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For normalized vectors , The result of dot multiplication is equal to the cosine of the angle between two vectors
Calculate the angle between two vectors
public Transform t1, t2;
float dot = Vector3.Dot(t1.position.normalized, t2.position.normalized);
float angle = Mathf.Acos(dot) * Mathf.Rad2Deg;

Relationship between result and angle

For normalized vectors , In exactly the same direction , The dot multiplication result is 1, The opposite , The dot multiplication result is -1, Perpendicular to each other is 0

If the angle between two vectors is greater than 60 degree , be

if(angle>60){} or if(dot<0.5f){}// advantage ： Save CPU Then calculate the angle

private void Update()

    {

        Debug.DrawLine(Vector3.zero, t1.position,Color.red);

        Debug.DrawLine(Vector3.zero, t2.position,Color.red);

        float dot = Vector3.Dot(t1.position.normalized, t2.position.normalized);

        float  angle = Mathf.Acos(dot) * Mathf.Rad2Deg;

}

Cross riding

also called “ Cross product ” or “ Exoproduct ”

The formula

[x1,y1,z1]*[x2,y2,y3]=[y1*z2-z1*y2,z1*x2-x1*z2,x1*y2-y1*x2]

Geometric meaning

The result is the vertical vector of the two faces , The module length is the product of two vector module lengths and then multiplied by the sine of the included angle

API:Vector3 vector=Vector3.Cross（a,b）;

application ：

· Create a vector perpendicular to the plane
· Judge the relative position of two vectors , That is, clockwise and counterclockwise

 private void Update()

    {

       Debug.DrawLine(Vector3.zero, t1.position, Color.red);

        Debug.DrawLine(Vector3.zero, t2.position, Color.red);

        dot = Vector3.Dot(t1.position.normalized, t2.position.normalized);

        angle = Mathf.Acos(dot) * Mathf.Rad2Deg;

        Vector3 cross = Vector3.Cross(t1.position, t2.position);

        if (cross.y < 0)

            angle = 360 - angle;

        Debug.DrawLine(Vector3.zero, cross, Color.green);

}

The relationship between the module length and angle of the vector obtained by cross multiplication ：

0~90 Degree angle

Vector3 cross=Vector3.Cross(a.normalized,b.normalized);

float angle=Mathf.Asin(cross.magnitude)*Mathf.Rad2Deg;

( Rarely used )

Euler Angle

What is the angle of Euler

Use three angles to save the orientation
x And z Rotate along its own coordinate system ,y Rotate along the world coordinate system
API:Vector3 eulerAngle=this.transform.eulerAngles;

advantage ：

Use only three numbers to express orientation , Small footprint
The unit of rotation along the coordinate axis is angle , In line with people's way of thinking
Any three numbers are legal , There is no illegal Euler angle

shortcoming ：

1. The expression of orientation is not unique

There are multiple Euler angle descriptions for a bearing , Therefore, it is impossible to judge whether the angular displacements represented by multiple Euler angles are the same

example ： angle 0,5,0 And angle 0,365,0

angle 250,0,0 And angle 290,180,180

In order to ensure that there is only a unique representation of any direction ,unity Limits the range of angles , Namely along x Axis rotation is limited to -90 To 90 Between , Along the y And x Axis rotation is limited to 0 To 360 Between （ stay unity There are no restrictions in the compiler , Limit in code ）

Vector3 euler=this.transform.eulerAngles;
Be careful ： Euler angle has no direction 、 The concept of size
Because three-dimensional vectors , contain x,y,z, So in Unity The data type of Central European pull angle is Vector3
Euler angle x,y,z Represents the rotation angle on each axis
Vector3 por=this.transform.position;
Location , There is a direction （ Point to the current position from the world origin ）, There are sizes （ Distance from the current position to the world origin ）
Vectorial x,y,z, Indicates the directional displacement on each axis

It can be used to detect Euler angle

public Vector3 euler;

    // It can be used to detect Euler angle 

    public void OnGUI()

    {

        euler = this.transform.eulerAngles;

        if (GUILayout.RepeatButton(" Along the X Shaft rotation "))

            this.transform.eulerAngles += new Vector3(1, 0, 0);

        if (GUILayout.RepeatButton(" Along the y Shaft rotation "))

            this.transform.eulerAngles += Vector3.up;

        if (GUILayout.RepeatButton(" Along the z Shaft rotation "))

            this.transform.eulerAngles += new Vector3(0, 0,1);

}

2. Universal joint deadlocks

Objects along x Shaft rotation +（-）90 degree , Self coordinate system z Axis and world coordinate system y The axes will coincide , Now go along y or z When axis rotates , Will lose a degree of freedom .
In case of cardan deadlock , Specified edge z The axis completes all rotation around the numerical axis , Is this time y The axis rotates to 0

Four yuan number

What is quaternion

Quaternion stay 3D In graphics stands for rotation , By a three-dimensional vector （x,y,z） And a scalar （w） form
The rotation axis is V, The rotation radian is θ, If you use quaternions to represent , Then the four components are ：
x=sin(θ/2)*V.x y=sin(θ/2)*V.y
z=sin(θ/2)*V.z w=cos(θ/2)
x,y,z,w The range of phi is zero -1 To 1
API：Quaternion qt=this.transform.rotation;

 public void OnGUI()

    {

        euler = this.transform.eulerAngles;

        if (GUILayout.Button(" Set the rotation angle of the object "))

        {

            // Rotation axis 

            Vector3 axis = t1.position - t2.position;

            // The arc of rotation 

            float rad = 50 * Mathf.Deg2Rad;

            Quaternion qt = new Quaternion();

            qt.x = Mathf.Sin(rad / 2) * axis.x;

            qt.y = Mathf.Sin(rad / 2) * axis.y;

            qt.z = Mathf.Sin(rad / 2) * axis.z;

            qt.w = Mathf.Cos(rad / 2);

            this.transform.rotation = qt;

        }

}

advantage ：

Avoid U-joint deadlock

this.transform.rotation（ Four yuan number ）*=Quaternion.Euler(0,1,0);

Can make an object move along its own coordinates Y Shaft rotation

public void OnGUI()

    {

        euler = this.transform.eulerAngles;

        if (GUILayout.RepeatButton(" Along the X Shaft rotation "))

            this.transform.rotation *= Quaternion.Euler(1, 0, 0);

        if (GUILayout.RepeatButton(" Along the y Shaft rotation "))

            this.transform.rotation *= Quaternion.Euler(0, 1, 0);

        if (GUILayout.RepeatButton(" Along the z Shaft rotation "))

            this.transform.rotation *= Quaternion.Euler(0, 0, 1);

    }

this.transform.Rotate(Vector3 eulerAngles)

Internally, quaternion multiplication is used to achieve

shortcoming

· Difficult to use , It is not recommended to modify a value separately

· There is an illegal quaternion

Quaternion operations

· Multiply by vector

Quaternion left multiplication vector , Indicates that the vector is rotated according to the angle represented by quaternion
example ：Vector3 point =new Vector3(0,0,10);// Walk forward 10 rice
Vector3 newpoint=Quaternion.Euler(0,30,0)（ Along the y Shaft rotation 30 degree ）*point;

· Multiply with quaternions

Multiply two quaternions to combine the rotation effect
example ：Quaternion rotation01=Quaternion.Euler(0,30,0)*Quaternion.Euler(0,20,0);
Quaternion rotation02=Quaternion.Euler(0,50,0);
rotation01 And rotation02 identical