This tutorial discusses off-axis perspective projection in Unity. It is based on Section “Vertex Transformations”. No shader programming is required since only the view matrix and the projection matrix are changed, which is implemented in JavaScript.

The main application of off-axis perspective projection are virtual reality environments such as the CAVE shown in the photo. Usually, the user's head position is tracked and the perspective projection for each display is computed for a camera at the tracked position such that the user experiences the illusion of looking through a window into a three-dimensional world instead of looking at a flat display.

### Off-Axis vs. On-Axis Perspective ProjectionEdit

On-axis projection refers to camera positions that are on the symmetry axis of the view plane, i.e. the axis through the center of the view plane and orthogonal to it. This case is discussed in Section “Vertex Transformations”.

In virtual reality environments, however, the virtual camera often follows the tracked position of the user's head in order to create parallax effects and thus a more compelling illusion of a three-dimensional world. Since the tracked head position is not limited to the symmetry axis of the view plane, on-axis projection is not sufficient for most virtual reality environments.

Off-axis perspective projection addresses this issue by allowing for arbitrary camera positions. While some low-level graphics APIs (e.g. older versions of OpenGL) supported off-axis projection, they had much better support for on-axis projection since this was the more common case. Similarly, many high-level tools (e.g. Unity) support off-axis projection but provide much better support for on-axis projection, i.e. you can specify any on-axis projection with some mouse clicks but you need to write a script to implement off-axis projection.

### Computing Off-Axis Perspective ProjectionEdit

Off-axis perspective projection requires a different view matrix and a different projection matrix than on-axis perspective projection. For the computation of the on-axis view matrix, a specified view direction is rotated onto the *z* axis as described in Section “Vertex Transformations”. The only difference for an off-axis view matrix is that this “view direction” is computed as the orthogonal direction to the specified view plane, i.e. the surface normal vector of the view plane.

The off-axis projection matrix has to be changed since the edges of the view plane are no longer symmetric around the intersection point with the (technical) “view direction.” Thus, the four distances to the edges have to be computed and put into a suitable projection matrix. For details, see the description by Robert Kooima in his publication “Generalized Perspective Projection”. The next section presents an implementation of this technique in Unity.

### Camera ScriptEdit

The following script is based on the code in Robert Kooima's publication. There are very few implementation differences. One is that, in Unity, the view plane is more easily specified as a built-in Plane object, which has corners at (±5, 0, ±5) in object coordinates. Furthermore, the original code was written for a right-handed coordinate system while Unity uses a left-handed coordinate system; thus, the result of the cross product has to be multiplied with -1. Another difference is that the rotation of the camera's GameObject and the parameter `fieldOfView`

are used by Unity for view frustum culling; therefore, the script should set those values to appropriate values. (These values have no meaning for the computation of the matrices.) Unfortunately, this might cause problems if other scripts (namely the script that sets the tracked head position) are also setting the camera rotation. Therefore, this estimation can be deactivated with the variable `estimateViewFrustum`

(at the risk of incorrect view frustum culling by Unity).

```
// This script should be attached to a Camera object
// in Unity. Once a Plane object is specified as the
// "projectionScreen", the script computes a suitable
// view and projection matrix for the camera.
// The code is based on Robert Kooima's publication
// "Generalized Perspective Projection," 2009,
// http://csc.lsu.edu/~kooima/pdfs/gen-perspective.pdf
#pragma strict
public var projectionScreen : GameObject;
public var estimateViewFrustum : boolean = true;
function LateUpdate() {
if (null != projectionScreen)
{
var pa : Vector3 =
projectionScreen.transform.TransformPoint(
Vector3(-5.0, 0.0, -5.0));
// lower left corner in world coordinates
var pb : Vector3 =
projectionScreen.transform.TransformPoint(
Vector3(5.0, 0.0, -5.0));
// lower right corner
var pc : Vector3 =
projectionScreen.transform.TransformPoint(
Vector3(-5.0, 0.0, 5.0));
// upper left corner
var pe : Vector3 = transform.position;
// eye position
var n : float = camera.nearClipPlane;
// distance of near clipping plane
var f : float = camera.farClipPlane;
// distance of far clipping plane
var va : Vector3; // from pe to pa
var vb : Vector3; // from pe to pb
var vc : Vector3; // from pe to pc
var vr : Vector3; // right axis of screen
var vu : Vector3; // up axis of screen
var vn : Vector3; // normal vector of screen
var l : float; // distance to left screen edge
var r : float; // distance to right screen edge
var b : float; // distance to bottom screen edge
var t : float; // distance to top screen edge
var d : float; // distance from eye to screen
vr = pb - pa;
vu = pc - pa;
vr.Normalize();
vu.Normalize();
vn = -Vector3.Cross(vr, vu);
// we need the minus sign because Unity
// uses a left-handed coordinate system
vn.Normalize();
va = pa - pe;
vb = pb - pe;
vc = pc - pe;
d = -Vector3.Dot(va, vn);
l = Vector3.Dot(vr, va) * n / d;
r = Vector3.Dot(vr, vb) * n / d;
b = Vector3.Dot(vu, va) * n / d;
t = Vector3.Dot(vu, vc) * n / d;
var p : Matrix4x4; // projection matrix
p[0,0] = 2.0*n/(r-l);
p[0,1] = 0.0;
p[0,2] = (r+l)/(r-l);
p[0,3] = 0.0;
p[1,0] = 0.0;
p[1,1] = 2.0*n/(t-b);
p[1,2] = (t+b)/(t-b);
p[1,3] = 0.0;
p[2,0] = 0.0;
p[2,1] = 0.0;
p[2,2] = (f+n)/(n-f);
p[2,3] = 2.0*f*n/(n-f);
p[3,0] = 0.0;
p[3,1] = 0.0;
p[3,2] = -1.0;
p[3,3] = 0.0;
var rm : Matrix4x4; // rotation matrix;
rm[0,0] = vr.x;
rm[0,1] = vr.y;
rm[0,2] = vr.z;
rm[0,3] = 0.0;
rm[1,0] = vu.x;
rm[1,1] = vu.y;
rm[1,2] = vu.z;
rm[1,3] = 0.0;
rm[2,0] = vn.x;
rm[2,1] = vn.y;
rm[2,2] = vn.z;
rm[2,3] = 0.0;
rm[3,0] = 0.0;
rm[3,1] = 0.0;
rm[3,2] = 0.0;
rm[3,3] = 1.0;
var tm : Matrix4x4; // translation matrix;
tm[0,0] = 1.0;
tm[0,1] = 0.0;
tm[0,2] = 0.0;
tm[0,3] = -pe.x;
tm[1,0] = 0.0;
tm[1,1] = 1.0;
tm[1,2] = 0.0;
tm[1,3] = -pe.y;
tm[2,0] = 0.0;
tm[2,1] = 0.0;
tm[2,2] = 1.0;
tm[2,3] = -pe.z;
tm[3,0] = 0.0;
tm[3,1] = 0.0;
tm[3,2] = 0.0;
tm[3,3] = 1.0;
// set matrices
camera.projectionMatrix = p;
camera.worldToCameraMatrix = rm * tm;
// The original paper puts everything into the projection
// matrix (i.e. sets it to p * rm * tm and the other
// matrix to the identity), but this doesn't appear to
// work with Unity's shadow maps.
if (estimateViewFrustum)
{
// rotate camera to screen for culling to work
var q : Quaternion;
q.SetLookRotation((0.5 * (pb + pc) - pe), vu);
// look at center of screen
camera.transform.rotation = q;
// set fieldOfView to a conservative estimate
// to make frustum tall enough
if (camera.aspect >= 1.0)
{
camera.fieldOfView = Mathf.Rad2Deg *
Mathf.Atan(((pb-pa).magnitude + (pc-pa).magnitude)
/ va.magnitude);
}
else
{
// take the camera aspect into account to
// make the frustum wide enough
camera.fieldOfView =
Mathf.Rad2Deg / camera.aspect *
Mathf.Atan(((pb-pa).magnitude + (pc-pa).magnitude)
/ va.magnitude);
}
}
}
}
```

To use this script, choose **Create > Javascript** in the **Project** view, double-click the new script to edit it, and copy & paste the code from above into it. Then attach the script to your main camera (drag it from the **Project** view over the camera object in the **Hierarchy** view). Furthermore, create a Plane object (**GameObject > Create Other > Plane** in the main menu) and place it into the virtual scene to define the view plane. Deactivate the **Mesh Renderer** of the Plane in the **Inspector** to make it invisible (it is only a placeholder), but make sure that the front face of the plane is always facing the camera. Select the camera object and drag the plane object to **Projection Screen** in the **Inspector**. The script will be active when the game is started. Add the line

```
@script ExecuteInEditMode()
```

at the top of the code to make the script also run in the editor.

There are a couple of limitations of this implementation in Unity:

- To keep the script simple, it assumes that the front face of the plane (the face that is visible when backface culling is active) is facing the camera.
- The camera preview in the Scene view doesn't take the changed projection matrix into account and is therefore no longer useful.
- The built-in skybox system also doesn't take the changed projection matrix into account and is therefore also no longer useful. (See Section “Skyboxes” for an alternative way of implementing skyboxes.)
- There are probably further parts of Unity that ignore the new projection matrix and that therefore are unusable in combination with this script.

### SummaryEdit

In this tutorial, we have looked at:

- uses of off-axis perspective projection and differences to on-axis perspective projection
- the computation of view and projection matrices for off-axis perspective projection
- an implementation of this computation and its limitations in Unity

### Further ReadingEdit

If you want to know more

- about the on-axis view matrix and the on-axis projection matrix, you should read the description in Section “Vertex Transformations”.
- about the implemented algorithm, you should read Robert Kooima's publication “Generalized Perspective Projection”.