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Composite transformations

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Composite transformations

1. 1. Composite TransformationMore complex geometric & coordinate transformations can be built from the basic transformation by using the process of composition of function.Example: Scaling about a fixed point.Transformation sequence to produce scalingw.r.t a selected fixed position (h, k) using ascaling function that can only scale relative tothe coordinate origin are:-
2. 2. 2 1 P P(h,k)1 1 1 2 1 2 P 1 1 1 2
3. 3. Steps for doing composite transformation:-1.) Translate the object so that its centre concides with the origin.2.) Scale the object with respect to origin.3.) Translate the scale object back to the original position.Thus the scaling with respect to the pointcan be formed by transformation. S sx,sy , P = Tv . S sx,sy . Tv-1
4. 4. Rotation about a fixed pointWe can generate rotation about any selectedpivot point (xr,yr) by performing followingsequence of translate-rotate-translate opn.1.) Translate the object so that pivot point position is moved to the co-ordinate origin.2.) Rotate the object about the co-ordinate origin.3.) Translate the object so that the pivot point is returned to the original position.
5. 5. yy (xr,yr) x x y y x x
6. 6. Thus the rotation about a point P can beformed by the transformation R θ ,P = Tv . Rθ . Tv-1Mirror reflection about a line:Let line L has a y intercept (0,b) & an angle ofinclination θ. Then the reflection of an objectabout a line L needs to follow the following:1.) Translate the intersection point to the origin.
7. 7. 2.) Rotate by -θ° so that line L align with x- axis.3.) Mirror reflect about the x-axis.4.) Rotate back by θ°.5.) Translate the origin back to the point (0,b). y P’ L (0,b) P θ x
8. 8. In translation notation, we haveML = Tv Rθ Mx R-θ T-vNoteWe must be able to represent the basictransformation as 3x3 homogeneouscoordinate matrices so as to compatible withthe matrix of transformation. This isaccomplished by augmenting the 2 x 2 matrixwith the third column 0 i.e x y 0 0 a b 0 1 0 0 1
9. 9. Ques1 What is the relation between Rθ, R -θ & Rθ-1?Ques2 (a) Find the matrix that represents rotation of an object by 30° about the origin. (b) What are the new coordinates of the point P(2,-4) after the rotation?Ques3 Perform a 45° rotation of triangle A(0,0), B(1,1), C(5,2) (a) about the origin (b) about P(-1,-1).
10. 10. Ques1 What is the relation between Rθ, R -θ & Rθ-1?Ques2 (a) Find the matrix that represents rotation of an object by 30° about the origin. (b) What are the new coordinates of the point P(2,-4) after the rotation?Ques3 Perform a 45° rotation of triangle A(0,0), B(1,1), C(5,2) (a) about the origin (b) about P(-1,-1).