Detailed quaternion

Quaternion Visualization

Understand the complex number dimension by dimension

One dimensional villains understand the plural

Little people can only understand operations in the range of real numbers , Just to i Take the square of itself as -1 Special symbols of .

Geometrically , Multiplication in the plural range , Consider the former complex number as a function acting on the second complex number

Fixed origin , take （1,0） This point is pulled to z Point location , After transformation w The point is the product result , There is only one transformation in two-dimensional space that can get this result

This process involves stretching and rotation , Stretching is very easy to understand , Because the one-dimensional number axis can also be . What about rotation ？ Isn't this the concept of a two-dimensional plane ？ All rotations correspond to a unique degree of rotation , We can project it onto a one-dimensional number axis

Imagine a point moving on a unit circle , This point is similar to （-1,0） The connecting line of and the complex axis intersect , One dimensional villain observes the complex axis

This point will first reach the infinite distance above , In the infinite distance from below

The two-dimensional man understands the three-dimensional rotation

Introduce a new unit j, A unit of distance from a complex plane , When we do not regard it as Z Axis , Instead, real numbers are treated as Z The axes allow us to better understand quaternions .

It is worth mentioning that , There is no good multiplication rule for such a definition , for example i*j and j*i Result

alike , We define a point on the unit sphere , And it and （-1,0） The intersection point of the line of on the plane

We rotate the unit ball , At the same time, how does the projection on the plane change

Understand quaternions

Quaternions contain three imaginary dimensions , They are all perpendicular to the real number axis , And they are perpendicular to each other

The corresponding mathematical property is

alike , Quaternion multiplication corresponds to the corresponding transformation

This sphere represents that all real parts are 0 Unit quaternion of

The yellow and red lines represent two concentric lines with the same radius 、 Circles that are perpendicular to each other and do not intersect （ In four dimensions , It's very common ）

The formula in the lower left corner is derived from the right-hand rotation theorem

multiply i Projection transformation of

More common cases , Multiply the left by a quaternion -0.5+0.5i+0.5j+0.5k

Rotate the first circle （ yellow ）, Give Way 1 Came to q The location of

Then rotate the vertical circle by the same angle according to the right-hand rule （ white ）

Three dimensional rotation

The easiest way to think of is that an object defines circles in three directions to control the direction , But then there will be “ Omnidirectional lock ” The problem of , That is to lose a degree of freedom

Two dimensional rotation ：

Let's first consider how to rotate in the two-dimensional plane , Let's say we're going to （4,1） This point rotates counterclockwise 30°, How to express by vector operation ？

We extend to three-dimensional rotation , We define a unit vector in the direction of the axis of rotation

Multiply the sine of the same rotation angle by the imaginary part , But this imaginary part has three parts —— Used to determine the rotation unit vector

This is rotation in a given direction 40°,** But in fact we use half angle ！！！ ** Why? ？

We want to know the position of a point after this transformation , What we need to do is not to multiply these two quaternions directly , Instead, a special “ Sandwich ” form

A quaternion multiplication visual website , It's also 3Blue1Brown The teacher made it himself (eater.net)—— Need to climb over the wall

There is a super cool function in it , That is, you can directly operate the visual case in his demonstration video. Through this, you can better understand why you need to right multiply an inverse matrix

Why right multiply

First, perform the normal rotation operation , Along i Direction of rotation

Then we do not right multiply an inverse matrix

You can see clearly , The four-dimensional hypersphere has an obvious displacement on the axis of rotation ——j（i） spot , This yellow green line is its projection in three-dimensional space

Want to eliminate this **“ side effect ”** You have to right multiply the inverse matrix

Why half angle

We can see clearly from the video website , Left multiplication not only rotates the 3D points , Migration and deformation are also carried out

$$i\ast j=-(j\ast i)$$
Left and right multiplication rotate in opposite directions , There is a minus sign for reverse
$$i\ast j=j\ast （-i)$$
That is, left multiply i And multiply right i The inverse matrix of is the same , The two directions of rotation are consistent ; But offset and deformation are the opposite , So only half angle

Unity The use of

Unity 3D in Rotation

stay Unity in , Rotation can usually be a three-dimensional vector (x,y,z) Express . Actually this is Euler's angle . The three components are respectively around x Axis 、y Axis and z The rotation Angle of the axis .

To one GameObject Rotate , You can use the following code directly ：

transform.Rotate(xAngle, yAngle, zAngle);

Editor in Transform The rotation axis of the component is the model space coordinate axis of the parent node , If there is no parent node , Then the rotation axis is the world coordinate axis .

stay Script Use in Rotate function , stay Space.Self Middle rotation

public void Rotate(Vector3 eulerAngles, Space relativeTo = Space.Self);
public void Rotate(float xAngle, float yAngle, float zAngle, Space relativeTo = Space.Self);
public void Rotate(Vector3 axis, float angle, Space relativeTo = Space.Self);

stay Space.Self Rotate in , The rotation axis is the coordinate axis of the local coordinate system .

stay Space.World Rotate in , The axis of rotation is the axis of the world coordinate system .

Different rotation order will lead to different results , As you can see from the documentation , Its transform.Rotate() It uses Z-X-Y Compliance

gimbal lock

The real problem with the universal joint lock is the interpolation animation , The object will rotate along an arc , This is because the lack of a degree of freedom must be adjusted in this way

 transform.eulerAngles = Vector3.Lerp(transform.eulerAngles, new Vector3(150, 90, 90), 0.5f);

Rotate with quaternions

​ Unity Quaternions in support of and Vector3 Multiply , If you take this Vector3 Think of it as a vector , that Multiply the left by a quaternion , It's equivalent to rotating this vector .

transform.rotation = Quaternion.FromToRotation(transform.up, toDirection) * transform.rotation;

The first parameter is what to rotate , The second parameter is the direction to be rotated. The three-dimensional coordinates here all represent a vector

The above code , Change it to quaternion

  transform.rotation = Quaternion.Slerp(transform.rotation, Quaternion.Euler(new Vector3(150,90,90)), dTime);