Quaternion -- basic concepts (Reprint)

from :https://zhuanlan.zhihu.com/p/27471300

1. Why use quaternions

Maybe everyone has seen the origin of quaternion many times . Long ago , An old man is sitting by the bridge , Look at the passing ships , All of a sudden a flash of inspiration , Engrave a few lines of big characters on the stone tablet beside the bridge , Quaternion was born ！ Everyone loves to hear stories , So why do we need quaternions ？ One way is to solve vector multiplication , We know that multiplication between vectors has inner product and outer product , But neither of these two operations is perfect , That is, the condition of the group is not satisfied （ Of course, when quaternion was born, there was no saying of inner product and outer product ）. Is there such a perfect multiplication between vectors , So the problems that cannot be solved in three-dimensional space are mapped to four-dimensional space . This is the opportunity for the birth of quaternions .

So here comes the question , Since quaternion is only for matrix multiplication , So why do we now use quaternions to rotate , Even replaced Euler angle 、 Shaft angle and other forms ？ First , Quaternions are not born to solve three-dimensional rotation , But its nature is very conducive to the expression of rotation information （ We'll talk more about that later ）, Therefore, to understand the properties of quaternions is prior to the application of quaternions in rotation . As for Quaternion replacing Euler angle and other forms , It needs to involve some other knowledge points , Let me first list the advantages of quaternion over other forms ：

 Solve the universal joint deadlock （Gimbal Lock） problem 
 Just store 4 A floating point number , Lighter than matrix 
 Quaternion, whether it's inverse 、 Series operation, etc , More efficient than matrix 

So take it into consideration , Now mainstream games or animation engines will scale vectors + Rotating quaternion + The motion data of the character is stored in the form of translation vector .

2. How did you come up with quaternion
Mediocre tutorials will directly propose the definition of quaternion 、 Operation rules and so on , Then the reader is at a loss . contrary , More systematic tutorials generally start with the plural （Complex Number） To guide , Gradually put forward the definition of quaternion , This will make it easier for readers to understand , It is also better to form a picture in the brain . The relationship between complex numbers and quaternions 、 And how to extend the concept of complex number to quaternion is an idea we need to clarify .

Let's start with a few concepts .

Subspace in space ： generally speaking , Space （ dimension >2） There are lower dimensional subspaces , For example, one-dimensional subspace in two-dimensional space , It's a straight line ; One dimensional subspace and two-dimensional subspace in three-dimensional space , That is, lines and faces . When we go beyond the concept of three-dimensional, it is difficult to imagine what it looks like , But there must be three-dimensional subspace or two-dimensional subspace in four-dimensional space . We'll add a prefix here （hyper） To describe these spaces , For example, a plane in a high-dimensional space is called a hyperplane .

Mapping between space and subspace ： We express two-dimensional space as (x,y), When y=0 when , In fact, it can be regarded as one-dimensional , It just means (x,0) This form . Push to the fourth dimension ,(w,x,y,z), When w=0 when ,(0,x,y,z) It's a three-dimensional subspace , That's why we can manipulate three-dimensional vectors with unit quaternions , In fact, we are mapping a three-dimensional vector to a four-dimensional three-dimensional subspace （w=0, This form also becomes a pure quaternion ）, Then rotate it , The final vector result is still in this three-dimensional subspace , So it can be mapped back to three-dimensional space .

Guangyiqiu ： The ball here is generalized . We're on a two-dimensional plane , The generalized ball actually refers to circle, Three dimensional space is our cognitive ball , be called two-sphere, The generalized sphere in four-dimensional space is actually a hypersphere （hyper-sphere）, Also known as three-sphere. The unit vector is actually the point on the generalized Ball , And the unit quaternion is three-sphere The point above .

