Projection

Perspective and orthographic projections

2 min read | Computer Graphics

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Projection

Projection transforms 3D coordinates into 2D coordinates for display. There are two main types: orthographic (parallel) and perspective.

Orthographic Projection

Projects objects onto a plane parallel to the view direction. Parallel lines remain parallel, and there is no depth distortion. Used in engineering and architectural drawings.

Orthographic Projection Matrix:

  [ 2/(r-l)    0       0      -(r+l)/(r-l) ]
  [ 0          2/(t-b) 0      -(t+b)/(t-b) ]
  [ 0          0      -2/(f-n) -(f+n)/(f-n)]
  [ 0          0       0       1            ]

Where:
  l, r = left, right
  t, b = top, bottom
  n, f = near, far

Simplified (centered):
  [ 1  0  0  0 ]
  [ 0  1  0  0 ]
  [ 0  0  1  0 ]
  [ 0  0  0  1 ]

Perspective Projection

Objects farther away appear smaller, creating a sense of depth. Parallel lines converge at vanishing points. This mimics how the human eye and cameras work.

Perspective Projection Matrix:

  [ 2n/(r-l)    0       (r+l)/(r-l)   0          ]
  [ 0           2n/(t-b) (t+b)/(t-b)   0          ]
  [ 0           0      -(f+n)/(f-n)  -2fn/(f-n)   ]
  [ 0           0      -1             0          ]

After division by w:
  x_screen = x_clip / w_clip
  y_screen = y_clip / w_clip
  z_screen = z_clip / w_clip

This division creates the perspective effect.

Field of View (FOV)

  fov_y = vertical field of view angle

  aspect = width / height

  f = 1 / tan(fov_y / 2)

  [ f/aspect  0    0           0 ]
  [ 0         f    0           0 ]
  [ 0         0   (f+n)/(n-f) -1 ]
  [ 0         0   2fn/(n-f)    0 ]

Example: 90° FOV, 16:9 aspect
  f = 1/tan(45°) = 1
  matrix[0][0] = 1/(16/9) = 0.5625

View Volume (Frustum)

  Perspective frustum:

        Near plane
       +---------+
      /|        /|
     / |       / |
    +---------+  |  Far plane
    |  +------|--+
    | /       | /   The visible 3D region
    |/        |/    is shaped like a
    +---------+     truncated pyramid