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![rotation
scaling
translation
perspective transform
P 'P
T '
where
' [ ', ', '] ,
[ , , ] .
T
T
P TP
P x y w
P x y w
=
=
=
2D Transformation Matrices
=
ihg
fed
cba
T
X 'X](https://image.slidesharecdn.com/cs580-052dtransformations-140720144044-phpapp01/85/2d-3D-transformations-in-computer-graphics-Computer-graphics-Tutorials-21-320.jpg)
![]][]...[[][
][
]][]...[['
12
12
TTTT
PT
PTTTP
n
n
=
=
=
composite transformation matrix
HomeworkHomework
4. Composite Transformations](https://image.slidesharecdn.com/cs580-052dtransformations-140720144044-phpapp01/85/2d-3D-transformations-in-computer-graphics-Computer-graphics-Tutorials-23-320.jpg)
![( , )T
F Fx y
(a) Original position of
object and fixed point
(b) Translate object so that
fixed point (xF, yF) is at origin
(c) Scale object with
respect to origin
(d) Translate object so that
fixed point is returned to position (xF, yF)T
( , )T
F Fx y
][ rT
][ cS 1
][ −
rT](https://image.slidesharecdn.com/cs580-052dtransformations-140720144044-phpapp01/85/2d-3D-transformations-in-computer-graphics-Computer-graphics-Tutorials-24-320.jpg)
![( , )T
R Rx y
(a) Original position of
object and pivot point
(b) Translation of object so that
the pivot point (xR, yR) is at origin
(c) Rotation about origin (d) Translation of object so that
the pivot point is returned to
position (xR, yR)T
( , )T
R Rx y
][ rT
][ tR 1
][ −
rT](https://image.slidesharecdn.com/cs580-052dtransformations-140720144044-phpapp01/85/2d-3D-transformations-in-computer-graphics-Computer-graphics-Tutorials-25-320.jpg)
![x
y
θ
Sy
Sx
[Rt]-1
[Sc][Rt]](https://image.slidesharecdn.com/cs580-052dtransformations-140720144044-phpapp01/85/2d-3D-transformations-in-computer-graphics-Computer-graphics-Tutorials-26-320.jpg)
![Final
Position
Final
Position
(a) [Rt][Tr] (b) [Tr][Rt]](https://image.slidesharecdn.com/cs580-052dtransformations-140720144044-phpapp01/85/2d-3D-transformations-in-computer-graphics-Computer-graphics-Tutorials-29-320.jpg)
![45°
[Rt] [Rf
x
]
45°
[Rt]-1
Reflection about y = x](https://image.slidesharecdn.com/cs580-052dtransformations-140720144044-phpapp01/85/2d-3D-transformations-in-computer-graphics-Computer-graphics-Tutorials-32-320.jpg)
2d/3D transformations in computer graphics(Computer graphics Tutorials)
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2d/3D transformations in computer graphics(Computer graphics Tutorials)
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- 2. CS580 Computer Graphics 2DTransformations2D Transformations Sung Yong Shin Dept. of Computer Science KAIST
- 3.
- 4. Outline 1. Preliminary 2. Basic2D Transformations 3. Homogeneous Coordinate system 4. Composite Transformations 5. Reflection and Shearing
- 5. per-vertex operations & primitiveassembly rasterization frame buffer texture assembly display lists evaluators pixel operations per-fragment operations geometric data pixel data OpenGL Rendering Pipeline Window-to-viewport mapping
- 6. 2. Basic Transformations translation scaling rotation (absolute)transformation relative transformation
- 7. = ′ ′ =′ =′ y x S S y x ySy xSx y x y x 0 0 x y ( , )T FFx y : fixed point 1p 2p 3p y x ( , )T F Fx y Scaling ( , )T x y ( ', ')T x y
- 8.
- 9. θθ θφθφ θφ sincos sinsincoscos )cos( yx rr rx −= −= +=′ θθ sincos xyy+=′ Similarly, ( , )T x y ( , )T x y′ ′ r θ φ r φ φ sin cos ry rx = = Rotation
- 10. θθ θθ sincos sincos xyy yxx +=′ −=′ − = ′ ′ y x y x θθ θθ cossin sincos ( ,)T R Rx y ( , )T x y ( , )T x y′ ′ r θ φ r θ ( , )T x y ( , )T x y′ ′ r φ r ( , )T R Rx y : Pivot point
- 11. = ′ ′ +=′ +=′ 1 10 01 y x T T y x Tyy Txx y x y x 50 100 150 150 100 50 50100 150 150 100 50 (a) (b) Translation
- 12. Scaling andRotation Translation How to uniformly treat transformations? = ′ ′ y x dc ba y x Observation = ′ ′ 1 10 01 y x f e y x
- 13. 1 ( , ,) ( , , 1) ( / , / ) for 0 ( , , ) ( / , / ,1) , 0 T T T T T P AP a b c x P d e f y g h i P x y w x w y w x w y w w x y w x w y w Rα α α ′ = ′ = ′ ′ ′ ′ ′ ′ ′≡ ⇔ ≡ ≠ ′ ′ ′ ′⇔ ∀ ∈ ≠ 3. Homogeneous Coordinate System w x y ( , , )T x y wα ′ ′ 1=w ( , ,1)T x w y w′ ′ (0,0,1)T
- 14. ( , )( , ,1) ( , , )T T T x y x y wx wy w= ⇔ 0≠w normalized(standard) No unique homogeneous representation of a Cartesian point !!!
- 15. What if ?( , ,0)T x y0=w lim lim( , ,1) lim( , ,1 ) ( , ,0) T t t t T t T p at bt a b t a b →∞ →∞ →∞ = = = ( , ,1)T tp at bt= w • Ideal points !!! points at infinity well, ... ( , ,1)T a b(0,0,1)T 0 0 0 0 1 1 1 0 1 1 t t a a at p s t b p t b bt s t ÷ ÷ ÷ ÷ ÷ = + ⇒ = + = ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ + =
- 16. Projective Space Points inthe Euclidean space + Points at infinity Positions Vectors Positions and vectors are treated homogeneously!!! ( , , ) , 0T x y w w ≠ ( , , ) , 0T x y w w =
- 17. 0 0 0 0 limlim( , ,1) lim( , ,1 ) ( , ,0) T t t t T t p at x bt y a x t b y t t a b →∞ →∞ →∞ ′ = + + = + + = 0 0( , ,1)T tp at x bt y′ = + + 0 0( , ,1)T x y ( , ,1)T tp at bt= (0,0,1)T ( , ,1)T a b lim lim( , ,1) lim( , ,1 ) ( , ,0)T T T t t t t p at bt a b t a b →∞ →∞ →∞ = = =
- 18. In a Euclideanspace two lines intersect each other If they are not parallel Non-homogeneous treatment !!!∴ However, in a projective space, two lines do always intersects. ∴ Homogeneous treatment !!!
- 19. A B No orientation !!! BA= ∴ However, Why?
- 20. For detailtreatment of this subject, see M. Penna and R. Patterson, Projective Geometry and its Application to Computer Graphics, 1986.
- 21. rotation scaling translation perspective transform P 'P T' where ' [ ', ', '] , [ , , ] . T T P TP P x y w P x y w = = = 2D Transformation Matrices = ihg fed cba T X 'X
- 22. Now, uniformtreatment !!! Rotation Scaling Translation − = 1100 0cossin 0sincos ' ' ' y x w y x θθ θθ = 1100 00 00 ' ' ' y x S S w y x y x = 1100 10 01 ' ' ' y x T T w y x y x
- 23.
- 24. ( , )T FFx y (a) Original position of object and fixed point (b) Translate object so that fixed point (xF, yF) is at origin (c) Scale object with respect to origin (d) Translate object so that fixed point is returned to position (xF, yF)T ( , )T F Fx y ][ rT ][ cS 1 ][ − rT
- 25. ( , )T RRx y (a) Original position of object and pivot point (b) Translation of object so that the pivot point (xR, yR) is at origin (c) Rotation about origin (d) Translation of object so that the pivot point is returned to position (xR, yR)T ( , )T R Rx y ][ rT ][ tR 1 ][ − rT
- 26.
- 27. Why if Sx= Sy ?
- 28. x y (0, 0)T (1, 0)T (1,1)T (0, 1)T x y (0, 0)T (3/2, 1/2)T (2, 2)T (1/2, 3/2)T 45° Sx = 1 Sy = 2
- 29.
- 30.
- 31.
- 32.
- 33.
- 34. (a) (b) (c) (0,0) (0, 0) (0, 0)(1, 0) (1, 1) (0, 1) (1, 0) (3, 1)(2, 1) (0, 1) (1, 2) (1, 3) x y x y x y 100 010 01 xSH 100 01 001 ySH Shearing
- 35. (a) (b) Raster methodfor transformation