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**Question**by amit_nath30 · Jul 24, 2015 at 07:11 PM · cameramatrixworldtoscreenpointmatrix4x4worldtoscreen

# Calculation behind camera.WorldToScreenPoint

Hello all,

Is there any way to map world point to screen point using matrix, which output value same as one get when using Camera.main.WorldToScreenPoint function?

**Answer** by zach-r-d
·
Jul 25, 2015 at 12:30 PM

Unfortunately, this is mathematically impossible. One of the operations involved in computing the screen point of a world coordinate (dehomogenizing a homogeneous coordinate) cannot be represented as a linear transformation, so there cannot exist a matrix (which is really just a convenient encoding of a linear map) that can represent it.

If you're curious how the function actually works, it probably looks something like this:

```
Vector3 manualWorldToScreenPoint(Vector3 wp) {
// calculate view-projection matrix
Matrix4x4 mat = cam.projectionMatrix * cam.worldToCameraMatrix;
// multiply world point by VP matrix
Vector4 temp = mat * new Vector4(wp.x, wp.y, wp.z, 1f);
if (temp.w == 0f) {
// point is exactly on camera focus point, screen point is undefined
// unity handles this by returning 0,0,0
return Vector3.zero;
} else {
// convert x and y from clip space to window coordinates
temp.x = (temp.x/temp.w + 1f)*.5f * cam.pixelWidth;
temp.y = (temp.y/temp.w + 1f)*.5f * cam.pixelHeight;
return new Vector3(temp.x, temp.y, wp.z);
}
}
```

Glad it helped! Remember to mark an answer as accepted if it answers your question. :)

How do you invert this? (i.e., what is manualScreenToWorldPoint)

Inverted function - ManualScreenToWorldPoint

To get the required matricies you can pass in camera.cameraToWorldMatrix and camera.projectionMatrix.inverse.

Hope this helps someone someday :)

```
public static Vector3 ManualScreenToWorldPoint(Vector2 screenPoint, float distance, Matrix4x4 cameraToWorldMatrix, Matrix4x4 projectionMatrixInverse) {
// here we are converting screen point in screen space to camera space point placed on a plane "distance" away from the camera
// screen point is in range [(0,0) - (Screen.Width, Screen.Height)]
Vector2 pointViewportSpace = screenPoint / new Vector2(Screen.width, Screen.height); // convert space [(0,0) - (Screen.Width, Screen.Height)] to [(0,0) - (1,1)]
Vector2 pointCameraSpaceNormalized = (pointViewportSpace * 2.0f) - Vector2.one; // convert space [(0,0) - (1,1)] to [(-1,-1) - (1,1)]
Vector2 pointCameraSpace = pointCameraSpaceNormalized * distance; // convert space [(-1,-1) - (1,1)] to [(-dist,-dist) - (dist, dist)]
Vector4 planePoint = new Vector4(pointCameraSpace.x, pointCameraSpace.y, distance, distance); // define the point (don't know why z and w components need to be set to distance)
// calculate convertion matrix from camera space to world space
Matrix4x4 matrix = cameraToWorldMatrix * projectionMatrixInverse;
// multiply world point by VP matrix
Vector4 worldPoint = matrix * planePoint;
return worldPoint;
}
```

The convertion other way around could look something like this. Tested and works without issues.

ManualScreenPointToWorldPoint function

```
public static Vector3 ManualScreenPointToWorldPoint(Vector2 screenPoint, float distance, Matrix4x4 cameraToWorldMatrix, Matrix4x4 projectionMatrixInverse) {
// here we are converting screen point in screen space to camera space point placed on a plane "distance" away from the camera
// screen point is in range [(0,0) - (Screen.Width, Screen.Height)]
Vector2 pointViewportSpace = screenPoint / new Vector2(Screen.width, Screen.height); // convert space [(0,0) - (Screen.Width, Screen.Height)] to [(0,0) - (1,1)]
Vector2 pointCameraSpaceNormalized = (pointViewportSpace * 2.0f) - Vector2.one; // convert space [(0,0) - (1,1)] to [(-1,-1) - (1,1)]
Vector2 pointCameraSpace = pointCameraSpaceNormalized * distance; // convert space [(-1,-1) - (1,1)] to [(-dist,-dist) - (dist, dist)]
Vector4 planePoint = new Vector4(pointCameraSpace.x, pointCameraSpace.y, distance, distance); // define the point (don't know why z and w components need to be set to distance)
// calculate convertion matrix from camera space to world space
Matrix4x4 matrix = cameraToWorldMatrix * projectionMatrixInverse;
// multiply world point by VP matrix
Vector4 worldPoint = matrix * planePoint;
return worldPoint;
}
```

Example usage (place on the camera gameobject and play around with screen point and distance fields)

```
using UnityEngine;
public class DrawManualScreenPointToWorldPoint : MonoBehaviour {
Camera cam;
[SerializeField]
float distance = 1000;
[SerializeField]
Vector2 screenPoint = Vector2.zero;
void Start() {
cam = GetComponent<Camera>();
}
void Update() {
var point = MyUtilityClass.ManualScreenPointToWorldPoint(
screenPoint,
distance,
cam.cameraToWorldMatrix,
cam.projectionMatrix.inverse);
Debug.DrawLine(cam.transform.position, point, Color.green);
}
}
```

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