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2d transformations
- 1. CAP4730: Computational Structures in Computer Graphics 2D Transformations
- 2. 2D Transformations • World Coordinates • Translate • Rotate • Scale • Viewport Transforms • Putting it all together
- 3. Transformations • Rigid Body Transformations - transformations that do not change the object. • Translate – If you translate a rectangle, it is still a rectangle • Scale – If you scale a rectangle, it is still a rectangle • Rotate – If you rotate a rectangle, it is still a rectangle
- 4. Vertices • We have always represented vertices as (x,y) • An alternate method is: • Example: = y x yx ),( = 8.4 1.2 )8.4,1.2(
- 6. Matrix * Matrix = ++ ++ = = = 10 01 ? * , dc ba dwcydzcx bwaybzax BA wz yx B dc ba A Does A*B = B*A? What does the identity do?
- 8. Translation • Translation - repositioning an object along a straight-line path (the translation distances) from one coordinate location to another. (x,y) (x’,y’) (tx,ty)
- 9. Translation • Given: • We want: • Matrix form: TPP t t y x y x tyy txx ttT yxP y x y x yx += + = += += = = ' ' ' ' ' ),( ),( 1.4' 4.3' 2.8 1.7 1.4 7.3 ' ' 2.81.4' 1.77.3' )2.8,1.7( )1.4,7.3( = = + − − = +−= +−= = −−= y x y x y x T P
- 10. Translation Examples • P=(2,4), T=(-1,14), P’=(?,?) • P=(8.6,-1), T=(0.4,-0.2), P’=(?,?) • P=(0,0), T=(1,0), P’=(?,?)
- 11. Which one is it? (x,y) (x’,y’) (tx,ty) (x,y) (tx,ty)
- 12. Recall • A point is a position specified with coordinate values in some reference frame. • We usually label a point in this reference point as the origin. • All points in the reference frame are given with respect to the origin.
- 14. What do we have here? • You know how to:
- 15. Scale • Scale - Alters the size of an object. • Scales about a fixed point (x,y) (x’,y’)
- 16. Scale • Given: • We want: • Matrix form: PSP y x s s y x ysy xsx ssS yxP y x y x yx ⋅= = = = = = ' 0 0 ' ' ' ' ),( ),( 6.6' 2.4' 2.2 4.1 30 03 ' ' 2.2*3' 4.1*3' )3,3( )2.2,4.1( = = = = = = = y x y x y x S P
- 18. Rotation • Rotation - repositions an object along a circular path. • Rotation requires an Θ and a pivot point
- 21. What is the difference? Revisited V(-0.6,0) V(0,-0.6) V(0.6,0.6) Translate (1.2,0.3) V(0,0.6) V(0.3,0.9) V(0,1.2) Translate (1.2,0.3) V(0.6,0.3) V(1.2,-0.3) V(1.8,0.9) V(0,0.6) V(0.3,0.9) V(0,1.2)
- 22. Rotations V(-0.6,0) V(0,-0.6) V(0.6,0.6) Rotate -30 degrees V(0,0.6) V(0.3,0.9) V(0,1.2)
- 23. Combining Transformations Q: How do we specify each transformation?
- 24. Specifying 2D Transformations • Translation – T(tx, ty) – Translation distances • Scale – S(sx,sy) – Scale factors • Rotation – R(θ) – Rotation angle
- 25. Combining Transformations • Using translate, rotation, and scale, how do we get:
- 26. Combining Transformations • Note there are two ways to combine rotation and translation. Why?
