Quaternion

The quaternions , noted $\backslash \; mathbb\; H$, are a type of numbers hypercomplexes, constituting an extension of the complex numbers, extension similar to that which had led real numbers $\backslash \; mathbb\; R$ to the complex numbers $\backslash \; mathbb\; C$.

Origins and principles

History

The quaternions “were discovered” by William Rowan Hamilton in 1843 starting from work of Carl Friedrich Gauss and with the previous century Leonhard Euler. He then studied the geometrical interpretation of arithmetic of complex numbers in the plan and sought to obtain results similar in space to three dimensions.

After years of research on the construction of a algebra with “triplets” of three real numbers, it butted against the multiplication, and in particular the conservation of the standard S (Georg Ferdinand Frobenius showed in 1877 that such a multiplication of triplets was impossible to define).

It had then the idea to use “quadruplets” by employing a additional Dimension. According to its dires, it went, on October 16th, 1843, along the royal channel, with his wife when suddenly to mind the solution in the form came to him from the relations: $i^2\; =\; j^2\; =\; k^2\; =\; ijk\; =\; -1\; \backslash ,$. It then engraved promptly these relations with a knife in a stone of the bridge of Brougham (now called Broom Bridge) with Dublin. This inscription, unfortunately erased by time, was replaced by a Plaque.jpg with the memory of Sir William Rowan Hamilton.

The theory was generalized, other whole like the Octonion S discovered thereafter. An element of a whole of this nature was described as Nombre hypercomplexe until the First World War. These units are now regarded as examples of semi-simple algebras. The Théorème of Artin-Wedderburn provides a method of construction generic, it is based on the theory of the Représentations of a group finished. The construction of the quaternions is given in the article Représentations of the group of the quaternions. It corresponds to the single faithful simple algebra of the representation of the Groupe of quaternions on the body of the real numbers.

Principle

Hamilton described a quaternion like quadruplet real numbers, the first element being a “scalar”, and the three elements remaining forming a “vector”, or “imaginary pure”.

It could thus define a multiplication with the good properties. This one can be summarized with this multiplication table:

Any quaternion H can be regarded as a linear combination of the four quaternions " unités" 1 , I , J , and K :

$H\; =\; has\; \backslash \; cdot\; 1\; +\; B\; \backslash \; cdot\; I\; +\; C\; \backslash \; cdot\; J\; +\; D\; \backslash \; cdot\; K\; \backslash ,$
(where has , B , C , D is real numbers).

H can be also written: H = Z + z'·J (with Z and z' of the complex numbers of the form has + B·I )

The real numbers has , B , C and D is characteristic of H : there exists only one way of writing H in this form, and any quaternion comprising these same 4 characteristics is necessarily equal to H (the reciprocal one is true).

has is called the component real or scalar of H , while B , C and D is the complex components of H . It is also said that has is the scalar of H and that the triplet { B , C , D } or I + C \ cdot J + D \ cdot K \, is the vector of H (or its vectorial part).

This discovery involved the abandonment of the exclusive use of the commutative laws, a radical projection for the time. The Vector S and the matrix S still formed part of the future, but Hamilton to some extent had just introduced the vector Product and the scalar Produit vectors.

Mathematical properties

Relative to the other classes of numbers

The algebra of the quaternions is not any more commutative, but partially anticommutative: 1 · I = I · 1 = I but I · J = K and J · I = - K .

• One carries out a rotation around the axis Y followed by a rotation around axis X:

• the two cubes underwent same rotations, but in a different order. The end result is different, which expresses in a graphic way not-commutation of rotations.

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Summon

The sum of 2 quaternions $Q\_1\; \backslash ,$ and $Q\_2\; \backslash ,$, noted $Q\_1\; +\; Q\_2\; \backslash ,$ is defined as follows:

if $Q\_1\; =\; has\; +\; B\; I\; +\; C\; J\; +\; D\; K\; \backslash ,$ and

if $Q\_2\; =\; a\text{'}+\; b\text{'}\; I\; +\; it\; J\; +\; of\; K\; \backslash ,$, then:

$Q\_1\; +\; Q\_2\; =\; (a+a\text{'})\; +\; (b+b\text{'})\; I\; +\; (c+c\text{'})\; J\; +\; (d+d\text{'})\; K\; \backslash ,$

The sum is commutative and associative.

