What Is the Projection Transformation?

The projection matrix is typically a scale and perspective projection. The projection transformation converts the viewing frustum into a cuboid shape. Because the near end of the viewing frustum is smaller than the far end, this has the effect of expanding objects that are near to the camera; this is how perspective is applied to the scene.

In the viewing frustum, the distance between the camera and the origin of the viewing transformation space is defined arbitrarily as D, so the projection matrix looks like:

The viewing matrix translates the camera to the origin by translating in the z direction by -D. The translation matrix is as follows:

Multiplying the translation matrix by the projection matrix (T*P) gives the composite projection matrix. It looks like:

The following illustration shows how the perspective transformation converts a viewing frustum into a new coordinate space. Notice that the frustum becomes cuboid and also that the origin moves from the upper-right corner of the scene to the center.

In the perspective transformation, the limits of the x- and y-directions are -1 and 1. The limits of the z-direction are 0 for the front plane and 1 for the back plane.

This matrix translates and scales objects based on a specified distance from the camera to the near clipping plane, but it doesn't consider the field of view (fov), and the z-values that it produces for objects in the distance can be nearly identical, making depth comparisons difficult. The following matrix addresses these issues, and it adjusts vertices to account for the aspect ratio of the viewport, making it a good choice for the perspective projection.

In this matrix, Z_n is the z-value of the near clipping plane. The variables w, h, and Q have the following meanings. Note that fov_w and fov_h represent the viewport's horizontal and vertical fields of view, in radians.

For your application, using field-of-view angles to define the x and y scaling coefficients might not be as convenient as using the viewport's horizontal and vertical dimensions (in camera space). As the math works out, the following two formulas for w and h use the viewport's dimensions, and are equivalent to the preceding formulas.

In these formulas, Z_n represents the position of the near clipping plane, and the V_w and V_h variables represent the width and height of the viewport, in camera space.