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IITD CSE Summer Workshop Computer Graphics PowerPoint Presentation

IITD CSE Summer Workshop Computer Graphics

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IITD CSE Summer Workshop Computer Graphics

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IITD CSE Summer WorkshopComputer Graphics

Subodh Kumar

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

- 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

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

zp-zl xp-xl

=

zr-zl xr-xl

Attribute Interpolationx0,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

- 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)

GLX

OpenGL

GLU

Application

GDI

WGL

OGL/DX

GLU

Application

Graphics SW ArchitectureTop ViewWindows

Unix

OpenGL Architecture

Client

State Update / Action

Pushbuffer

Machine

state

Frame

Buffer

Must Respect PB order