Opposed

The quaternions $Q\_1\; \backslash ,$ and $Q\_2\; \backslash ,$ are known as opposite if their amount is null:

$Q\_1\; +\; Q\_2\; =\; 0\; +\; 0\; I\; +\; 0\; J\; +\; 0\; K\; \backslash ,$

In this case, one writes:

$Q\_2\; =\; -\; Q\_1\; \backslash ,$

Product

The product of 2 quaternions $Q\_1$ and $Q\_2$, noted $Q\_1\; \backslash \; cdot\; Q\_2$ is defined as follows:

if $Q\_1\; =\; has\; +\; B\; I\; +\; C\; J\; +\; D\; K\; \backslash ,$ and

if $Q\_2\; =\; a\text{'}+\; b\text{'}\; I\; +\; it\; J\; +\; of\; K\; \backslash ,$, then:

$Q\_1\; \backslash \; cdot\; Q\_2\; =\; aa\text{'}-bb\text{'}-cc\text{'}-dd\text{'}\; +\; (ab\text{'}+ba\text{'}+cd\text{'}-dc\text{'})\; I\; +\; (ac\text{'}+ca\text{'}+db\text{'}-bd\text{'})\; J\; +\; (ad\text{'}+da\text{'}+bc\text{'}-cb\text{'})\; K\; \backslash ,$

which can be still written:

$Q\_1\; \backslash \; cdot\; Q\_2\; =\; aa\text{'}-\; (bb\text{'}+cc\text{'}+dd\text{'})\; +\; (ab\text{'}+ba\text{'})\; I\; +\; (ac\text{'}+ca\text{'})\; J\; +\; (ad\text{'}+da\text{'})\; K\; +\; (bi+cj+dk)\; \backslash \; wedge\; (b\text{'}\; i+c\text{'}\; j+d\text{'}\; K)\; \backslash ,$, $Q\_1\; \backslash \; cdot\; Q\_2\; =\; aa\text{'}-\; (\backslash \; vec\; V\_1\; \backslash \; bullet\; \backslash \; vec\; V\_2)\; +\; (ab\text{'}+ba\text{'})\; I\; +\; (ac\text{'}+ca\text{'})\; J\; +\; (ad\text{'}+da\text{'})\; K\; +\; (\backslash \; vec\; V\_1\; \backslash \; wedge\; \backslash \; vec\; V\_2)\; \backslash ,$,

the last formula using at the same time the produces scalar (symbol $\backslash \; bullet\; \backslash ,$) and the vector product (symbol $\backslash \; wedge\; \backslash ,$) of the vectorial components $\backslash \; vec\; V\_1\; =\; (bi+cj+dk)\; \backslash ,$ and $\backslash \; vec\; V\_2\; =\; (b\text{'}\; i+c\text{'}\; j+d\text{'}\; K)\; \backslash ,$ of the two quaternions.

The product is associative but, as one mentioned above, it is not (except exceptions) commutative!

The product is distributive compared to the addition:

Scalar

A scalar $\backslash \; lambda\; \backslash ,$ can be regarded as a quaternion whose 3 complex components are null (just as a real number can be regarded as a complex number of which the imaginary part is null). One can thus define the sum and the product of a scalar and a quaternion. In this particular case, the product is commutative:

$\backslash \; lambda\; \backslash \; Q\; =\; Q\; \backslash \; \backslash \; lambda\; \backslash ,$

Conjugation

One defines the Conjugué (noted * ) of the quaternion: $Q\; =\; has\; +\; B\; I\; +\; C\; J\; +\; D\; K\; \backslash ,$ of components has , B , C , D by:

$Q^*\; =\; has\; -\; B\; I\; -\; C\; J\; -\; D\; K\; \backslash ,$

The product of $Q\; \backslash ,$ by its combined $Q^*\; \backslash ,$ gives:

$Q\; \backslash \; cdot\; Q^*\; =\; (+\; B\; has\; I\; +\; C\; J\; +\; D\; K).\; (-\; B\; I\; -\; C\; J\; has\; -\; D\; K)\; =\; a^2\; +\; b^2\; +\; c^2\; +\; d^2\; \backslash ,$

who is the square of the standard $\backslash |Q\; \backslash |\backslash ,$ of $Q\; \backslash ,$.

Remarks

• Attention this conjugation is not a Morphisme, it is a antimorphism, considering that $\backslash \; forall\; (has,\; b)\; \backslash \; in\; \backslash \; mathbb\; H^2,\; (a*b)^*=\; (b^*)\; *\; (a^*)$.
• the invariants numbers by this conjugation, such as $a*=a$, are the realities number.

