Parameters:
        `q` - Quaternion.
      Returns:
        the product of this instance with `q`, in that order.

    - #### add

          public static Quaternion add(Quaternion q1,
                       Quaternion q2)

      Computes the sum of two quaternions.
      Parameters:
        `q1` - Quaternion.
        `q2` - Quaternion.
      Returns:
        the sum of `q1` and `q2`.

    - #### add

          public Quaternion add(Quaternion q)

      Computes the sum of the instance and another quaternion.
      Parameters:
        `q` - Quaternion.
      Returns:
        the sum of this instance and `q`

    - #### subtract

          public static Quaternion subtract(Quaternion q1,
                            Quaternion q2)

      Subtracts two quaternions.
      Parameters:
        `q1` - First Quaternion.
        `q2` - Second quaternion.
      Returns:
        the difference between `q1` and `q2`.

    - #### subtract

          public Quaternion subtract(Quaternion q)

      Subtracts a quaternion from the instance.
      Parameters:
        `q` - Quaternion.
      Returns:
        the difference between this instance and `q`.

    - #### dotProduct

          public static double dotProduct(Quaternion q1,
                          Quaternion q2)

      Computes the dot-product of two quaternions.
      Parameters:
        `q1` - Quaternion.
        `q2` - Quaternion.
      Returns:
        the dot product of `q1` and `q2`.

    - #### dotProduct

          public double dotProduct(Quaternion q)

      Computes the dot-product of the instance by a quaternion.
      Parameters:
        `q` - Quaternion.
      Returns:
        the dot product of this instance and `q`.

    - #### getNorm

          public double getNorm()

      Computes the norm of the quaternion.
      Returns:
        the norm.

    - #### normalize

          public Quaternion normalize()

      Computes the normalized quaternion (the versor of the instance). The norm of the quaternion must not be zero.
      Returns:
        a normalized quaternion.
      Throws:
        `ZeroException` - if the norm of the quaternion is zero.

    - #### equals

          public boolean equals(Object other)

      Overrides:
        `equals` in class `Object`

    - #### hashCode

          public int hashCode()

      Overrides:
        `hashCode` in class `Object`

    - #### equals

          public boolean equals(Quaternion q,
                       double eps)

      Checks whether this instance is equal to another quaternion within a given tolerance.
      Parameters:
        `q` - Quaternion with which to compare the current quaternion.
        `eps` - Tolerance.
      Returns:
        `true` if the each of the components are equal within the allowed absolute error.

    - #### isUnitQuaternion

          public boolean isUnitQuaternion(double eps)

      Checks whether the instance is a unit quaternion within a given tolerance.
      Parameters:
        `eps` - Tolerance (absolute error).
      Returns:
        `true` if the norm is 1 within the given tolerance, `false` otherwise

    - #### isPureQuaternion

          public boolean isPureQuaternion(double eps)

      Checks whether the instance is a pure quaternion within a given tolerance.
      Parameters:
        `eps` - Tolerance (absolute error).
      Returns:
        `true` if the scalar part of the quaternion is zero.

    - #### getPositivePolarForm

          public Quaternion getPositivePolarForm()

      Returns the polar form of the quaternion.
      Returns:
        the unit quaternion with positive scalar part.

    - #### getInverse

          public Quaternion getInverse()

      Returns the inverse of this instance. The norm of the quaternion must not be zero.
      Returns:
        the inverse.
      Throws:
        `ZeroException` - if the norm (squared) of the quaternion is zero.

    - #### getQ0

          public double getQ0()

      Gets the first component of the quaternion (scalar part).
      Returns:
        the scalar part.

    - #### getQ1

          public double getQ1()

      Gets the second component of the quaternion (first component of the vector part).
      Returns:
        the first component of the vector part.

    - #### getQ2

          public double getQ2()

      Gets the third component of the quaternion (second component of the vector part).
      Returns:
        the second component of the vector part.

    - #### getQ3

          public double getQ3()

      Gets the fourth component of the quaternion (third component of the vector part).
      Returns:
        the third component of the vector part.

    - #### getScalarPart

          public double getScalarPart()

      Gets the scalar part of the quaternion.
      Returns:
        the scalar part.
      See Also:
        `getQ0()`

    - #### getVectorPart

          public double[] getVectorPart()

      Gets the three components of the vector part of the quaternion.
      Returns:
        the vector part.
      See Also:
        `getQ1()`, `getQ2()`, `getQ3()`

    - #### multiply

          public Quaternion multiply(double alpha)

      Multiplies the instance by a scalar.
      Parameters:
        `alpha` - Scalar factor.
      Returns:
        a scaled quaternion.

    - #### toString

          public String toString()

      Overrides:
        `toString` in class `Object`

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