quaternion transformations is which course fof math

Full Answer

Is there a single quaternion that represents a combination of transforms?

This means that we cant necessarily derive a single quaternion that can represent a combination of the above transforms. This is obvious since a quaternion contains 4 scalar values so there is no way it can represent say the 6 degrees of freedom of a solid object.

What are quaternions in math?

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative.

How to use quaternions for rotation and translation?

Use quaternions for the rotation part and handle the translation part separately ( see affine translations ). If we want to rotate, reflect or scale around a point other than the origin, this is the same as doing the operation around the origin combined with a translation.

What is the quaternion to represent a 90 degree rotation?

As an example of using quaternions for 3D rotations, what is the quaternion to represent a 90 degree (PI/2 radian) rotation about the z axis: q = cos (t/2) + i ( x1 * sin (t/2)) + j (y1 * sin (t/2)) + k ( z1 * sin (t/2))

What class are quaternions taught in?

The subject of quaternions is often taught in engineering and physics courses due to its applications in 3D rotations and quantum mechanics.

Are quaternions an algebra?

Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that generate the elements of order two in the Brauer group of F. For some fields, including algebraic number fields, every element of order 2 in its Brauer group is represented by a quaternion algebra.

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.

Are quaternions linear algebra?

Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear algebra over skew fields.

What are quaternions used for?

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.

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.

How do i learn quaternion?

3:225:59Quaternions and 3d rotation, explained interactively - YouTubeYouTubeStart of suggested clipEnd of suggested clipNumber that's 30 degrees off the horizontal. With magnitude 1 cosine 30 degrees plus sine of 30MoreNumber that's 30 degrees off the horizontal. With magnitude 1 cosine 30 degrees plus sine of 30 degrees. Times I and then you just multiply this by your point represent it as a complex number.

What is a rotation in math?

A rotation is a type of transformation that takes each point in a figure and rotates it a certain number of degrees around a given point.

What is q8 group?

In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation.

Is quaternion multiplication associative?

Quaternion multiplication is associative; not every operation is (look up the "octonions" to see that the "next generalization" gives you a non-commutative, non-associative mutliplication.

Is quaternion 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).

Is quaternion a vector?

Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space.

How to get a quaternion?

multiply to get a quaternion to represent a combination of two separate rotations. Cant combine reflections in this way. Combining two reflections gives a rotation, but we cant do it by simply multiplying the two quaternions. real part. The real part depends on the amount of rotation. The real part is zero.

What do the blue arrows on the left and right of the quaternions do?

The blue arrows, going from left to right, show different rotations which take a point P in and moves it to point P out. The red arrow, from top to bottom, takes two rotations and combines them to give a third. We will see here how to use quaternions to do both of these algebras.

What is the rotation of Q1 followed by Q2?

The other important result is that a rotation of q1 followed by a rotation of q2 is equivalent to a single rotation of q2*q1. Note the reversal of order, that is, we put the first rotation on the right hand side of the multiplication.

What is category theory?

Category theory (see Page here) allows us to study algebraic structures by looking at their external properties. Other approaches build up structures from simpler elements. Here we look at morphisms between whole structures and represent these as diagrams like this:

What is the quaternion in number theory?

Quaternions are also used in one of the proofs of Lagrange's four-square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm .

When were quaternions introduced?

Quaternions were introduced by Hamilton in 1843 . Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues ' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.

How to represent quaternions as matrices?

There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 real matrices. In each case, the representation given is one of a family of linearly related representations. In the terminology of abstract algebra, these are injective homomorphisms from#N#H {displaystyle mathbb {H} }#N#to the matrix rings M (2,ℂ) and M (4,ℝ), respectively.

What is the quaternion group?

P.R. Girard's 1984 essay The quaternion group and modern physics discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely SO (3), the Lorentz group, the general theory of relativity group, the Clifford algebra SU (2) and the conformal group, can easily be related to the quaternion group in modern algebra. Girard began by discussing group representations and by representing some space groups of crystallography. He proceeded to kinematics of rigid body motion. Next he used complex quaternions ( biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He cited five authors, beginning with Ludwik Silberstein, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions ( split-biquaternions) as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. Among the fifty references, Girard included Alexander Macfarlane and his Bulletin of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions.

What is the center of a noncommutative ring?

The center of a noncommutative ring is the subring of elements c such that cx = xc for every x. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if q = a + b i + c j + d k belongs to the center, then

What is the scalar part of a quaternion?

is called the vector part (sometimes imaginary part) of q, and a is the scalar part (sometimes real part) of q. A quaternion that equals its real part (that is, its vector part is zero) is called a scalar or real quaternion, and is identified with the corresponding real number.

What are the symbols of a quaternion?

where a, b, c, d, are real numbers, and i, j, k, are symbols that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of a, b, c, d is 0, the corresponding term is omitted; if a, b, c, d are all zero, the quaternion is the zero quaternion, denoted 0; if one of b, c, d equals 1, the corresponding term is written simply i, j, or k .

What is the root of quaternions?

The root of quaternions is based on the concept of the complex number system. In addition to the well-known number sets ( Natural, Integer, Real, and Rational ), the Complex Number system introduces a new set of numbers called imaginary numbers.

Why do we use quaternions in computer graphics?

One of the most important reasons for using quaternions in computer graphics is that quaternions are very good at representing rotations in space. Quaternions overcome the issues that plague other methods of rotating points in 3D space such as Gimbal lock which is an issue when you represent your rotation with euler angles.

How to find the inverse of a quaternion?

The inverse of a quaternion is denoted q − 1. To compute the inverse of a quaternion, we take the conjugate of the quaternion and divide it by the square of the norm:

What is a transformation matrix?

In computer graphics, we use transformation matrices to express a position in space (translation) as well as its orientation in space (rotation). Optionally, a single transformation matrix can also be used to express the scale or “shear” of an object. We can think of this transformation matrix as a “basis space” where if you multiply a vector or a point (or even another matrix) by a transformation matrix you “transform” that vector, point or matrix into the space represented by that matrix.

Who discovered the orientation of an object in space?

The concept of quaterinions was realized by the Irish mathematician Sir William Rowan Hamilton on Monday October 16th 1843 in Dublin, Ireland.

Overview

Conjugation, the norm, and reciprocal

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let be a quaternion. The conjugate of q is the quaternion . It is denoted by q , q , , or q. Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. …

History

Definition

Algebraic properties

Quaternions and the space geometry

Matrix representations

Lagrange’s four-square theorem