Normalizes and unit quaternions

When the standard $(a^2\; +\; b^2\; +\; c^2\; +\; d^2)\; ^\; \{\backslash \; frac\; \{1\}\; \{2\}\}\; \backslash ,$ of a quaternion $Q\; =\; has\; +\; B\; I\; +\; C\; J\; +\; D\; K\; \backslash ,$ is worth 1, it is said that the quaternion is normalized or that it is about a unit quaternion .

We will see below that one can establish a kind of correspondence between a unit quaternion and a vectorial rotation in the Euclidean space of dimension 3, and that this characteristic allows a very simple representation of the product of two vectorial rotations.

Opposite

If the quaternion $Q\; \backslash ,$ is not null, it has single opposite, $Q^\; \{-\; 1\}\; =\; \backslash \; frac\; \{1\}\; \{a^2+b^2+c^2+d^2\}\; \backslash \; Q^*\; \backslash ,$.

Division

The product being noncommutative, one can define two ways of dividing the quaternion $P\; \backslash ,$ by the quaternion $Q\; \backslash ,$ (not no one):

première way: $P\; \backslash \; cdot\; Q^\; \{-\; 1\}\; \backslash ,$
seconde way: $Q^\; \{-\; 1\}\; \backslash \; cdot\; P\; \backslash ,$

Combined of a reverse, combined sum and product of two quaternions

The equalities easily are shown:

The notation (has, V)

The quaternion $Q\; =\; has\; \backslash \; cdot\; 1\; +\; B\; \backslash \; cdot\; I\; +\; C\; \backslash \; cdot\; J\; +\; D\; \backslash \; cdot\; K\; \backslash ,$peut to be broken up (and in a single way) into a formed couple of reality $a\; \backslash ,$ and vector $\backslash \; vec\; V$ of $\backslash \; mathbb\; R^3$ whose coordinates are ( B , C , D ).

One writes: $Q\; =\; (has\; \backslash ,\; \backslash \; \backslash \; vec\; V)\; \backslash ,$.

This notation makes it possible to define the sum and the product in the following way:

It also makes it possible to redefine or define the 3 following concepts:

• combined the $Q^*\; =\; (has\; \backslash ,\; \backslash \; -\; \backslash \; vec\; V)\; \backslash ,$ of $Q\; \backslash ,$,
• the scalar product of two quaternions: $Q\_1\; \backslash \; bullet\; Q\_2\; =\; (a\_1\; \backslash ,\; \backslash \; \backslash \; vec\; V\_1)\; \backslash \; bullet\; (a\_2\; \backslash ,\; \backslash \; \backslash \; vec\; V\_2)\; =\; a\_1\; \backslash \; cdot\; a\_2\; +\; \backslash \; vec\; V\_1\; \backslash \; bullet\; \backslash \; vec\; V\_2$
  

from where one deduces:

• the standard of a quaternion: $\backslash |Q\; \backslash |\; =\; \backslash \; sqrt\; \{Q\; \backslash \; bullet\; Q\}\; =\; \backslash \; sqrt\; \{Q.Q^*\}\; =\; (Q.Q^*)\; ^\; \backslash \; frac\; \{1\}\; \{2\}\; =\; (a^2\; +\; \backslash \; vec\; V\; \backslash \; bullet\; \backslash \; vec\; V)\; ^\; \backslash \; frac\; \{1\}\; \{2\}\; =\; (a^2+\; \backslash |\backslash \; vec\; V\; \backslash |^2)\; ^\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash ,$

foot-note : le produces scalar definite above is commutative and it is thus of course différent product of quaternions defined higher.

That is to say now a quaternion $Q\; =\; (has\; \backslash ,\; \backslash \; \backslash \; vec\; V)\; \backslash ,$ unspecified; let us note $q\; =\; \backslash |Q\; \backslash |\backslash ,$ and $v\; =\; \backslash |\backslash \; vec\; V\; \backslash |\backslash ,$. If reality $v\; \backslash ,$ positive is not null, reality $q\; \backslash ,$ is not it either and one can thus always write:

