Analysis of four Yuan

What is a four-dollar number? Mathematically speaking, the four-dollar number is one of the plural. The so-called plural is the number of real and imaginary parts. The plural (A, a) defines the number A + bi, I is the so-called imaginary, A is the real part, B is the imaginary part, satisfies the i²=-1. For a complex number (a, a), B can be considered as the y-axis, and A is the x-axis. Four USD has three imaginary parts, a real part. For a four-dollar number (W, x, Y, z), W is its real part, and X, Y, Z is its imaginary part. Know the mathematical properties of the four-dollar number, but in order to facilitate our use of it. For example, the conjugate of a complex number, by making the imaginary part of the complex variable negative, it can calculate its conjugate. What is the meaning of conjugation for a four-dollar number? When the modulus of four is equal to 1, its conjugate is equal to its inverse. This makes it very easy to reverse the four-dollar number. Because we usually only care about the four-dollar number of modules that are 1.

Why use only $ four for modulo 1? First look at how to use the axis-angle relationship to describe a four-dollar number. Assuming that the four-tuple q = (w, x, Y, z) represents the rotation angle θ around the specified axis, we can represent it in this way using the axis angle: q= (cos (Θ/2), sin (Θ/2) n). This is one of the most important basic concepts of the four-dollar number-its four-digit relationship with the angle and axis. As you can see, the W component is related to the Cos (Θ/2), and the imaginary part of the vector is related to sin (Θ/2). When n is a unit vector, the modulus of four ²= w²+x²+y²+z²= cos (θ/2) ²+ sin (θ/2) ²n²= cos (Θ/2) ²+ sin (θ/2) ²= 1. (According to trigonometric theorem, sinθ²+cosθ²= 1.) Since n represents the axis of rotation, we certainly want to make the N unit vector easy to calculate, which is why we only use a modulus of 1 for the four-dollar number. The inverse of a four-dollar number is defined by dividing its conjugate by its modulus. This is why it is said that when the modulus of four is one, its conjugate equals its inverse.

The operation of four-dollar number is a cross-multiplication, a point multiplication, and a power operation. You can use a cross-multiply when we want to connect the rotation represented by two four-dollar numbers as if we were connecting two matrices. The point multiplication of the four-tuple is very similar to the point multiplication of the vector, and the result is a scalar, which is multiplied by the real part and the result of the dot multiplication of the imaginary part, and the resulting scalar is the result of the multiplication of the four-dollar number. The point multiplier is used to calculate the angle between two four-dollar numbers.  The angle of four Q1 and Q2 θ satisfies the cosθ= Q1 Q2. The four-tuple exponentiation is analogous to a real-power, such as Q, which rotates around the x-axis and 10°,q² 20 ° around the x-axis. For the four-dollar exponentiation we need to be aware that we can ask for q², but do not make further calculations on this basis, such as (q²) ², the result may not be the four of the Q we expect. In addition, if the known q1,q2,q1q0 = Q2, how to ask Q0? It's simple, multiply the inverse of Q1 with Q2. q0 = q1^ ( -1) Q2. Note that four-dollar cross-multiplication does not satisfy the commutative law, and we usually mean that the four-tuple-multiplier does not use a *, using AB, which indicates that the point is multiplied by a dot (•) symbol, such as a B.

One of the important uses of the four-dollar number is the spherical linear interpolation, slerp. The Slerp is able to smooth the difference between two and four, and there is no Euler angle interpolation for those problems. Calculation interpolation can be referred to:

, there are two non-0 constants K0 and K1, which makes VT = K0v0 + k1v1. The next triangle formula can be used to obtain the VT. Imagine a spherical surface in which all four of the modulo 1 are present on the plane.

The four-dollar number provides a unique way to provide smooth interpolation, which is not possible with Euler angles and matrices. Four-dollar cross-multiplication can be connected to multiple rotations, the conjugate can be more convenient to obtain reverse rotation, more convenient than matrix transpose. The four-dollar number is more economical than the matrix by using four numbers. The main problem with the four-dollar number is that it is very non-intuitive. Also, rounding of floating-point numbers and incorrect data entry may make it illegal.

It is usually recommended to use a value of four when using incremental rotation and smooth interpolation. When we do vector transformations between coordinate systems, we can only use matrices. For a given azimuth, the matrix is unique, there are countless representations of Euler's angle, and there are two four-dollar numbers, each of which is negative. The advantage of Euler's angle is that it is always legal and very intuitive and easy to use. When assigning an orientation to an object in the world, it is advisable to use an intuitive Euler angle.

Analysis of four Yuan