Skip to content

Latest commit

 

History

History
102 lines (84 loc) · 2.87 KB

05-projections.md

File metadata and controls

102 lines (84 loc) · 2.87 KB
Raw

← Return to Index

Projections

  • Center of Projection — the point at which rays of projection meet
  • Direction of Projection — when the center of projection reaches a distance nearing infinity, the rays of projection become parallel

Orthogonal Projection

  • Orthogonal Projection is a special case of parallel projection
  • x_p = x
  • y_p = y
  • z_p = z
|x_p|   |1 0 0 0| |x|
|y_p| = |0 1 0 0| |y|
|z_p|   |0 0 1 0| |z|
|1  |   |0 0 0 1| |1|

Projection Normalisation

  • We work in 4 dimensions using homogenous coordinates
  • Depth information (distance along a projector) is retained for hidden surface removal (later in the pipeline)
  • Projection Normalization:
    • Distort the object
    • Project object orthogonally
    • Projection is desired projection
    • Translate → Scale
  • x = ±1
  • y = ±1
  • z = ±1
  • T = T( -[right+left]/2, -[top+bottom]/2, [far+near]/2 )
  • S = S( 2/[right-left], 2/[top-bottom], 2/[near-far] )
  • z = -near → z = -1
  • z = -far → z = 1
  • And as such, the final Projection Normalization Matrix is:
N = ST = | 2/[right-left] 0 0 -[left+right]/[right-left] |
         | 0 2/[top-bottom] 0 -[top+bottom]/[top-bottom] |
         | 0 0 -2/[far-near] -[far+near]/[far-near] |
         | 0 0 0 1 |
         
Na^(→) = | 2a_x/[right-left] - [right+left]/[right-left] |
         | 2a_y/[top-bottom] - [top+bottom]/[top-bottom] |
         | 2a_z/[far-near] - [far+near]/[far-near] |

Oblique Projection

  • Oblique Projection — same as orthogonal, except the direction of projection is skewed by an angle θ
  • tanθ = z/(x_p - x)
  • x_p = x + zcotθ
  • y_p = y + zcotθ
  • z_p = 0
  • Projection = M_orth × S × T × H(θ, ɸ)
  • Where H is the oblique projection matrix:
H(θ, ɸ) = |1 0 cotθ 0|
          |0 1 cotɸ 0|
          |0 0 1    0|
          |0 0 0    1|

Perspective Projection

  • Perspective Projection — Given a plane d, we use the following formulas to determine the position of the points (x, y, z) onto the plane.
  • x/z = x_p/d → x_p = x/(z/d) → x_p = -xd/z (where the scaling factor is -d/z)
  • y/z = y_p/d → y_p = y/(z/d) → 'y_p = -yd/z`
  • x' = x
  • y' = y
  • z' = az + b
  • w' = -z
  • To convert to 3D, we divide by w':
    • x'' = -x/z
    • y'' = -y/z
    • z'' = -([az + b]/z)
  • And in matrix form:
N = |1 0 0 0|
    |0 1 0 0|
    |0 0 a b|
    |0 0 -1 0|
  • z'' = -(a × -near + b)/(-near) = -1
  • z'' = -(a × -far + b)/(-far) = 1
  • a = (n+f)(n-f) → a = -(f+n)/(f-n)
  • b = n × ([n+f]/[n-f] - [n-f]/[n-f]) → b = 2nf/(n-f) → b = -2nf/(f-n)
  • And finally, we get the Perspective Projection Matrix:
| 2near/[right-left] 0 [right+left]/[right-left] 0 |
| 0 2near/[top-bottom] [top+bottom]/[top-bottom] 0 |
| 0 0 -[far+near]/[far-near] [-2far × near]/[far-near] |
| 0 0 -1 0 |