$Q\; =\; Q\; \backslash \; cdot\; \backslash \; left\; (\backslash \; frac\; \{has\}\; \{Q\}\; \backslash ,\; \backslash \; \backslash \; frac\; \{1\}\; \{Q\}\; \backslash \; cdot\; \backslash \; vec\; V\; \backslash \; right)\; =\; Q\; \backslash \; cdot\; \backslash \; left\; (\backslash \; frac\; \{has\}\; \{Q\}\; \backslash ,\; \backslash \; \backslash \; frac\; \{v\}\; \{Q\}\; \backslash \; frac\; \{1\}\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; V\; \backslash \; right)\; \backslash ,$

However $\backslash \; frac\; \{1\}\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; V$ is a normalized vector and one can write: $q^2\; =\; a^2\; +\; v^2\; \backslash ,$, or: $\backslash \; left\; (\backslash \; frac\; \{has\}\; \{Q\}\; \backslash \; right)\; ^2\; +\; \backslash \; left\; (\backslash \; frac\; \{v\}\; \{Q\}\; \backslash \; right)\; ^2=\; 1$.

It results from it that there exists:

• an angle $\backslash \; varphi\; \backslash ,$ (whose cosine and sine are worth $\backslash \; frac\; \{respectively\; has\}\; \{Q\}\; \backslash ,$ and $\backslash \; frac\; \{v\}\; \{Q\}\; \backslash ,$) and
• a vector normalized $\backslash \; vec\; U\; =\; \backslash \; frac\; \{1\}\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; V$

who are such as one can write the quaternion $Q\; \backslash ,$ (of vector $\backslash \; vec\; V$ not no one) in the form:

This way of writing a quaternion is important: the terms of the couple, $q\; \backslash \; cos\; \backslash \; varphi\; \backslash ,$ and $q\; \backslash \; sin\; \backslash \; varphi\; \backslash \; cdot\; \backslash \; vec\; U\; \backslash ,$, are indeed respectively the scalar product and the vector product of two vectors $\backslash \; vec\; orthogonal\; V\_1$ and $\backslash \; vec\; V\_2$ with $\backslash \; vec\; V$, these 2 vectors forming between them an angle equal to $\backslash \; varphi\; \backslash ,$. And this writing makes it possible to build the multiplication of the quaternions thanks to the composition of the similarities of $\backslash \; mathbb\; R$ ³ as one can see it while clicking here

Similarities of space and quaternions

To demystify the quaternions, we will make a small instructive turning by the elementary geometry and in particular by the similarities in space. A Similitude in $\{\backslash \; mathbb\; R\}\; ^3$ is entirely defined by triple given:

• of an axis of rotation well directed (an unit vector U),
• of a Angle 2φ defined in 2kπ near and
• of a report/ratio of Homothety K, a reality strictly positif.
  the effect of a similarity on all the vectors can be regarded coarsely as a screwing with expansion .

Voyage and ways

More precisely, the image of transformed of a vector V (of which the origin is supposed to be located on the axis U) is obtained initially by a multiplication (homothety) of this vector per K, followed by a rotation of angle 2φ around the axis of rotation (one could also start with rotation and make it follow homothety, but it would be necessary to modify a little the explanations which will follow…). This rotation makes turn of an angle 2φ the end of vector Kv on a circle (C) centered on the axis and located in a plan perpendicular to U. However on this circle, there are two ways of carrying out the way: either by using an arc, or by using its complementary, these arcs not being able unfortunately to be distinguished by only 2φ measurement + 2kπ.
It is precisely this difficulty which the concept of quaternion makes it possible to solve. Schematically, one can say that a quaternion, it is as a similarity which could distinguish the 2 ways that can borrow associated rotation.

In the everyday life, so for a voyage between two localities L and L, you have a priori two possible ways, the distinction between these ways can be made by indicating two intermediate site-stages S and S. And while speaking about the way S and the way S, you will imply arrival and the starting localities L and L.

By preserving this analogy, it is necessary for us thus to define two intermediate points out of the two arcs of the way.

Halfway

The points located halfway are perfect for this mission. Indeed, if I divide the angle of vectors 2φ + 2kπ by 2, I obtain two distinct angles φ + 2kπ and φ-π + 2kπ. However, if I use the rotation of axis U and angle φ + 2kπ, I definite a site-stage different from that which I obtain with rotation φ-π + 2kπ. Thus to the similarity sim (U, 2φ, K), it corresponds two distinct ways which are represented by the two distinct quaternions quat (U, φ, K) and quat (U, −π+φ, K).

the formalism

The triplet (U, φ, K) can be written in an equivalent way in the shape of the couple (kcos (φ), ksin (φ) ∙U) of the notation (has, V). And by using vectors has and B orthogonal with U suitable, it is easy to show that this couple takes the form (a.b, a^b). Thus, our site-stages enable us to return to very simple operations on vectors. And as these operations are rich remarkable properties, one can define (as one saw above) a multiplication and an addition of the quaternions. You can “see” these two operations on the quaternions here: http://www.alcys.com

To announce that a promising way of research can be consulted on the site. One precisely defines in it the similarity in space in three dimensions by a Bivecteur which is with the couple of vectors what the vector is with the couple of points. The law of composition introduced into these bivecteurs is indeed noncommutative, and the restriction of this unit on the plan is the whole of the complexes.

