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![1) Translation- changing the position
P(x,y)-point before translation
P’(x’,y’)-point after translation
tx- translation parameter with respect to x-axis
ty- translation parameter with respect to y-axis
x’= x + tx
y’= y + ty
P’= P + T
Matrix Form- [x’ y’]=[x y]+[tx ty]](https://image.slidesharecdn.com/cgunit-32d3dtransformationsandprojections-250725141546-729910ca/75/CG-Unit-3-2D-3D-Transformations-and-Projections-pptx-3-2048.jpg)
CG Unit-3 2D,3D Transformations and Projections.pptx
- 1. Computer Graphics Unit-3 2D,3DTransformation and Projections
- 2. 2D Transformation- ● Manipulationdone on object. ● Translation - changing the position tx- translation parameter with respect to x-axis ty- translation parameter with respect to y-axis ● Scaling - Resizing the object Sx - Scaling parameter with respect to x-axis Sy - Scaling parameter with respect to y-axis ● Rotation - Rotate object Rotate clockwise & anticlockwise
- 3. 1) Translation- changingthe position P(x,y)-point before translation P’(x’,y’)-point after translation tx- translation parameter with respect to x-axis ty- translation parameter with respect to y-axis x’= x + tx y’= y + ty P’= P + T Matrix Form- [x’ y’]=[x y]+[tx ty]
- 4. Exa. Consider square(0,0),(2,0), (0,2), (2,2) with tx=2, ty=3 (0,0) x’=0 + 2 =2 y’=0 + 3=3 (0,0) →(2,3) (2,0) x’=2 + 2 =4 y’=0 + 3=3 (2,0) →(4,3) (0,2) x’=0 + 2 =2 y’=2 + 3=5 (0,2) →(2,5) (2,2) x’=2 + 2 =4 y’=2 + 3=5 (2,2) →(4,5)
- 5. 2) Scaling-Resize theobject ● Sx - Scaling parameter with respect to x-axis ● Sy - Scaling parameter with respect to y-axis ● Both if Sx & Sy in between 0 & 1 ○ Point is closer to origin ○ Size decreases ● if Sx & Sy are greater than 1 ○ Point is away from origin ○ Size increases ● If Sx & Sy are equal Scaling will be done uniformly x’=x.Sx y’=y.Sy Matrix form-
- 6. Exa 1). Considersquare (0,0),(2,0), (0,2), (2,2) with Sx=2, Sy=3 (0,0) x’=0 * 2 =0 y’=0 * 3=0 (0,0) →(0,0) (2,0) x’=2 * 2 =4 y’=0 * 3=0 (2,0) →(4,0) (0,2) x’=0 * 2 =0 y’=2 * 3=6 (0,2) →(0,6) (2,2) x’=2 * 2 =4 y’=2 * 3=6 (2,2) →(4,6)
- 7. Exa 2). Considersquare (0,0),(2,0), (0,2), (2,2) with Sx=0.5, Sy=0.5 (0,0) x’=0 * 0.5 =0 y’=0 * 0.5=0 (0,0) →(0,0) (2,0) x’=2 * 0.5 =1 y’=0 * 0.5=0 (2,0) →(1,0) (0,2) x’=0 * 0.5 =0 y’=2 * 0.5=1 (0,2) →(0,1) (2,2) x’=2 * 0.5 =1 y’=2 * 0.5=1 (2,2) →(1,1)
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- 10. Example 1) Note-Ifnot mention direction in exa. Consider Anticlockwise.
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- 12. 4) Shearing- x-shear -y same only x distorted y’=y x’=x +Shx.y x shearing parameter y-shear - x same only y distorted x’=x y’=y +Shy.x y shearing parameter
- 14. Exa 1). Considersquare (0,0),(2,0), (0,2), (2,2) with Shx=2 units (0,0) x’=0 + 2.0 =0 y’=0 (0,0) →(0,0) (0,2) x’=0+2.2=4 y’= 2 (0,2) →(4,2) (2,0) x’=2 + 2.0 =2 y’=0 (2,0) →(2,0) (2,2) x’=2 +2.2=2+4 =6 y’=2 (2,2) →(6,2)
- 15. Exa 2). Considersquare (0,0),(2,0), (0,2), (2,2) with Shy=2 units (0,0) x’= 0 y’=0 + 2.0=0 (0,0) →(0,0) (0,2) x’=0 y’= 2 + 2.0=2 (0,2) →(0,2) (2,0) x’=2 y’=0 + 2.2=4 (2,0) →(2,4) (2,2) x’=2 y’=2+2.2=2+4=6 (2,2) →(2,6)
- 16. Homogeneous Coordinates Representation- ●Homogeneous coordinates means 2D matrix converted into 3D matrix for simplification. (x,y) → (xh,yh,h) where h- any non-zero for simplification, consider h=1 1) Translation- Homogeneous matrix 2) Scaling-Homogeneous matrix representation of Translation representation of Scaling
- 17. Homogeneous Coordinates… 3) Rotation-Homogeneous matrix 2) Shearing-Homogeneous matrix representation of Rotation representation of Shearing (anticlockwise)
- 18. Rotation about anarbitrary point- ● We have derived rotation matrices with respect to origin but suppose reference point of rotation is other than origin we must follow 3 steps- Step 1- Arbitrary point translate to origin (0,0) tx=-xp, ty=-yp
- 19. Rotation about anarbitrary point… Step 2- Rotate at angle 𝝷 Step 3- Translate back at same Arbitrary position tx=xp, ty=yp
- 20. Rotation about anarbitrary point… Now let us form a combined matrix This is transformation matrix is overall transformation matrix for rotation about arbitrary point (xp,yp) by an angle in anticlockwise direction. 𝝷
- 21. 3D Transformation- ● Translation- changing the position/moving the object tx,ty,tz ● Scaling - Resizing the object/changing the size Sx,Sy,Sz ● Rotation - Rotate object with 𝝷 Rotate clockwise & anticlockwise ● Shearing - Tilting the object x-shear, y-shear, z-shear
- 22. 1) 3D Translation- tx-translation parameter with respect to x-axis x’= x + tx ty- translation parameter with respect to y-axis y’= y + ty tz- translation parameter with respect to z-axis z’= y + tz
- 23. 2) 3D Scaling- Sx-Scaling parameter with respect to x-axis x’= x . Sx Sy- Scaling parameter with respect to y-axis y’= y . Sy Sz- Scaling parameter with respect to z-axis z’= y . Sz
- 24. 3) 3D Rotation- A.X-axis rotation B. Y-axis rotation C. Z-axis rotation
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- 28. 4) 3D Shearing- A.Shearing in X-axis B. Shearing in Y-axis C. Shearing in Z-axis
- 29. A. Shearing inX-axis
- 30. B. Shearing inY-axis
- 31. C. Shearing inZ-axis
- 32. 3D Homogeneous CoordinatesRepresentation- 1) Translation- 2) Scaling- 3) Rotation-anticlockwise 4) Shearing-
- 33. 3D Rotation aboutan arbitrary point- ● If we want to rotate a point (x,y,z) by an angle which is in space and lying 𝝷 on an arbitrary axis then we have to follow 7 steps as follows: Step 1- Translate the arbitrary axis so that it will pass through the origin. When we translating axis, a point also gets translated.