Four elements or vectors?

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The four elements are weapon used to describe the posture of a moving object. It was discovered in Ireland in 1843 by Sir William Rowan Hamilton.

Chinese mathematicians
Hamilton was born in Ireland. He was taken care of by James, his uncle. Hamilton liked literature very much. During his college years, he not only studied mathematics, but also classical literature. Therefore, he is proficient in 12 languages in total. In addition to the European language, he also knows Persian, Malay, Arabic, Sanskrit and Chinese. His wife is the daughter of a priest. The greatest achievement of Hamilton is the discovery of the Quaternary element and its wide application in various aspects of physics. Hamilton has made significant contributions to the development of optics, dynamics, and algebra. His achievements became the backbone of quantum mechanics.

The treasure rock

One day in the autumn of 1843, Hamilton and his wife walked along the royal canal in Dublin. At that time, he was studying how to extend the two-dimensional plural to a higher dimension. He can't think of a 3D example, but the four-dimensional model can. He made a very simple but extremely creative assumption:
Ling
I ^ 2 =-1
J ^ 2 =-1
K ^ 2 =-1
However
I! = J! = K
In addition, his equation completely abandoned the multiplication exchange rate, which was an extreme idea at the time.
I * j = K; j * I =-K;
J * k = I; K * j =-I;
K * I = J; I * k =-J;
Hamilton immediately engraved the equation on the nearby broom bridge. Later, the National University of Ireland, Manus, began to host a trip from the dunsink observatory to the Royal Canal on 1989. Unfortunately, the Hamilton's engraved stone was not found on the way.

Four elements or vectors?
After the four elements are found, Hamilton continues to promote the four elements. He has written many related monographs, including elements of quaternions, which is more than eight hundred pages. But soon, vectors were found and popular. Some Hamilton supporters have spared no effort to oppose the development of vector algebra of Oliver heiveiser and Vector Analysis of josia Willard gark to maintain the detachment of the element. In fact, in the middle of the 20th century, in the scientific and engineering world, vectors have almost completely replaced the position of the Quaternary.
Mathematicians believe that the Quaternary element is a subalgebra called Clifford Algebra of geometric algebra, which has been well studied and applied, especially in theoretical physics. For example, the special theory of relativity and classical electrodynamics can be expressed in a very beautiful form by using geometric algebra. The bubble matrix commonly used in quantum mechanics is actually a matrix representation of a subalgebra of geometric algebra, A similar example shows the non-interchangeable rotation of a rigid body in classical mechanics.
Now, the Quaternary element is used to represent the rotation and orientation of a three-dimensional object in computer graphics (and related image analysis. It is also seen in control theory, signal processing, attitude control, physical and orbital mechanics, and is used to indicate rotation and orientation. This is due to the fact that the conversion and combination of four elements is more stable than many matrix conversion combinations in numbers.
However, even today, the use of the four elements is still under debate.

Rotate with four elements
Four elements can be regarded as vectors with four elements:
Q = I * x + J * Y + K * z + W
Inverse Q = I * x + J * Y + K * z-W
Each element of Q indicates a rotation.
Q1 * Q2 indicates that a Q2 rotation and a Q1 rotation are performed first.
Q and-q are the same rotation.
If you want to round an object along the normalized vector (x, y, z) Around degrees of parallelism, then Q is created accordingly:
Q = sin (Cosine/2) * (x * I + y * j + K * z) + cos (Cosine/2)

Advantages of four elements
Interpolate)
Completely eliminate gimbal lock (or no singularity)
Although the four elements are restored to a matrix at last, the efficiency of the four elements in concatenation is much better than that of the matrix.

Convert four elements into a matrix
1-2 * (y * Y + z * z) 2 * (x * Y-z * W) 2 * (z * x + y * w) 0
2 * (x * Y + z * w) 1-2 * (z * z + x * X) 2 * (y * z-x * w) 0
2 * (z * x-y * W) 2 * (y * z + x * w) 1-2 * (x * x + y * Y) 0
0 0 0 1
Q = (w, x, y, z)

Four elements or vectors?