Yang Eninala
Links: http://www.zhihu.com/question/23005815/answer/33971127
Source: Know
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According to my understanding, most people use Hamilton four yuan to do just three-dimensional rotation transformation (I have not seen any other usage). Then you do not have to study group theory, even do not review linear algebra, look at my next few pictures can be.
First, define a rotation that you need to do. The rotation axis is a vector, and the rotational angle is the rotation of the right manipulation. As shown in the following:
In this diagram,
Then the corresponding four-dollar number (the next three lines are a meaning, just a different expression)
Then its conjugate (the next three lines are a meaning, just a different form of expression),
If you want to calculate a point in this rotation under the new coordinates, you need to do the following,
1. Definition of pure four yuan
2. Perform a four-dollar operation
3. The generated must be a pure four yuan, that is, its first item is 0, has the following form:
4. The following three items are:
This completes the four-dollar rotation operation.
Similarly, if you have a four-dollar number:
So, it corresponds to a rotation operation with a vector-axis rotation (right-hand rotation).
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If you want a more in-depth understanding of the four-dollar number, please look down.
Four dollars invented by Hamilton, the invention originated in the day of 19th century. On this morning, Hamilton went downstairs to have breakfast. Then his son asked him, "Dad, can we do multiplication on ternary arrays (triplet, which can be understood as three-dimensional vectors)?" "Hamilton said," No, I can only add them. ”
At this point, the narrator from 21st Century said, "We are quick to see how the mathematicians of 19th century are two, even the inner product and the outer product are not known." ”
19th century Hamilton may not know the inner product and the outer product, but he knows that the three-dimensional vector multiplication He wants is a lot more "tall" than the inner product and the outer product operation. This multiplication operation satisfies the following four properties:
1. The result of the operation is also a three-dimensional vector
2. There is a unary operation, and the result of any three-dimensional vector being a meta-operation is itself
3. For any one operation, there is a inverse operation, the product of the two operations is the meta-operation
4. Arithmetic satisfies the binding law
In other words, what Hamilton wants to define is not a simple mapping relationship, but a group! (later we know that the four-dollar group is S3, and the four-dollar number represents the three-dimensional rotation is so (3), the former is the latter twice times the coverage) of the product of the property 1 is not satisfied, the outer product does not meet the nature of 3.
Mr. Hamilton was baffled by the question his son had raised. Through countless days and nights, he racked his brains and didn't want to understand the problem. Finally, one day (1843), Mr. Hamilton realized that the operation he needed was impossible in three-dimensional space, but that it was possible in the four-dimensional space that he was so excited that he carved a four-dollar formula on a bridge in Ireland.
Narrator: "WTF, I let you talk about the rotation of three-dimensional objects, you give me four-dimensional space to go up." ”
(no explanation, the following said four yuan is the total unit of four yuan)
In fact, the four-dollar number has four variables, which can be considered as a four-dimensional vector. A unit of four yuan (norm=1) is present in a sphere of four-dimensional space. , a four-dollar multiplied by four-dollar number is actually considered (1) for the left rotation, or (2) for the right rotation. So from the beginning to the end, the four-dollar definition is four-dimensional rotation, not three-dimension rotation! Any four-dimensional rotation can only be split into a left rotation and a right rotation, the expression is. Here, we have a left rotation and a right rotation for the four-dollar (four-dimensional vector). The result is of course a four-dollar number, which conforms to the nature of 1. This operation also conforms to the nature 2,3,4.
Well, after talking about four-dimensional rotation, we can finally talk about three-dimensional rotation. To be blunt, three-dimensional rotation is a special case of four-dimensional rotation, just as a two-dimension rotation is a special case of three-dimensional rotation. It is not accurate to say that the exception is a subset or a subgroup. In order to perform the three-dimensional rotation operation, Hamilton first carved out a three-dimensional space in four-dimensional space. Hamilton defines a pure four-quaternion, which has an expression of. The first item of pure four Yuan is zero, which exists in three-dimensional hyper-plane of four-dimensional space, and corresponds to three-dimensional vector one by one in three dimension space. Then, we have the common this is the left Multiply unit four, right multiply its conjugate expression. I really don't know how Lewis Hamilton came up with this, but looking back, this is a form of computing to limit the space in which the results of the operation are located. Simply put, when a three-dimensional vector is rotated three-dimensional, we want to get a three-dimensional vector. (If you can really get a four-dimensional vector, you do not dare to turn around at home, turn round, entered four times!) Then the left multiply unit four, right multiply its conjugate operation to ensure that the result is a three-dimensional super-plane in the pure four-dollar number.
The expression of the left and right multiplication as matrices will let us see more clearly. According to the definition, the matrix form is
Obviously, the previous matrix is a 4x4 four-dimensional rotation matrix, but it is only in the lower right corner of the 3x3 area and a unit matrix is different. So, it's a four-dimensional rotation that is limited to three dimensions. If the right side of the expression is not a conjugate, but an arbitrary four-dollar number, then what we do is a very normal four-dimensional rotation. If it's just a four-dollar number on the left and nothing on the right, then we get a subset of the four-dimensional rotation, and this subset doesn't guarantee that the results will be confined to the three-dimensional hyper-plane. If only the right multiply, not the left multiply is the same.
Say so much, for sticking to the last of you, a piece, to the table thanks.
In fact, this picture explains a long-standing question. Why four dollars is used instead of. This is because a rotation is done, and a rotation is made. We rotated two times, not once, and the result of the two rotations was a rotation of the rotation angle.
Four yuan (turn to know)
