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Why is quaternion used?
Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis.
Is a quaternion a matrix?
The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix. Multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for −1.
How do you get quaternion from vector?
One solution is to compute a vector half-way between u and v, and use the dot and cross product of u and the half-way vector to construct a quaternion representing a rotation of twice the angle between u and the half-way vector, which takes us all the way to v!
Where does the word quaternion come from in English?
From Middle English quaternioun, from Late Latin quaterniō, from quaternī, ablative form of quater (“four times”) + -iōn. The mathematics sense was coined by William Rowan Hamilton in 1843.
How to use a quaternion for a 3D rotation?
This matches the matrix representation of a 3D rotation so we use this ‘sandwich’ form: P out = q * P in * conj (q) to use a quaternion to apply a 3D rotation. Note that because of this double multiplication ‘i’ now represents a 90° for each multiplication, that is, 90°+90°=180° and similarly for ‘j’ and ‘k’.
What is the formula for a quaternion transform?
Probably the most important result on this page is the formula for representing rotations in 3 dimensions using quaternions (although we will also discuss how to use quaternions for other transforms on this page). This formula for 3D rotations is:
What kind of calculations can a quaternion be used for?
Quaternions find uses in both pure and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis.