**Related Topics:** OpenGL Transformation

**Updates:** The MathML version is available here.

A computer monitor is a 2D surface. A 3D scene rendered by OpenGL must be projected onto the computer screen as a 2D image. GL_PROJECTION matrix is used for this projection transformation. First, it transforms all vertex data from the eye coordinates to the clip coordinates. Then, these clip coordinates are also transformed to the normalized device coordinates (NDC) by dividing with *w* component of the clip coordinates.

Therefore, we have to keep in mind that both clipping (frustum culling) and NDC transformations are integrated into **GL_PROJECTION** matrix. The following sections describe how to build the projection matrix from 6 parameters; *left*, *right*, *bottom*, *top*, *near* and *far* boundary values.

Note that the frustum culling (clipping) is performed in the clip coordinates, just before dividing by w_{c}. The clip coordinates, x_{c}, y_{c} and z_{c} are tested by comparing with w_{c}. If any clip coordinate is less than -w_{c}, or greater than w_{c}, then the vertex will be discarded.

Then, OpenGL will reconstruct the edges of the polygon where clipping occurs.

In perspective projection, a 3D point in a truncated pyramid frustum (eye coordinates) is mapped to a cube (NDC); the range of x-coordinate from [l, r] to [-1, 1], the y-coordinate from [b, t] to [-1, 1] and the z-coordinate from [-n, -f] to [-1, 1].

Note that the eye coordinates are defined in the right-handed coordinate system, but NDC uses the left-handed coordinate system. That is, the camera at the origin is looking along -Z axis in eye space, but it is looking along +Z axis in NDC. Since **glFrustum()** accepts only positive values of *near* and *far* distances, we need to negate them during the construction of GL_PROJECTION matrix.

In OpenGL, a 3D point in eye space is projected onto the *near* plane (projection plane). The following diagrams show how a point (x_{e}, y_{e}, z_{e}) in eye space is projected to (x_{p}, y_{p}, z_{p}) on the *near* plane.

From the top view of the frustum, the x-coordinate of eye space, x_{e} is mapped to x_{p}, which is calculated by using the ratio of similar triangles;

From the side view of the frustum, y_{p} is also calculated in a similar way;

Note that both x_{p} and y_{p} depend on z_{e}; they are inversely propotional to -z_{e}. In other words, they are both divided by -z_{e}. It is a very first clue to construct GL_PROJECTION matrix. After the eye coordinates are transformed by multiplying GL_PROJECTION matrix, the clip coordinates are still a homogeneous coordinates. It finally becomes the normalized device coordinates (NDC) by divided by the w-component of the clip coordinates. (*See more details on OpenGL Transformation.*)

,

Therefore, we can set the w-component of the clip coordinates as -z_{e}. And, the 4th of GL_PROJECTION matrix becomes (0, 0, -1, 0).

Next, we map x_{p} and y_{p} to x_{n} and y_{n} of NDC with linear relationship; [l, r] ⇒ [-1, 1] and [b, t] ⇒ [-1, 1].

Then, we substitute x_{p} and y_{p} into the above equations.

Note that we make both terms of each equation divisible by -z_{e} for perspective division (x_{c}/w_{c}, y_{c}/w_{c}). And we set w_{c} to -z_{e} earlier, and the terms inside parentheses become x_{c} and y_{c} of the clip coordiantes.

From these equations, we can find the 1st and 2nd rows of GL_PROJECTION matrix.

Now, we only have the 3rd row of GL_PROJECTION matrix to solve. Finding z_{n} is a little different from others because z_{e} in eye space is always projected to -n on the near plane. But we need unique z value for the clipping and depth test. Plus, we should be able to unproject (inverse transform) it. Since we know z does not depend on x or y value, we borrow w-component to find the relationship between z_{n} and z_{e}. Therefore, we can specify the 3rd row of GL_PROJECTION matrix like this.

In eye space, w_{e} equals to 1. Therefore, the equation becomes;

To find the coefficients, *A* and *B*, we use the (z_{e}, z_{n}) relation; (-n, -1) and (-f, 1), and put them into the above equation.

To solve the equations for *A* and *B*, rewrite eq.(1) for B;

Substitute eq.(1') to *B* in eq.(2), then solve for A;

Put *A* into eq.(1) to find *B*;

We found *A* and *B*. Therefore, the relation between z_{e} and z_{n} becomes;

Finally, we found all entries of GL_PROJECTION matrix. The complete projection matrix is;

This projection matrix is for a general frustum. If the viewing volume is symmetric, which is and , then it can be simplified as;

Before we move on, please take a look at the relation between z_{e} and z_{n}, eq.(3) once again. You notice it is a rational function and is non-linear relationship between z_{e} and z_{n}. It means there is very high precision at the *near* plane, but very little precision at the *far* plane. If the range [-n, -f] is getting larger, it causes a depth precision problem (z-fighting); a small change of z_{e} around the *far* plane does not affect on z_{n} value. The distance between *n* and *f* should be short as possible to minimize the depth buffer precision problem.

Constructing GL_PROJECTION matrix for orthographic projection is much simpler than perspective mode.

All x_{e}, y_{e} and z_{e} components in eye space are linearly mapped to NDC. We just need to scale a rectangular volume to a cube, then move it to the origin. Let's find out the elements of GL_PROJECTION using linear relationship.

Since w-component is not necessary for orthographic projection, the 4th row of GL_PROJECTION matrix remains as (0, 0, 0, 1). Therefore, the complete GL_PROJECTION matrix for orthographic projection is;

It can be further simplified if the viewing volume is symmetrical, and .