Constraints and eigenvectors ： A point in space is represented by x, y, z And other parameters , In general, the number of parameters is equal to the dimension , Points in two-dimensional space {x, y} Parameters , Points in four-dimensional space {x, y, z, w} Parameters . But to constrain the points in space , This will reduce the number of parameters , For example, a point in three-dimensional space is on a unit sphere , The original three parameters {x, y, z} The point that can be expressed now only needs two parameters {u, v} It can express . If {u, v} It's the unit vector , It can also be called {u, v} yes {x, y, z} Eigenvector of .

The above concept gives you an idea , Quaternion is not a thing that can be achieved overnight , In terms of space , It is essentially no different from the well-known low dimensional space , Or it has a lot in common . Many properties of quaternions are extended from low dimensions , More specifically, it is extended from the concept of plural .

Then what is plural ？

The concept of plural number is very simple , It is actually to meet the obsessive-compulsive disorder of mathematicians ,-1 The square root of must be meaningful . Seemingly meaningless plural , But it is very useful in practice . Let's first look at its geometric expression , Make x It is the real part ,y It is the imaginary part , be x+iy It can be represented as the following figure .

The proposition of complex number directly adds a dimension to the original one-dimensional numerical range , The increase of data volume can be said to be huge . Let's think about complex numbers from the perspective of polar coordinates ,x+i·y It can be written. r·cosθ+i·r·sinθ, We only consider the unit plural ,|r|=1, And you get cosθ+i·sinθ. Speaking of this , It is necessary to propose a formula commonly used in algebra —— Euler formula , Basically, this formula will be used in all operations of complex numbers .

Next, I won't talk about the multiplication of complex numbers , Unit complex multiplication can achieve a two-dimensional rotation effect , If you are interested, you can draw a picture here , Push... Push . Let me first introduce how to derive quaternions from complex numbers , Just as a complex number is composed of a real part and an imaginary part , Then we will replace one imaginary part with three imaginary parts , That is, two intersect {i, j, k}.

among n Is a three-dimensional unit vector ,i²=j²=k²=i·j·k=-1. This is the conventional expression of quaternion , But unit quaternions have a lot of constraints , Not all four-dimensional vectors are quaternions .

Here is another brain hole , Maybe you heard about quaternion 、 Octet , But there are ternary numbers 、 Pentagram ？ In fact, objectively speaking , Ternary number 、 Four yuan number 、n All elements exist （ That is, there are infinite levels of complex numbers ）, But not all number systems satisfy modular operation . And with the improvement of dimension , Features will gradually be sacrificed , Compared with complex number operation , Quaternions sacrifice the commutative law ; Compared with quaternion operation , Octuples sacrifice the law of Association . Of course, there are also sixteen elements , But hexadecimals will have fewer features , It is even more meaningless in Mathematics , So quaternions are the most widely used .

3. How to understand quaternion
Now let's summarize the quaternion properties that have been obtained .

Four yuan number （ It doesn't refer specifically to Four yuan number = The unit is 4 yuan ） Is the point above a hypersphere in four-dimensional space , Satisfy w²+x²+y²+z²=1; A pure quaternion is a four-dimensional space in w=0 Point of a subspace of time , In the form of {0, q}, In particular, pure quaternion and quaternion are different concepts .
Quaternions are the result of the extension of the imaginary part of complex numbers , The imaginary part of a complex number is 1 individual , And the imaginary part of a quaternion has 3 individual , And they are orthogonal to each other , The real part is cosθ/2, The imaginary part is a potential axis multiplied by sinθ/2.
Quaternion degrees of freedom have no four dimensions , Due to the existence w²+x²+y²+z²=1 This constraint , Its degree of freedom is actually only 3, And each quaternion can correspond to an eigenvector , namely n. But remember that quaternions do not correspond to eigenvectors one by one , It will be said later .
Because quaternions exist in four-dimensional space , Therefore, how to use low dimensional information to understand high-dimensional information is particularly important . Let's take three-dimensional as an example , The three-dimensional sphere is expressed algebraically as x²+y²+z²=1, Although the point on the ball is made by x,y,z Three parameters to determine , But actually we only need two . Suppose to take x and z Express , among y Can pass x and z To solve the . that , We will y Axis information is hidden , Just look at the projection plane , As shown in the figure below . This picture means , The whole ball is XOZ The projection on the plane is a circle , When a point of the sphere is projected on a circle ,y=0; When the position of the projection is in the circle , There are two cases ,y>0 In the northern hemisphere ,y<0 In the southern hemisphere . So we can restore the whole sphere only through the projected circle .

