science

Quaternion

So, we have finally reached this misterious combination of vectors and complex numbers. As it have been told in 3Blue1brown video, “Quaternions are an absolutely fascinating and often underuppreciated number system from math”.

Basic usage

use irrevion\science\Math\Entities\{Scalar, Fraction, Imaginary, Complex, ComplexPolar, Vector, QuaternionComponent, Quaternion};

$Q = new Quaternion([4.1, 8.2, 16.4, 32.8]); // array of 4 numbers passed wich defines a, b, c and d coefficients as in [ a + bi + cj + dk ]

$Q = new Quaternion(new Scalar(-5.75), new Vector([7, 13, 17]));
print "[-5.75 + [7, 13, 17]] -> $Q (".($Q::class).")\n"; // outputs [-5.75 + [7, 13, 17]] -> [-5.75 + 7i + 13j + 17k] (irrevion\science\Math\Entities\Quaternion)

$Q = new Quaternion(-5.75); // create quaternion with only real part without imaginary
$Q = new Quaternion(new Scalar(-5.75));

$Q = new Quaternion(new Imaginary(-9.999999e-13)); // create quaternion with only i component using b coefficient

$Q = new Quaternion(new QuaternionComponent(Math::PI, 'k')); // create quaternion with only k component using d coefficient

$Q = new Quaternion(new Complex(7.21, -135.296)); // create quaternion from complex number [ a + bi ] in form [ a + bi + 0j + 0k ]

$Q = new Quaternion(new ComplexPolar(26.3, 0.0974));
print "[r 26.3, φ 0.0974] -> $Q (".($Q::class).")\n"; // outputs [r 26.3, φ 0.0974] -> [26.175347698301 + 2.5575716750597i + 0j + 0k] (irrevion\science\Math\Entities\Quaternion)

$Q = new Quaternion(new Vector([
	new QuaternionComponent(-1, 'i'),
	new QuaternionComponent(17.3, 'j'),
	new QuaternionComponent(-2, 'k'),
], QuaternionComponent::class)); // only vector part of quaternion is acceptable as well

$Q_clone = new Quaternion($Q); // reinstantiate quaternion object

Constructor

You can create an instance of a Quaternion in different ways. Some of them was already shown above:

$Q = new Quaternion(new Scalar(-5.75), new Vector([7, 13, 17]));
$Q = new Quaternion(new ComplexPolar(26.3, 0.0974));

You can also create a copy of another Quaternion:

$q1 = new Quaternion(new Scalar(1), new Vector([2, 3, 4]));
$q2 = new Quaternion($q1);

Addition / Subtraction

There are methods for addition and subtraction of quaternions:

$q1 = new Quaternion(new Scalar(1), new Vector([2, 3, 4]));
$q2 = new Quaternion(new Scalar(5), new Vector([6, 7, 8]));

$sum = $q1->add($q2); // result is [6 + 8i + 10j + 12k]
$diff = $q1->subtract($q2); // result is [-4 - 4i - 4j - 4k]

Multiplication / Division

Quaternion multiplication and division methods are also available:

$prod = $q1->multiply($q2); // result is [-60 + 12i + 30j + 24k]
$quotient = $q1->divide($q2); // result is [0.033333333333333 + 0.066666666666667i + 0.1j + 0.13333333333333k]

Other methods

Conjugate:

$conjugate = $q1->conjugate(); // result is [1 - 2i - 3j - 4k]

Magnitude:

$magnitude = $q1->abs(); // result is 5.4772255750517

Inverse:

$inverse = $q1->inverse(); // result is [0.033333333333333 - 0.066666666666667i - 0.1j - 0.13333333333333k]