That would be a course in abstract algebra, as a few users suggested. The statement means that any rotation in R3 can be written as fq for some unit quaternion q, which is uniquely determined up to a sign. Indeed, q and −q both map to the same rotation.
Are quaternions used in physics?
Physics: The quaternions have found use in a wide variety of research. – They can be used to express the Lorentz Transform making them useful for work on Special and General Relativity[9].
Are quaternions an algebra?
In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, being both a division ring and a domain.
What are quaternions in math?
quaternion, in algebra, a generalization of two-dimensional complex numbers to three dimensions. Quaternions and rules for operations on them were invented by Irish mathematician Sir William Rowan Hamilton in 1843. He devised them as a way of describing three-dimensional problems in mechanics.
What is a quaternion in the Bible?
noun. a group or set of four persons or things. Bookbinding. four gathered sheets folded in two for binding together. Mathematics.
Does Roblox use quaternions?
Quaternions is a Roblox game developer notable for developing the games surf and bhop, which are recreations of popular game modes in the FPS and Sandbox-style games Counter Strike: Source, Counter Strike 1.6, Half Life, Counter Strike: Global Offensive and Garry’s Mod.
Who invented quaternions?
In the case of quaternions, however, we know that they were discovered by the Irish mathematician, William Rowan Hamilton on October 16*#, 1843 (we will see later how we come to be so precise). The early 19*# century was a very exciting time for Complex Analysis.
Is a quaternion a matrix?
The quaternions form a noncommutative ring, and therefore addition and multiplication can be defined for quaternionic matrices as for matrices over any ring. Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors when computing the product of matrices.
Are quaternions complex numbers?
The quaternions are structured like the complex numbers, but with additional square roots of –1, which Hamilton called j and k. Every quaternion has the form a + bi + cj +dk, where a, b, c and d are real numbers, and i^2=j^2=k^2=-1.
Are quaternions 4D?
Complex algebra is 2D and what is known as quaternion algebra jumps to 4D. Using 1,i,j, and k as the base (where complex uses 1 and i (or j if you are an EE)) which results in a 4-axis space.
Are quaternions a complex algebra?
I just began to study about algebras over rings and quickly came across the fact that the quaternions are not an algebra over the complex numbers. I would prefer an answer as elementary as possible.
What does four quaternions of soldiers mean?
A group of four soldiers in the Roman legion.
How does a quaternion work?
Quaternions are an alternate way to describe orientation or rotations in 3D space using an ordered set of four numbers. They have the ability to uniquely describe any three-dimensional rotation about an arbitrary axis and do not suffer from gimbal lock.
How do you calculate quaternions?
- Identify the vector defining the axis of rotation.
- If needed, find its unit equivalent.
- The quaternion of rotation is q = cos(θ/2) + (xₐ*i + yₐ*j + zₐ*k) * sin(θ/2) , where:
- If needed, rotate v using the formula qᵥ’ = q * qᵥ * q⁻¹ , where:
Why are quaternions useful?
Quaternions are very efficient for analyzing situations where rotations in R3 are involved. A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo- metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.
What are 4 squads soldiers?
(A slightly larger unit is a section, which consists of 10 to 40 soldiers but is usually used only within headquarters or support organizations.) Three or four squads make up a platoon,… … two or more sections, or squads, led by noncommissioned officers.
What happened to the guards who were chained to Peter?
In the morning, there was no small commotion among the soldiers as to what had become of Peter. After Herod had a thorough search made for him and did not find him, he cross-examined the guards and ordered that they be executed.
What is the inverse of a quaternion?
The inverse of a quaternion refers to the multiplicative inverse (or 1/q) and can be computed by q-1=q’/(q*q’) for any non-zero quaternion.
Are quaternions a vector space?
The n-dimensional quaternionic space is a 4n-dimensional vector space over the real numbers.
What is a pure quaternion?
A pure quaternion is defined as a quaternion with a zero for the scalar value (q0=0). A standard 3D vector can be readily stored in a pure quaternion. Pure quaternions can then be used to rotate vectors or transform the vector coordinates between different rotated reference frames. Vector Transformation.
Do quaternions form a field?
The quaternions almost form a field. They have the basic operations of addition and multiplication, and these operations satisfy the associative laws, (p + q) + r = p + (q + r), (pq)r = p(qr).
Do quaternions commute?
Some quaternions do commute. Any quaternion corresponding to a real number commutes with every other quaternion. The quaternions corresponding to complex numbers will commute with one another, because the complex numbers are commutative.
Are quaternions continuous?
Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning.
Are imaginary numbers used in physics?
Imaginary numbers are what you get when you take the square root of a negative number, and they have long been used in the most important equations of quantum mechanics, the branch of physics that describes the world of the very small.
Are there 3D numbers?
There are no three dimensional numbers because it’s impossible to construct such a system that behaves like ‘numbers’. The real, complex, quaternion and octonion numbers are the only ‘normed division algebras’.