Double product of quaternions

Just as one can calculate a double vector product, it is possible to calculate a double product of quaternions.

Correspondence between unit quaternion and vectorial rotation

One can show that transformed the $\backslash \; vec\; V\text{'}\; =\; \backslash \; mathbf\; R\_\; \{\backslash \; left\; \backslash \; vec\; NR\; \backslash \; right\}\; (\backslash \; vec\; V)\; \backslash ,$ of any vector $\backslash \; vec\; V\; \backslash ,$ unspecified (of the Euclidean space of dimension 3) in the rotation $\backslash \; mathbf\; R\; \backslash \; left\; NR\; \backslash \; right$ of angle $2\; \backslash ,\; \backslash \; varphi\; \backslash ,$ and of axis $\backslash \; vec\; NR\; \backslash ,$ ($\backslash \; vec\; NR\; \backslash ,$ being a normalized vector) can be calculated thanks to the product of following quaternions:

where $(\backslash \; cos\; \backslash \; varphi,\; \backslash \; \backslash \; sin\; \backslash \; varphi\; \backslash \; \backslash \; vec\; NR)$ and $(\backslash \; cos\; \backslash \; varphi,\; \backslash \; -\; \backslash \; sin\; \backslash \; varphi\; \backslash \; \backslash \; vec\; NR)$ are two combined unit quaternions and where $(0,\; \backslash \; \backslash \; vec\; V)$ and $(0,\; \backslash \; \backslash \; vec\; V\text{'})$ are quaternions whose scalar component is null.

One can also write this transformed with the notation $Q\; =\; has\; \backslash \; cdot\; 1\; +\; B\; \backslash \; cdot\; I\; +\; C\; \backslash \; cdot\; J\; +\; D\; \backslash \; cdot\; K\; \backslash ,$. If rotation is around an axis directed according to the vector $\backslash \; vec\; V\; \backslash ,$ of coordinates (X, there, Z) (the vector being normalized) and of angle $\backslash \; varphi$, the associated quaternion is worth:

Composition of vectorial rotations and produced quaternions

The preceding property justifies the fact that one has habit to say, but in a not very rigorous way, that the quaternion $(\backslash \; cos\; \backslash \; varphi,\; \backslash \; \backslash \; sin\; \backslash \; varphi\; \backslash \; \backslash \; vec\; NR)$ represents rotation $\backslash \; mathbf\; R\; \backslash \; left\; NR\; \backslash \; right$.

By using the same approximate language, one can say that the composition of two successive rotations $\backslash \; mathbf\; R\_1$ puis $\backslash \; mathbf\; R\_2$ is a rotation $\backslash \; mathbf\; R$ which are represented by the quaternion $Q\; =\; Q\_2\; \backslash \; cdot\; Q\_1\; \backslash ,$, the quaternions $Q\_1\; \backslash ,$ and $Q\_2\; \backslash ,$ being the respective representatives rotations $\backslash \; mathbf\; R\_1$ and $\backslash \; mathbf\; R\_2$.

Let us show it!

While posing: $\backslash \; vec\; V\text{'}\; =\; \backslash ,\; \backslash \; mathbf\; R\_1\; \backslash ,\; (\backslash \; vec\; V)\; \backslash ,$, then $\backslash \; vec\; V$ = \, \ mathbf R_2 \, (\ vec V') \, , the formula framed above gives us, written in a condensed way, the 2 equalities:
$(0,\; \backslash \; \backslash \; vec\; V\text{'})\; =\; Q\_1\; \backslash \; cdot\; (0,\; \backslash \; \backslash \; vec\; V)\; \backslash \; cdot\; Q^*\_1$ et
$(0,\; \backslash \; \backslash \; vec\; V$) = Q_2 \ cdot (0, \ \ vec V') \ cdot Q^*_2, which can thus be still written:

