iitd cse summer workshop computer graphics
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IITD CSE Summer Workshop Computer Graphics. Subodh Kumar. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0. x y z 1. Perspective Transformation. x,y,z,w. x,y,z,z. 1. Linear and Affine transform. Linear: x’ = Ax Scale, Rotation T(a x + b y ) = aT( x ) + bT( y ) Affine: Translate x ’ = A x + b

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perspective transformation
1 0 0 0

0 1 0 0

0 0 1 0

0 0 1 0

x

y

z

1

Perspective Transformation

x,y,z,w

x,y,z,z

1

linear and affine transform
Linear and Affine transform
  • Linear: x’ = Ax
    • Scale, Rotation
    • T(ax + by) = aT(x) + bT(y)
  • Affine: Translate
    • x’ = Ax + b
    • Transforms a line to a line
    • Respects parallelism
    • Does not respect angles and lengths
projective transformation
Projective Transformation
  • n-dimensional projective space is an n+1-dimensional vector space
    • 0,0,0,… is not a part of this space
    • x = kx
    • xn+1 == 0 => Point at infinity {= Direction}
      • Actually any coordinate may be chosen
  • Affine space is a projection of Projective space
    • Typically on the plane xn+1 = 1
  • Projective transformation
    • xj = ∑aixi
    • Ax, A is a 4x4 matrix (15 DOF!)
normal transformation
Normal Transformation

n.p = 0

nTp = 0

=> n’T Mp must be 0

=> (M’n)T Mp = 0

=> nTM’T Mp = 0

=> M’T M = kI

=> M’T = M-1 (k = 1)

=> M’ = k(M-1)T

attribute interpolation
zp-zl xp-xl

=

zr-zl xr-xl

Attribute Interpolation

x0,y0,z0

y = mx + c

x = 1/m(y+c)

=ay+b

x2,y2,z2

x+1 = ay+a + b

= x + a

x1, y1,z1

zp+1 = zp + k

perspective correction
Perspective Correction
  • Linear interpolation:
    • p(t) = p1 + (p2 – p1) u, 0 <= u<= 1
  • Screen space interpolation:
    • ys = y1s + t (y2s – y1s)
    • same as: y/z = y1/z1 + t (y2/z2 – y1/z1)
  • But really:
    • y = y1 + u (y2 – y1)
    • z = z1 + u (z2 – z1)
graphics sw architecture top view
Xlib

GLX

OpenGL

GLU

Application

GDI

WGL

OGL/DX

GLU

Application

Graphics SW ArchitectureTop View

Windows

Unix

opengl architecture
OpenGL Architecture

Client

State Update / Action

Pushbuffer

Machine

state

Frame

Buffer

Must Respect PB order

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