Unity - Fundamentals of 3D mathematics

It took less than half a day today to get a preliminary understanding 3D Mathematical basis ！ Enter tomorrow unity Game script writing ！

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Catalog

3D Coordinate system ：

· overall situation （ The world ） Coordinate system ：

· Local （ Model 、 object ） Coordinate system ：

· Camera coordinate system ：

· Screen coordinate system ：

vector

Vector operation ：

Vector3 class

Vector3 application ：

· Calculate the distance between two objects

· Slow motion

Homogeneous coordinates

Four yuan number

Coordinate transformation principle

Conversion between world coordinates and local coordinates ： Change global and local coordinates

Conversion between world coordinates and screen coordinates ： Set world coordinates

· How to set the world coordinate position of an object by clicking the mouse

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3D Coordinate system ：

· overall situation （ The world ） Coordinate system ：

A benchmark that describes the position and direction of all objects in the game scene

Hierrarchy In the view Position attribute by （1,2,3） It means that it is at... From the origin of the global coordinate system x There is... In the direction of the axis 1 Unit coordinates , stay y There is... In the direction of the axis 2 Unit coordinates , stay z There is... In the direction of the axis 3 Unit coordinates

When the game object Yes When the parent object , Its Hierrarchy In the view Position Is shown as Local coordinates , When the game object nothing When the parent object , Its Hierrarchy In the view Position Show Both local and global coordinates

· Local （ Model 、 object ） Coordinate system ：

Every object has its own coordinate system , And move or rotate with the object

Model mesh The saved vertex coordinates are Coordinates in the local coordinate system

The coordinate system of the parent-child object is related , The child object will take a coordinate point of the parent object as its coordinate origin

· Camera coordinate system ：

The coordinate system established according to the observation position and direction

Judge whether the object is in front of the camera , And the sequence of occlusion between objects

· Screen coordinate system ：

Two dimensional coordinate system established on the screen , Used to describe the position of pixels on the screen , Where the origin position （0,0） In the lower left corner of the screen

vector

Used to describe physical quantities with two attributes of size and direction

Vector operation ：

· Addition and subtraction

· Number multiplication

A vector is multiplied by a scalar

· Point multiplication

Multiply two vectors to get a scalar , The value is equal to the cosine of the angle between the two vectors multiplied by the length of the two vectors

· Cross riding

The cross multiplication of two vectors yields a new vector , The new vector is perpendicular to the original two vectors , And the length is equal to the sine of the angle multiplied by the length of the original vector

Vector3 class

Vector3 Member variables

Vector3 Method

 

Vector3 application ：

· Calculate the distance between two objects

public class Distance : MonoBehaviour
{
    // Start is called before the first frame update
    public Transform objTrans1;
    public Transform objTrans2;
    void Start()
    {
        float dis = Vector3.Distance(objTrans1.position, objTrans2.position);
        Debug.Log("Distance:" + dis);
    }

    // Update is called once per frame
    void Update()
    {
        
    }
}

(Debug.Log： Record the message to Unity Console )

Bind the script to the main camera in the scene Main Camera, And then again Inspector In the view objTrans1、objTrans2 Designated as 2 individual cube

function 、CTRL+SHIFT+C Call up the console

· Slow motion

Vector3.Lerp Method

The slow motion effect of moving an object from its current position to another position in the coordinate system

public class MoveToTarget : MonoBehaviour
{
    public Transform endTrans;
    // Start is called before the first frame update
    void Start()
    {
        
    }
    // Update is called once per frame
    void Update()
    {
        transform.position = Vector3.Lerp(transform.position, endTrans.position, Time.deltaTime);
    }
}

Time.deltaTime Express Update() Time taken to complete the previous frame on the method

Bind the script to Cube On , And then in it Inspector In view endTrans Designated as Cube（1）

Homogeneous coordinates

Put the original three-dimensional vector （x,y,z） Use four-dimensional vector （wx,wy,wz,w） To express

Purpose of introduction ：

Better distinguish between vectors and points ;

The translation is uniformly expressed by matrix multiplication 、 rotate 、 Zoom transform

When the weight w=0 Time can be used to represent the point at infinity

Four yuan number

It contains a scalar component and a three-dimensional vector component , Four yuan number Q Write it down as Q=[w,（x,y,z)]

In the game object Transform In the component , Variable Transform.rotation Is the rotation of the object in the global coordinate system ; Variable Transform.localRotation Is the rotation of the object in the local coordinate system of the parent object

application ： Control the rotation of objects through quaternions

Coordinate transformation principle

Conversion between world coordinates and local coordinates ： Change global and local coordinates

ransform Component's Transform.TransformPoint Method can convert coordinate points from local coordinate system to global coordinate system ,Transform.InverseTransformPoint Method can convert coordinate points from global coordinate system to local coordinate system

Conversion between world coordinates and screen coordinates ： Set world coordinates

stay Unity in , Because the screen picture is made by the camera （Camera） Presented , Therefore, the conversion between world coordinates and screen coordinates needs to take the camera as the medium

· How to set the world coordinate position of an object by clicking the mouse

newly build C# Script

public class ScreenToWorldPoint : MonoBehaviour
{
    Vector3 screenPos;
    // Start is called before the first frame update
    void Start()
    {
        screenPos = Camera.main.WorldToScreenPoint(transform.position);
        // Print coordinate information 
        Debug.Log(" Screen coordinates ：" + screenPos);
    }

    // Update is called once per frame
    void Update()
    {
        if(Input.GetMouseButtonDown(0))
        {
            // Get the position coordinates of the mouse 
            Vector3 mousePos = Input.mousePosition;
            // Set the screen coordinates of the object z The axis value is assigned to the mouse coordinate z Axis 
            mousePos.z = screenPos.z;
            // Change the screen coordinates into world coordinates 
            Vector3 worldPos = Camera.main.ScreenToWorldPoint(mousePos);
            // Set object position 
            transform.position = worldPos;
        }
    }
}

Bind script to Cube On the cube

When the run starts , The console will output this Cube The world coordinates of are converted into the coordinates of the screen ; When the mouse pointer is Game View , When left clicking , You can see Cube The position of will follow the position of the mouse pointer

After the world coordinates are converted into screen coordinates , The screen coordinates are z The axis is larger than 0, It means that the coordinate is in front of the camera , conversely , It means that the coordinate is behind the camera