- 27. Let’s look at the equations θθ θθ θ cos'sin'" sin'cos'" ' ' ''' ' )( ),( ),( yxy yxx tyy txx PRP TPP R ttT yxP y x yx += −= += += •= += ( ) ( ) ( ) ( ) ( ) ( ) y x yx yx tyxy tyxx yxy yxx TPP PRP tytxy tytxx ++= +−= += −= += •= +++= +−+= θθ θθ θθ θθ θθ θθ cossin" sincos" cossin' sincos' '" ' cos'sin" sin'cos"
- 28. Combining them • We must do each step in turn. First we rotate the points, then we translate, etc. • Since we can represent the transformations by matrices, why don’t we just combine them? PSP PRP TPP •= •= += ' ' '
- 29. 2x2 -> 3x3 Matrices • We can combine transformations by expanding from 2x2 to 3x3 matrices. ( ) ( ) ( ) − = − = = = = + = 100 0cossin 0sincos cossin sincos 100 00 00 0 0 , 100 10 01 , θθ θθ θθ θθ θR s s s s ssS t t t t y x ttT y x y x yx y x y x yx
- 30. Homogenous Coordinates • We need to do something to the vertices • By increasing the dimensionality of the problem we can transform the addition component of Translation into multiplication. = = = → → = = = → = 2 2 2 14 7 2 6 3 2 14 6 7 3 ., 1 2 4 2 4 . ExEx h h h y y h x x h y x y x P h h h h
- 31. Homogenous Coordinates • Homogenous Coordinates - term used in mathematics to refer to the effect of this representation on Cartesian equations. Converting a pt(x,y) and f(x,y)=0 -> (xh,yh,h) then in homogenous equations mean (v*xh,v*yh,v*h) can be factored out. • What you should get: By expressing the transformations with homogenous equations and coordinates, all transformations can be expressed as matrix multiplications.
- 32. Final Transformations - Compare Equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) PRPPRP y x y x y x y x R PssSPPssSP y x s s y x y x s s y x ssS PttTPPttTP y x t t y x y x t t y x ttT yxyx y x y x yx yxyx y x y x yx •=→•= − = → − = = •=→•= = → = = •=→+= = → + = = θθ θθ θθ θθ θθ θ 1100 0cossin 0sincos 1 ' ' cossin sincos ' ' ,, 1100 00 00 1 ' ' 0 0 ' ' , ,, 1100 10 01 1 ' ' ' ' ,
- 33. Combining Transformations − = − = = − ••=→•=•= 1 ' ' 100 10 01 100 0cossin 0sincos 1 " " 1 ' ' 100 0cossin 0sincos 1 " " 1100 10 01 1 ' ' )60(),2,4641.0(),4,3( "'",' y x t t y x y x y x y x t t y x RTP PBAPPBPPAP y x y x θθ θθ θθ θθ − = − = + −− = 1100 cossin sincos 1 " " 1100 0cossin 0sincos 100 10 01 1 " " 1100 cossincossin sincossincos 1 " " y x t t y x y x t t y x y x tt tt y x y x y x yx yx θθ θθ θθ θθ θθθθ θθθθ
- 34. How would we get:
- 35. How would we get:
- 36. Coordinate Systems • Object Coordinates • World Coordinates • Eye Coordinates
- 40. Coordinate Hierarchy O b je c t # 1 O b je c t C o o r d in a t e s T r a n s f o r m a t io n O b je c t # 1 -> W o r ld O b je c t # 2 O b je c t C o o r d in a t e s T r a n s f o r m a t io n O b je c t # 2 - > W o r ld O b je c t # 3 O b je c t C o o r d in a t e s T r a n s f o r m a t io n O b je c t # 3 -> W o r ld W o r ld C o o r d in a t e s T r a n s f o r m a t io n W o r ld - > S c r e e n S c r e e n C o o r d in a t e s
- 41. Let’s reexamine assignment 2b
- 42. Transformation Hierarchies • For example:
- 43. Transformation Hierarchies • Let’s examine:
- 44. Transformation Hierarchies • What is a better way?
- 45. Transformation Hierarchies • What is a better way?
- 46. Transformation Hierarchies • We can have transformations be in relation to each other B lu e O b je c t C o o r d in a te s T ra n s fo r m a tio n B lu e -> R e d R e d O b je c t C o o r d in a te s T ra n s fo r m a tio n R e d - > G r e e n G r e e n O b je c t 's C o o r d in a te s T ra n s fo r m a tio n G re e n -> W o r ld W o r ld C o o r d in a t e s