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CSL 859: Advanced Computer Graphics. Dept of Computer Sc. & Engg. IIT Delhi. User Interfaces. Keyboard Mouse Widgets or Menus? Gesture based Optical/Magnetic tracking Visual language?. Virtual Trackball. Map mouse motion to rotation of a volume Imagine mouse dragged on ground plane

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csl 859 advanced computer graphics

CSL 859: Advanced Computer Graphics

Dept of Computer Sc. & Engg.

IIT Delhi

user interfaces
User Interfaces
  • Keyboard
  • Mouse
  • Widgets or Menus?
  • Gesture based
    • Optical/Magnetic tracking
  • Visual language?
virtual trackball
Virtual Trackball
  • Map mouse motion to rotation of a volume
  • Imagine mouse dragged on ground plane
    • from (x1, z1) to (x2, z2)
  • Imagine the points raised onto unit hemisphere
    • from (x1, y1, z1) to (x2, y2, z2)
trackball
Trackball
  • p1: x12+y12+z12 = 1
  • p2: x22+y22+z22 = 1
  • Ball rotates about the axis formed by p1 and p2
    • p1 X p2
  • Rotation angle proportional to the distance p1p2
    • Or simply ||p1 X p2||
  • What happens if the object is not at the origin?
graphics pipeline
Graphics Pipeline
  • Transform vertices
  • Light vertices
  • Clip to Window extremes
  • Setup edges
  • Rasterize
    • Fill and interpolate colors, depth
  • Test against Z-buffer
    • One entry per pixel: closest so far
  • Update Color and Z buffers
why transformation
Why Transformation?
  • Modeling?
  • Projection?
  • Viewport?
transform vertices
Transform Vertices?

T(p) = aT(p1) + bT(p2) + cT(p3))

p0

p = a p0 + b p1 + cp2

p1

p2

matrix transformations
Matrix Transformations
  • Composition
    • (A (B (C x))) = (A B C) x
    • Fixed frame of reference
  • Rotation
  • Scaling
  • Translation
  • Shear
  • Projection
composition fixed frame
Composition: Fixed frame
  • T(1,0)
  • R(90)
  • R T p
composition local frame
Composition: Local frame
  • T(1,0)
  • R(90)
  • R T p
rotation

cos θ -sin θ

sin θ cos θ

Rotation

r

r

θ

r cos β

r cos α

translation1

1 0 0 tx

0 1 0 ty

0 0 1 tz

0 0 0 1

x

y

z

w

Translation
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

transform change basis
Transform = Change Basis
  • New Axis: x’, y’, z’
  • New Origin: o
  • +Scale

xT 0

yT 0

zT 0

13T 0

T-o

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