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2D TRANSFORMATIONS

2D TRANSFORMATIONS. To be discussed…. Need of 2D-transformation. Types of transformation - Translation - Rotation - Scaling Composite transformation Reflection Shearing. 2D Transformations.

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2D TRANSFORMATIONS

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  1. 2D TRANSFORMATIONS

  2. To be discussed… • Need of 2D-transformation. • Types of transformation - Translation - Rotation - Scaling • Composite transformation • Reflection • Shearing

  3. 2D Transformations -The geometrical changes of an object from a current state to modified state. • Need of transformation: -To manipulate the initially created object and to display the modified object without having to redraw it.

  4. 2D Transformations • 2 ways • Object Transformation • Alter the coordinates descriptions of an object. • Translation, rotation, scaling etc. • Coordinate system remains unchanged. • Coordinate transformation • Produce a different coordinate system.

  5. ? =4 ty = 6 tx (2, 2) Translation • A translation moves all points in an object along the same straight-line path to new positions. • The path is represented by a vector, called the translation or shift vector. • We can write the components: p'x px + p'y = py + • or in matrix form: P' = P + T ’ ’ tx ty

  6. Matrix Representation • Point in column-vector: • Our point now has three coordinates. So our matrix is needs to be 3x3. • Translation: x y 1 BACK

  7. Rotation • A rotation repositions all points in an object along a circular path in the plane centered at the pivot point. • First, we’ll assume the pivot is at the origin. P’  P

  8. Rotation P’(x’, y’) • We can write the components: • p'x= pxcos – pysin  • p'y= pxsin  + pycos • or in matrix form: • P' = R • P •  can be clockwise (-ve) or counterclockwise (+ve as our example).  y’ P(x,y) r  y  x x’ BACK

  9. P’ P Scaling • Scaling changes the size of an object and involves two scale factors, Sx and Sy for the x- and y- coordinates respectively. • Scales are about the origin. • We can write the components: • p'x = sx • px • p'y = sy • py • or in matrix form: • P' = S • P • Scale matrix as:

  10. Matrix Representation • Rotation • Scaling BACK

  11. Composite Transformation • We can represent any sequence of transformations as a single matrix. • Composite transformations: • Rotate about an arbitrary point – translate, rotate, translate • Scale about an arbitrary point – translate, scale, translate • Change coordinate systems – translate, rotate, scale

  12. cos -sin 0 sin cos 0 0 0 1 1 0 -tx 0 1 -ty 0 0 1 . . 1 0 tx 0 1 ty 0 0 1 cos -sin -tx cos+ ty sin sin cos -tx sin - ty cos 0 0 1 1 0 tx 0 1 ty 0 0 1 . cos -sin -tx cos+ ty sin + tx sin cos -tx sin - ty cos + ty 0 0 1 Composite Transformation Matrix • Arrange the transformation matrices in order from right to left. • General Pivot- Point Rotation • Operation :- • Translate (pivot point is moved to origin) • Rotate about origin • Translate (pivot point is returned to original position) T(pivot) • R() • T(–pivot)

  13. Composite Transformation Matrix • General Fixed-Point Scaling • Operation :- • Translate (fixed point is moved to origin) • Scale with respect to origin • Translate (fixed point is returned to original position) BACK

  14. Other transformations Reflection: x-axis y-axis

  15. Other transformations Reflection: origin line x=y BACK

  16. Other transformations Shear: x-direction y-direction BACK

  17. Thankyou BACK

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