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GEOMETRIC TRANSFORMATIONS

GEOMETRIC TRANSFORMATIONS. Review of Mathematical Preliminaries Points:. Vectors : directional lines. P 1 (x 1 ,y 1 ). P 0 (x 0 ,y 0 ). 1 . A vector has a direction and a length: Length: |V|=(x 2 + y 2 ) 1/2 A unit vector: |V| = 1 e.g.: Normalize a vector:

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GEOMETRIC TRANSFORMATIONS

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  1. GEOMETRIC TRANSFORMATIONS Review of Mathematical Preliminaries Points: • Vectors: directional lines P1(x1,y1) P0(x0,y0)

  2. 1. A vector has a direction and a length: Length: |V|=(x2 + y2)1/2 A unit vector: |V| = 1 e.g.: • Normalize a vector: 2. Add two vectors • 3. Scalar Multiplication

  3. V •  W , • 4. Dot product of two vectors scalar • V  W if  = 90 => cos = 0 => V  W = 0. • if  < 90 => cos > 0 => V  W > 0. • if  > 90 => cos < 0 => V  W < 0. • 5.Normal: a unit vector perpendicular to a surface

  4. . • Lines y = mx+b e.g.: y = x; ax+by+c=0 Let f(x,y) = ax + by + c A point p(xp,yp) is on the line if f(xp,yp) = 0. When b < 0: p(xp,yp) is above the line if f(xp,yp) < 0. p(xp,yp) is below the line if f(xp,yp) > 0. When b > 0: p(xp,yp) is above the line if f(xp,yp) > 0. p(xp,yp) is below the line if f(xp,yp) < 0.

  5. Parametric Form: P(t)=P0+t(P1-P0); 0 <= t <= 1 P1(x1,y1) P0(x0,y0) • P(t)=P0+t(P1-P0)

  6. Where, • x(t) = x0 + t * (x1 – x0) • y(t) = y0 + t * (y1 – y0) • 0 <= t <= 1 • x(t) = (1-t) * x0 + t * x1 • y(t) = (1-t) * y0 + t * y1 • 0 <= t <= 1

  7. 2D Transformations • Translate a point The algebraic representation of translation of point P(x,y) by D(dx,dy) is x’= x + dx y’= y + dy Its “matrix” representation is • Its logical representation is P’ = P + D • Its visual representation is P’ (x’,y’) P(x,y) D(dx,dy)

  8. Y Y 10 10 5 5 X X 0 0 5 5 10 10 (4,5) (7,5) (10,1) (7,1) Before Translation After Translation • To move a shape: translate every vertex of the shape

  9. Scaling (relative to the origin) Scale a point P(x,y) by S(sx,sy) Algebraic: x’=sx* x y’=sy* y Matrix: Logic: P’ = SP or P’ = S(sx,sy)P Scale a line P0P1 (scale each point) P0’=SP0 • P1’=SP1

  10. Y Y 10 10 5 5 X X 0 0 5 5 10 10 (4,5) (7,5) (2,5/4) (7/2,5/4) Before Scaling After Scaling • Scale a shape: scale every vertex of the shape. Visual representation Uniform Scaling:sx=sy

  11. Rotate (around the origin) Positive angles are measured counterclockwise from x axis to y axis. • Rotate point (x,y) around the origin Algebraic representation: x’ = x * cos - y * sin y’ = x * sin + y * cos Matrix representation: Logic representation: P’ = RP

  12. (4.9, 7.8) (2.1, 4.9) (5,2) (9,2) Y Y Y Y Before Rotation After Rotation 10 10 10 10 5 5 5 5 X X X X 0 0 0 0 5 5 5 5 10 10 10 10 • Visual representation  Before After Rotate a shape: rotate every vertex of the shape

  13. Summary • Translation: P’ = P + D • Scaling: P’ = S P • Rotation: P’ = R P Homogeneous Coordinates: P(x,y,w) (x,y) to (x,y,w) : (x,y,w) to (x,y)

  14. y’= y + dy w’=1 x’= x + dx • P’’ = T(dx2 , dy2)P’ • = T(dx2 , dy2) T(dx1 , dy1) P • = T(dx2+dx1, dy2+dy1) P

  15. Scaling P’=S(Sx, Sy)P • P’’ = S(Sx2 , Sy2)P’ • = S(Sx2 , Sy2)S(Sx1 , Sy1) P • = S(Sx2Sx1, Sy2Sy1)P

  16. Rotation • P’=R()P P’’=R(2)R(1)PP”=R(2+1)P

  17. Shear Transformation:SHx(a) and SHy(b) P’=SHx(a)P • Shear in x against y by a (or an angle). • x’ = x+ay • y’ = y w’=1

  18. P’=SHy(b)P • Shear in y against x by b (or an angle). • x’=x • y’=y + bx • w’=1.

  19. Y Y Y 10 10 10 5 5 5 X X X 0 0 0 5 5 5 10 10 10 Sheared in x Before Shear Sheared in y

  20. Rigid-body Transformation: T and R. • change: location, orientation; • not change: size, angle between elements. Affine Transformation: S, SH • change: size, location, angle; • not change: line (parallelism). • Note: uniform scaling is between the two. It changes size but not angle. • So far, our S, R, SH are all around the origin.

  21. y x After translation of P1 to origin y y P1  x x After rotation After translation To P1 • Composition of 2D Transformation Rotate the house around P1 For every vertex (P) on the object: P’ = T(P1)R()T(-P1)P y

  22. y y y P1 x x x Original house P1 Translate P1 to origin Scale y y P2 x x Translate to final position P2 Rotate Scale and rotate the house around P1 and move it to P2

  23. For every vertex (P) on the object: P’ = T(P2) R()S(Sx,Sy)T(-P1)P. • Swap the order of operations:not permitted, except uniform scaling (sx=sy) can be swapped with rotation.

  24. y Maximum range of screen coordinates y Viewport 2 Window Viewport 1 x x World coordinates Window in world coordinates Window translated to origin Window scale to size of viewport Screen coordinates Translated by (u,v) to final position y y Window Window x x Screen coordinates World coordinates 5.4 Window-to-Viewport Transformation world-coordinate: inches, feet etc & screen-coordinate: pixels

  25. y x z • Matrix Representation of 3D Transformation Translation: Scaling:

  26. Rotation, around Z-axis: Rotation, around X-axis: Rotation, around Y-axis:

  27. P( x, y, z ) x xp P’(xp, yp, zp) z d z (0,0,0) Projection Plane (View Plane) Projection: Project 3D objects on to a 2D surface

  28. xp= x’ / w’ = x / (z/d + 1) • yp= y’ / w’ = y / (z/d + 1)

  29. P1 P2 P’1 z P’2 Foreshortening: The size of the projected object becomes smaller when the object moves away from the eye. Perspective Projection: the projection that has the foreshortening effect.

  30. 0 Parallel Projection: d  , lines of sight become parallel P1 0 z P2 View Plan (0,0,0)

  31. 0

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