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2hjsakhvchcvj hSKchvsABJChjSVCHjhvcvdxz.pptx

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Harikumar Rajasekar

1. The scaling transformation is represented by a scaling matrix S. 2. The scaling matrix contains the scale factors along the x and y axes. 3. To apply the scaling transformation, the point P is multiplied by the scaling matrix: P' = S * P.

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2D TRANSFORMATIONS
2D Transformations What is transformations? The geometrical changes of an object from a current state to modified state. Why the transformations is needed? To manipulate the initially created object and to display the modified object without having to redraw it.
2 ways Object Transformation Alter the coordinates descriptions an object Translation, rotation, scaling etc. Coordinate system unchanged Coordinate transformation Produce a different coordinate system 2D Transformations
Matrix Math Why do we use matrix? More convenient organization of data. More efficient processing Enable the combination of various concatenations Matrix addition and subtraction a b c d a c b d =
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 + tx p'y = py + ty or in matrix form: P' = P + T tx ty x y x y tx ty = + (2, 2) = 6 =4 ?
Rotation A rotation repositions all points in an object along a circular path in the plane centered at the pivot point. First, well assume the pivot is at the origin. P P
Rotation Review Trigonometry => cos = x/r , sin = y/r x = r. cos , y = r.sin P(x,y) x y r x y P(x, y) r => cos (+ ) = x/r x = r. cos (+ ) x = r.coscos -r.sinsin x = x.cos y.sin =>sin (+ ) = y/r y = r. sin (+ ) y = r.cossin + r.sincos y = x.sin + y.cos Identity of Trigonometry
Rotation We can write the components: p'x = px cos py sin p'y = px sin + py cos or in matrix form: P' = R P can be clockwise (-ve) or counterclockwise (+ve as our example). Rotation matrix P(x,y) x y r x y P(x, y) cos sin sin cos R
Example Find the transformed point, P, caused by rotating P= (5, 1) about the origin through an angle of 90. Rotation cos sin sin cos cos sin sin cos y x y x y x 90 cos 1 90 sin 5 90 sin 1 90 cos 5 0 1 1 5 1 1 0 5 5 1
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: y x s s S 0 0 P P
Scaling If the scale factors are in between 0 and 1 the points will be moved closer to the origin the object will be smaller. P(2, 5) P Example : P(2, 5), Sx = 0.5, Sy = 0.5 Find P ?
Scaling If the scale factors are in between 0 and 1 the points will be moved closer to the origin the object will be smaller. P(2, 5) P Example : P(2, 5), Sx = 0.5, Sy = 0.5 Find P ? If the scale factors are larger than 1 the points will be moved away from the origin the object will be larger. P Example : P(2, 5), Sx = 2, Sy = 2 Find P ?
Scaling If the scale factors are the same, Sx = Sy uniform scaling Only change in size (as previous example) P(1, 2) P If Sx Sy differential scaling. Change in size and shape Example : square rectangle P(1, 3), Sx = 2, Sy = 5 , P ? What does scaling by 1 do? What is that matrix called? What does scaling by a negative value do?
Combining transformations We have a general transformation of a point: P' = M P + A When we scale or rotate, we set M, and A is the additive identity. When we translate, we set A, and M is the multiplicative identity. To combine multiple transformations, we must explicitly compute each transformed point. Itd be nicer if we could use the same matrix operation all the time. But wed have to combine multiplication and addition into a single operation.
Homogenous Coordinates Lets move our problem into 3D. Let point (x, y) in 2D be represented by point (x, y, 1) in the new space. Scaling our new point by any value a puts us somewhere along a particular line: (ax, ay, a). A point in 2D can be represented in many ways in the new space. (2, 4) ---------- (8, 16, 4) or (6, 12, 3) or (2, 4, 1) or etc. y y x x w
Homogenous Coordinates We can always map back to the original 2D point by dividing by the last coordinate (15, 6, 3) --- (5, 2). (60, 40, 10) - ?. Why do we use 1 for the last coordinate? The fact that all the points along each line can be mapped back to the same point in 2D gives this coordinate system its name homogeneous coordinates.
Matrix Representation Point in column-vector: Our point now has three coordinates. So our matrix is needs to be 3x3. Translation x y 1 1 1 0 0 1 0 0 1 1 y x t t y x y x P=T(tx,ty) . P
Rotation:
Rotation P=R(). P Scaling Matrix Representation 1 1 0 0 0 cos sin 0 sin cos 1 y x y x 1 1 0 0 0 0 0 0 1 y x s s y x y x P=S(sx, sy). P
2hjsakhvchcvj hSKchvsABJChjSVCHjhvcvdxz.pptx
Problem: Scaling