Let's extend it to four dimensions ,w²+x²+y²+z²=1 To take x、y and z To represent the superball . As shown in the figure below , Project a four-dimensional space onto a three-dimensional hyperplane （w=0） It could be a two-sphere. When the projection point is in the whole two-sphere At the edge of ,w It must be 0, It is worth mentioning that the quaternion in this space is a pure quaternion . When the projection point falls on two-sphere When inside , There are also two situations ,w>0 and w<0. But we can find that the corresponding eigenvectors in these two cases are the same , So when we convert a rotation matrix to a quaternion , There are two corresponding values , The range of quaternions is 2 Times over 3D rotate （2:1 mapping）.

The above two paragraphs are relatively mysterious , But I think it is quite helpful to understand four-dimensional space and quaternion . This article has limited expressive ability , If you really don't understand , You can flip through 《Visualizing Quaternions》 Chapter viii. , Personally, I think this is one of the most wonderful parts of the book .

4. Quaternion “ Multiplication ” operation
The quaternion is mentioned above “ Multiplication ” The rule is to satisfy the property of the Group , What kind of multiplication is that , How to satisfy , Now let's take a look .

Because quaternions have i,j,k Three imaginary parts , So I have to be satisfied i²=j²=k²=i·j·k=-1 This condition . Here is the order * by “ Multiplication ” The operator , be p*q The formula is as follows , I won't write the specific derivation steps .

Let's start with the unit quaternion , Pass one by one according to the nature of the Group ：

Sealing property ： Easy to prove ,p and p Conjugate multiplication of ,|p*q|=1.
Associative law ： This is also a good proof , Just prove (p*q)*r=p*(q*r).
The above two proofs are not detailed here , After all, it's not easy to edit formulas . We can change our thinking , Think geometrically , Because the unit quaternion is still the unit quaternion after rotation , So it satisfies the closeness . meanwhile , Analogy matrix multiplication , The unit quaternion should also satisfy the law of Association .

Unit element ：e=(1,0,0,0), This is also an initial value of quaternion （ Equivalent to the identity matrix ）. It can be seen from the above formula ,p*(1,0,0,0) Or is it equal to p.
Inverse element ：p There is a quaternion ,* The result of the operation is e, Refer to the following formula for details , It can be seen that the inverse element is its conjugate divided by the square of the module .

If this is plural , It may also satisfy the commutative law , It is an Abelian group . But quaternions certainly do not satisfy the commutative law , From an algebraic point of view , Or from a geometric point of view . Come here , Basically, I have introduced the multiplication of quaternions , If you use its exponential form （exp form ）, It will be simpler to prove these properties .

5. The exponent of quaternion 、 Logarithm and inner product
According to Euler formula , The exponential form of quaternion is just another expression , All express the same thing . And its logarithmic form is the exponent of quaternion exponential form , As shown in the formula below . This formula is very useful when calculating quaternion interpolation , This will be discussed later .

So what is the relationship between logarithm and exponent ？ Let's refer to the figure below , In fact, the change of logarithm is actually the size of angle , So the relationship between exponential form and logarithmic form of quaternion can be seen intuitively with the figure below .

Finally, let's introduce the inner product of quaternions . Let's first look at three-dimensional space or two-dimensional space , If two unit vectors are inner products , The result of inner product is the angle between two vectors . The same is true of quaternions , The inner product of the quaternion of two units is its included angle , This conclusion is extended to n Wei is also established , The reason is simple , Because three points are coplanar , Even if n Dimensional space , The space of two vectors is only a two-dimensional subspace . As for the outer product , Because for complex numbers and quaternions , The meaning of outer product is not very obvious , There will be no too much discussion here .