Game Mathematics: calculate the points on the plane in the screen points (God's perspective)

Game math : Calculate the points on the plane in the points on the screen （ God's perspective ）

Application scenarios

God's perspective , Click on the screen , You need to calculate the position on the plane at this point , For example, move the object to the clicked position , Or you need to drag objects to move

Mathematical calculation

ad locum , It's actually a calculation with several premises , Or the intersection of a simplified version of a ray and a plane
Because it's God's perspective , So the starting point of the ray is y The axis coordinates must be positive , And y The included angle of the axis must be an acute angle , And we use xy The plane composed of axes is used as the plane for intersection calculation

We go through Unity The screen points of are converted into rays , You can get the starting point and direction unit vector of the ray , We assume that the starting point of the ray is A spot , And xy The intersection point of the axis plane is B spot ,A The point is xy The projection point of the axis plane is C spot , Then calculate according to the vector , Yes B Point coordinates = A Point coordinates + AB vector , So we just need to AB It's just as long as you want

Therefore, the solution formula is as follows ：
AC length = ay
AB length = ay / cos( The direction of the ray And AC The angle of the vector )
B spot = A spot + Ray direction unit vector * AB length

stay Unity Example in

We are Unity To complete the verification of this calculation

using System.Collections;
using System.Collections.Generic;
using UnityEngine;
using UnityEngine.UI;
using UnityEngine.EventSystems;
public class InteractiveTest : UIBehaviour, IPointerClickHandler
{
    
    public Camera cam;
    public GameObject ctrlObj;
    
    public void OnPointerClick(PointerEventData pointerEventData)
    {
    
        var ray = cam.ScreenPointToRay(pointerEventData.position);
        SetObjPos(ray.origin, ray.direction);
    }
    void SetObjPos(Vector3 origin, Vector3 direction)
    {
    
        float lengthAC = origin.y;  // Because the location of our camera must be xy Above the plane 
        float cosABAC = -direction.y; //direction and (0, -1, 0) The Angle between , That is to say AB and AC The Angle between , It must be an acute angle 
        float lengthAB = lengthAC / cosABAC;
        // Vector3 AB = lengthAB * direction;
        // Vector3 CB = new Vector3(AB.x, 0, AB.z);
        // Vector3 C = new Vector3(origin.x, 0, origin.z);
        // Vector3 B = new Vector3(C.x + CB.x, 0, C.z + CB.z);
        // ctrlObj.transform.position = B;
        Vector3 B = origin + direction * lengthAB;
        ctrlObj.transform.position = B;
    }
}

We simply use a plane , A ball as an example

Click anywhere on the screen , We can see that the ball can move to the clicked position accurately

summary

Although the calculation needs to be conditional , But for this kind of application scenario where rays and planes intersect , We use this calculation method to consume less , Of course Unity The screen click points converted to the rays of the world can also be calculated by themselves , In this way, the sub consumption can be reduced .