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2hjsakhvchcvj hSKchvsABJChjSVCHjhvcvdxz.pptx

  • 2. 2D Transformations What is transformations? The geometrical changes of an object from a current state to modified state. Why the transformations is needed? To manipulate the initially created object and to display the modified object without having to redraw it.
  • 3. 2 ways Object Transformation Alter the coordinates descriptions an object Translation, rotation, scaling etc. Coordinate system unchanged Coordinate transformation Produce a different coordinate system 2D Transformations
  • 4. Matrix Math Why do we use matrix? More convenient organization of data. More efficient processing Enable the combination of various concatenations Matrix addition and subtraction a b c d a c b d =
  • 5. 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 + tx p'y = py + ty or in matrix form: P' = P + T tx ty x y x y tx ty = + (2, 2) = 6 =4 ?
  • 6. Rotation A rotation repositions all points in an object along a circular path in the plane centered at the pivot point. First, well assume the pivot is at the origin. P P
  • 7. Rotation Review Trigonometry => cos = x/r , sin = y/r x = r. cos , y = r.sin P(x,y) x y r x y P(x, y) r => cos (+ ) = x/r x = r. cos (+ ) x = r.coscos -r.sinsin x = x.cos y.sin =>sin (+ ) = y/r y = r. sin (+ ) y = r.cossin + r.sincos y = x.sin + y.cos Identity of Trigonometry
  • 8. Rotation We can write the components: p'x = px cos py sin p'y = px sin + py cos or in matrix form: P' = R P can be clockwise (-ve) or counterclockwise (+ve as our example). Rotation matrix P(x,y) x y r x y P(x, y) cos sin sin cos R
  • 9. Example Find the transformed point, P, caused by rotating P= (5, 1) about the origin through an angle of 90. Rotation cos sin sin cos cos sin sin cos y x y x y x 90 cos 1 90 sin 5 90 sin 1 90 cos 5 0 1 1 5 1 1 0 5 5 1
  • 10. 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: y x s s S 0 0 P P
  • 11. Scaling If the scale factors are in between 0 and 1 the points will be moved closer to the origin the object will be smaller. P(2, 5) P Example : P(2, 5), Sx = 0.5, Sy = 0.5 Find P ?
  • 12. Scaling If the scale factors are in between 0 and 1 the points will be moved closer to the origin the object will be smaller. P(2, 5) P Example : P(2, 5), Sx = 0.5, Sy = 0.5 Find P ? If the scale factors are larger than 1 the points will be moved away from the origin the object will be larger. P Example : P(2, 5), Sx = 2, Sy = 2 Find P ?
  • 13. Scaling If the scale factors are the same, Sx = Sy uniform scaling Only change in size (as previous example) P(1, 2) P If Sx Sy differential scaling. Change in size and shape Example : square rectangle P(1, 3), Sx = 2, Sy = 5 , P ? What does scaling by 1 do? What is that matrix called? What does scaling by a negative value do?
  • 14. Combining transformations We have a general transformation of a point: P' = M P + A When we scale or rotate, we set M, and A is the additive identity. When we translate, we set A, and M is the multiplicative identity. To combine multiple transformations, we must explicitly compute each transformed point. Itd be nicer if we could use the same matrix operation all the time. But wed have to combine multiplication and addition into a single operation.
  • 15. Homogenous Coordinates Lets move our problem into 3D. Let point (x, y) in 2D be represented by point (x, y, 1) in the new space. Scaling our new point by any value a puts us somewhere along a particular line: (ax, ay, a). A point in 2D can be represented in many ways in the new space. (2, 4) ---------- (8, 16, 4) or (6, 12, 3) or (2, 4, 1) or etc. y y x x w
  • 16. Homogenous Coordinates We can always map back to the original 2D point by dividing by the last coordinate (15, 6, 3) --- (5, 2). (60, 40, 10) - ?. Why do we use 1 for the last coordinate? The fact that all the points along each line can be mapped back to the same point in 2D gives this coordinate system its name homogeneous coordinates.
  • 17. Matrix Representation Point in column-vector: Our point now has three coordinates. So our matrix is needs to be 3x3. Translation x y 1 1 1 0 0 1 0 0 1 1 y x t t y x y x P=T(tx,ty) . P
  • 19. Rotation P=R(). P Scaling Matrix Representation 1 1 0 0 0 cos sin 0 sin cos 1 y x y x 1 1 0 0 0 0 0 0 1 y x s s y x y x P=S(sx, sy). P