2D transformation (Computer Graphics)
- 2. 2D Transformations “Transformations are the operations applied to geometrical description of an object to change its position, orientation, or size are called geometric transformations”.
- 3. Translation Translation is a process of changing the position of an object in a straight-line path from one co- ordinate location to another. We can translate a two dimensional point by adding translation distances, tx and ty. Suppose the original position is (x ,y) then new position is (x’, y’). Here x’=x + tx and y’=y + ty.
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- 5. Translation
- 6. Matrix form of the equations: X’ = X + tx and Y’ = Y + ty is P = x P’ = x’ T= tx y y’ ty we can write it, P’= P + T
- 7. Translate a polygon with co-ordinates A(2,5) B(7,10) and C(10,2) by 3 units in X direction and 4 units in Y direction. A’ = A +T = 2 + 3 = 5 5 4 9 B’ = B + T = 7 + 3 = 10 10 4 14 C’ = C + T = 10 + 3 = 13 2 4 7
- 8. Rotation A two dimensional rotation is applied to an object by repositioning it along a circular path in the xy plane. Using standard trigonometric equations , we can express the transformed co-ordinates in terms of x’ = r cos( coscosr sinsin y’ = r sin( cosr sin The original co-ordinates of the point is x = r cos y = r sin
- 9. Click icon to add picture After substituting equation 2 in equation 1 we get x’=x cos Y’=x sin + y cos
- 10. Rotation
- 11. That equation can be represented in matrix form x’ y’ = x y cos -sin cos we can write this equation as, P’ = P . R Where R is a rotation matrix and it is given as R = cos -sin cos
- 12. A point (4,3) is rotated counterclockwise by angle of 45. find the rotation matrix and the resultant point. R = cos = cos45 sin45 - cos -sin45 cos45 = 1/√2 1/√2 - 1/√2 1/√2 P’ = 4 3 1/√2 1/√2 - 1/√2 1/√2 = 4/√2 – 3/√2 4/√2 + 3/√2 = 1/√2 7/√2
- 13. Scaling A scaling transformation changes the size of an object. This operation can be carried out for polygons by multiplying the co-ordinates values (x , y) of each vertex by scaling factors Sx and Sy to produce the transformed co-ordinates (x’ , y’). x’ = x . Sx y’ = y . Sy In the matrix form x’ y’ = x y Sx 0 0 Sy = P . S
- 14. Scaling • Uniform Scaling Un-uniform Scaling
- 15. Homogeneous co-ordinates for Translation The homogeneous co-ordinates for translation are given as T = 1 0 0 0 1 0 tx ty 1 Therefore , we have x’ y’ 1 = x y 1 1 0 0 0 1 0 tx ty 1 = x + tx y + ty 1
- 16. Homogeneous co-ordinates for rotation The homogeneous co-ordinates for rotation are given as R = cos sin -sin cos 0 0 1 Therefore , we have x’ y’ 1 = x y 1 cos sin -sin cos 0 0 1 = x cos - y sin x sin + y cos 1
- 17. Homogeneous co-ordinates for scaling The homogeneous co-ordinate for scaling are given as S = Sx 0 0 0 Sy 0 0 0 1 Therefore , we have x’ y’ 1 = x y 1 Sx 0 0 0 Sy 0 0 0 1 = x . Sx y . Sy 1
- 28. CONCLUSION To manipulate the initially created object and to display the modified object without having to redraw it, we use Transformations.
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