# Chapter 6: Projections

Definitions/Variables used

NameDescription
nDistance from camera position to near plane
Camera View directionPositive z axis
fDistance from camera position to far plane
sAspect ratio=width/height
gDistance from camera position to projection plane
Field of ViewAngle between left/right or top/bottom frustum planes.

fovY = 2tan-1(1/g)

g = tan-1(fovY/2)

To calculate camera-space frustum planes:

Frustum PlaneCamera-space (x,y,z,w)
Near(0,0,1,-n)
Far(0,0,-1,f)
Left(1/sqrt(g2+s2))(g,0,s,0)
Right(1/sqrt(g2+s2))(-g,0,s,0)
Top(1/sqrt(g2+1))(0,g,1,0)
Bottom(1/sqrt(g2+1))(0,-g,1,0)

To calculate the corner points on the near/far frustum planes q0,q1,q2,q3:

 q0 c + ((us)/g)x + (u/g)y + uz q1 c + ((us)/g)x - (u/g)y + uz q2 c - ((us)/g)x - (u/g)y + uz q3 c - ((us)/g)x + (u/g)y + uz

x = Mcamera[0] (column 0), y = Mcamera[0] (column 1), z = Mcamera[0] (column 2), c = Mcamera[3] (column 3), u = n for near plane points, f for far plane points.

Reversing the depth buffer value/z allows for less floating point errors/more accuracy at farther z-positions away from the camera as opposed to very close to the near clip plane:

Normal Perspective Projection Matrix:

Infinite Distance Perspective Projection Matrix:

Reversed depth infinite distance perspective projection matrix:

This is a good choice for the default projection matrix to use, just need to make sure to flip the depth test pass condition from "greater than" to "less than"

Orthographic Projection Matrix:

Extracting camera-space planes from "canonical view volume" (normalized device coordinates cube):

Frustum PlaneClip-SpaceCamera-Space
Left(1,0,0,1)P3 + P0
Right(-1,0,0,1)P3 - P0
Top(0,1,0,1)P3 + P1
Bottom(0,-1,0,1)P3 - P1
Regular projection matrix
Near(0,0,1,0)P2
Far(0,0,-1,1)P3-P2
Reversed Depth projection matrix
Near(0,0,-1,1)R3-R2
Far(0,0,1,0)R2

Pi means projection matrix zero-based index row i, Ri for reversed-depth projection matrix row i

To reflect scene about a world-space boundary plane k, Mreflection = Hreflect(k)Mcamera.

Then to modify a projection matrix to clip points by the boundary plane, replace the third row of P with mk, or R3 - mk for reversed depth proj mat.

m = 1/dot(k,vcamera)

vcamera = (sign(kx)-P02)/P00, (sign(ky)-P12)/P11, 1, (1-P22)/P23)