2D TRANSFORMATION Transformations in 2D:, Exams of Computer Graphics

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2D transformations and

homogeneous coordinates

Dr Nicolas Holzschuch

University of Cape Town e-mail: [email protected] Modified by Longin Jan Latecki [email protected]

Map of the lecture

• (^) Transformations in 2D:
  • (^) vector/matrix notation
  • (^) example: translation, scaling, rotation
• (^) Homogeneous coordinates:
  • (^) consistent notation
  • (^) several other good points (later)
• (^) Composition of transformations
• (^) Transformations for the window system

Translations

• (^) Each vertex is modified:
  • (^) x’ = x+tx
  • (^) y’ = y+ty Before (^) After

Translations: vector notation

• (^) Use vector for the notation:
  • (^) makes things simpler
• (^) A point is a vector:
• (^) A translation is merely a vector

sum:

P’ = P + T

x y      

Scaling in 2D, matrix

notation

• (^) Scaling is a matrix multiplication: P’ = S P x   y         s x 0 0 s y       x y      

Rotating in 2D

• (^) New coordinates depend on both x

and y

• (^) x’ = cos x - sin y
• (^) y’ = sin x + cos y Before (^) After 

2D transformations,

summary

• (^) Vector-matrix notation simplifies writing:
  • (^) translation is a vector sum
  • (^) rotation and scaling are matrix-vector multiplication
• (^) I would like a consistent notation:
  • (^) that expresses all three identically
  • (^) that expresses combination of these also identically
• (^) How to do this?

Homogeneous coordinates

• (^) Introduced in mathematics:
  • (^) for projections and drawings
  • (^) used in artillery, architecture
  • (^) used to be classified material (in the 1850s)
• (^) Add a third coordinate, w
• (^) A 2D point is a 3 coordinates vector: x y w

Translations with

homogeneous

x   y  w 

1 0 t x 0 1 t y 0 0 1

x y w

x  y  w 

x  wt x y  wt y w

x  

w 

y 

w 

x

w

 t

x

y

w

 t

y

Scaling with homogeneous

x   y  w             s x 0 0 0 s y 0 0 0 1           x y w           x   y  w 

s x x s y y w

x 

w 

y 

w 

s

x

x

w

s

y

y

w

Composition of

transformations

• (^) To compose transformations, multiply the matrices: - (^) composition of a rotation and a translation: M = RT
• (^) all transformations can be expressed as matrices - (^) even transformations that are not translations, rotations and scaling

Rotation around a point Q

• (^) Rotation about a point Q:
  • (^) translate Q to origin ( T Q),
  • (^) rotate about origin ( R )
  • (^) translate back to Q ( - T Q).

P’= ( - T

Q

)R



T

Q

P

From World to Window

• (^) Inside the application:
  • (^) application model
  • (^) coordinates related to the model
  • (^) possibly floating point
• (^) On the screen:
  • (^) pixel coordinates
  • (^) integer
  • (^) restricted viewport: umin/umax, vmin/vmax

From Model to Viewport

xmin xmax ymin ymax