$(0,\; \backslash \; \backslash \; vec\; V$) = Q_2 \ cdot \ left (0, \ \ vec V) \ cdot Q^*_1 \ right \ cdot Q^*_2 or, if one takes account of the associativeness of the product of quaternions:
$(0,\; \backslash \; \backslash \; vec\; V$) = (Q_2 \ cdot Q_1) \ cdot (0, \ \ vec V) \ cdot (Q^*_1 \ cdot Q^*_2) , or:
$(0,\; \backslash \; \backslash \; vec\; V$) = (Q_2 \ cdot Q_1) \ cdot (0, \ \ vec V) \ cdot (Q_2 \ cdot Q_1) ^*, by taking account of the value of combined of two quaternions.

What establishes the property announced for the composition of two rotations and that we will write:

$\backslash \; Bigg\; (0,\; \backslash \; \backslash \; mathbf\; R\_\; \{\backslash \; left\; \backslash \; vec\; N\_2\; \backslash \; right\}\; \backslash \; left\; (\backslash \; mathrm\; R\_\; \{\backslash \; left\; \backslash \; vec\; N\_1\; \backslash \; right\}\; (\backslash \; vec\; V)\; \backslash \; right)\; \backslash \; Bigg)\; =\; (\backslash \; cos\; \backslash \; varphi\_2,\; \backslash \; \backslash \; sin\; \backslash \; varphi\_2\; \backslash \; \backslash \; vec\; N\_2)\; \backslash \; cdot\; (\backslash \; cos\; \backslash \; varphi\_1,\; \backslash \; \backslash \; sin\; \backslash \; varphi\_1\; \backslash \; \backslash \; vec\; N\_1)\; \backslash \; cdot\; (0,\; \backslash \; \backslash \; vec\; V)\; \backslash \; cdot\; (\backslash \; cos\; \backslash \; varphi\_1,\; \backslash \; -\; \backslash \; sin\; \backslash \; varphi\_1\; \backslash \; \backslash \; vec\; N\_1)\; \backslash \; cdot\; (\backslash \; cos\; \backslash \; varphi\_2,\; \backslash \; -\; \backslash \; sin\; \backslash \; varphi\_2\; \backslash \; \backslash \; vec\; N\_2)$

Matric notations

Just as it is possible to put in correspondence the complex number $z\; =\; has\; +\; I\; B\; \backslash ,$ with the matrix: , it is possible to make correspond the quaternion $Q\; =\; has\; +\; B\; I\; +\; C\; J\; +\; D\; K\; \backslash ,$ with the following complex matrix:

or with the following real matrix:

There exist several matric representations of a quaternion. The preceding matrix is one. That which follows is more often used. Thus, the real matrix created starting from a quaternion is written this way (by keeping q=a+ib+jc+kd):

If the unit quaternion represents a rotation since the origin, one can represent it using a matrix 3x3

With these equivalences, the sum and the product of two quaternions correspond respectively to the sum and the product of the matrices which correspond to them.

Note::

The matrix complexes can be still written in the form:

where 4 matrices: , , and is the complex matrices which correspond to four quaternion-units 1, I , J and K evoked in the first definition of the quaternions.

Applications

Whereas that is debatable in dimensions three, the quaternions cannot be employed in other dimensions (although extensions like those of the Biquaternion S and the algebras of Clifford are usable). In any event, the concept of vector had almost universally replaced that of the quaternions in science and technology in the middle of the 20th century.

Today, the quaternions find their place in Infographie, Control theory, in the Treatment of the signal, in the ordering of movement and the orbital Mécanique, mainly to represent the Rotation S and the orientations in dimension three. For example, it is frequent that the control devices of displacement of a spaceship are governed in terms of quaternions. The reason is that to carry out much operations on the quaternions is numerically more stable than to carry out many operations on the matrices.

Interpolation of rotations

If one takes 2 rotations of space $r\_a$ and $r\_b$, the linear interpolation of these rotations is in general not a rotation. To be able to interpolate, it is necessary is

• to use the angles of Euler,
• to use the quaternions.

In the last case, 2 rotations are represented by 2 quaternions $q\_a$ and $q\_b$ on the sphere unit $S\_3$, and the interpolation corresponds to the Géodésique between these 2 points

See too

Related articles

• Group of quaternions
• Biquaternion S
• Matrix of Dirac
• Algebra of space time
• Adolf Hurwitz

External bonds

• Definition

• On the quaternions

Sources

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