Two dimentional transform
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Two dimentional transform
- 2. Transformation An operation that changes one configuration into another Types of Transformation: Geometric transformation Object itself is transformed relative to a stationary co- ordinate Co-ordinate transformation Co-ordinate system is transformed relative to an object. Object is held stationary
- 3. 2D Geometric Transformations A two dimensional transformation is any operation on a point in space (x, y) that maps that point's coordinates into a new set of coordinates (x1, y1). Instead of applying a transformation to every point in every line that makes up an object, the transformation is applied only to the vertices of the object and then new lines are drawn between the resulting endpoints.
- 4. 2D Geometric Transformations Translate Rotate Scale Shear
- 5. 2D Translation One of rigid-body transformation, which move objects without deformation Translate an object by Adding offsets to coordinates to generate new coordinates positions Set tx, ty be the translation distance, we have P’=P+T Translation moves the object without deformation P P’ T xtx'x += yty'y += = y x P = y x t t T = 'y 'x 'P
- 6. Basic 2D Translation To move a line segment, apply the transformation equation to each of the two line endpoints and redraw the line between new endpoints To move a polygon, apply the transformation equation to coordinates of each vertex and regenerate the polygon using the new set of vertex coordinates
- 7. Example Translate a polygon with coordinates A(2,5), B(7,10) and c(10,2) by 3 units in x direction and 4 units in y direction
- 8. 2D Rotation Object is rotated ϴ° about the origin. ϴ > 0 – rotation is counter clock wise ϴ < 0 – rotation is clock wise 6 π θ = y x0 1 1 2 2 3 4 5 6 7 8 9 10 3 4 5 6
- 9. 2-D Rotation x = r cos (φ) y = r sin (φ) x’ = r cos (φ + θ) y’ = r sin (φ + θ) Trig Identity… x’ = r cos(φ) cos(θ) – r sin(φ) sin(θ) y’ = r sin(φ) sin(θ) + r cos(φ) cos(θ) Substitute… x’ = x cos(θ) - y sin(θ) y’ = x sin(θ) + y cos(θ) θ (x, y) (x’, y’) φ
- 10. Basic 2D Geometric Transformations 2D Rotation matrix P’=R·P ΘΘ Θ−Θ = cossin sincos R ΦΦ (x,y)r r θ (x’,y’) − = y x y x θθ θθ cossin sincos ' '
- 11. Basic 2D Geometric Transformations 2D Rotation Rotation for a point about any specified position (xr, yr) x’=xr+(x - xr) cos θ – (y - yr) sin θ y’=yr+(x - xr) sin θ + (y - yr) cos θ
- 12. Rotations also move objects without deformation A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
- 13. Example A point (4,3) is rotated counterclockwise by an angle 45° find the rotation matrix and resultant point.
- 14. Basic 2D Geometric Transformations 2D Scaling Scaling is the process of expanding or compressing the dimension of an object Simple 2D scaling is performed by multiplying object positions (x, y) by scaling factors sx and sy x’ = x · sx y’ = y · sy ⋅ = y x s s y x y x 0 0 ' ' P(x,y) P’(x’,y’) xsx• x sy• y y
- 15. 2D Scaling Any positive value can be used as scaling factor Sf < 1 reduce the size of the object Sf > 1 enlarge the object Sf = 1 then the object stays unchanged If sx= sy, we call it uniform scaling If scaling factor <1, then the object moves closer to the origin and If scaling factor >1, then the object moves farther from the origin y x 0 1 1 2 2 3 4 5 6 7 8 9 10 3 4 5 6 1 2 1 3 3 6 3 9
- 16. Basic 2D Geometric Transformations 2D Scaling We can control the location of the scaled object by choosing a position called the fixed point (xf, yf) x’ – xf = (x – xf) sx y’ – yf = (y – yf) sy x’=x · sx+ xf (1 – sx) y’=y · sy+ yf (1 – sy) Polygons are scaled by applying the above formula to each vertex, then regenerating the polygon using the transformed vertices
- 17. Example Scale the polygon with co-ordinates A(2,5), B(7,10) and c(10,2) by 2 units in x direction and 2 units in y direction
- 18. Homogeneous Coordinates Expand each 2D coordinate (x, y) to three element representation (xh, yh, h) called homogenous coordinates h is the homogenous parameter such that x = xh/h, y = yh/h, A convenient choice is to choose h = 1
- 19. Homogeneous Coordinates for translation 2D Translation Matrix or, P’ = T(tx,ty)·P ⋅ = 1100 10 01 1 ' ' y x t t y x y x
- 20. Homogeneous Coordinates for rotation 2D Rotation Matrix or, P’ = R(θ)·P ⋅ ΘΘ Θ−Θ = 1100 0cossin 0sincos 1 ' ' y x y x
- 21. Homogeneous Coordinates for scaling 2D Scaling Matrix or, P’ = S(sx,sy)·P ⋅ = 1100 00 00 1 ' ' y x s s y x y x
- 22. Inverse Transformations 2D Inverse Translation Matrix − − =− 100 10 01 1 y x t t T
- 23. Inverse Transformations 2D Inverse Rotation Matrix ΘΘ− ΘΘ =− 100 0cossin 0sincos 1 R
- 24. Inverse Transformations 2D Inverse Scaling Matrix =− 100 0 1 0 00 1 1 y x s s S
- 25. 2D Composite Transformations We can setup a sequence of transformations as a composite transformation matrix by calculating the product of the individual transformations P’=M2·M1·P P’=M·P
- 26. 2D Composite Transformations Composite 2D Translations + + = ⋅ 100 10 01 100 10 01 100 10 01 21 21 1 1 2 2 yy xx y x y x tt tt t t t t
- 27. 2D Composite Transformations Composite 2D Rotations Θ+ΘΘ+Θ Θ+Θ−Θ+Θ = ΘΘ Θ−Θ ⋅ ΘΘ Θ−Θ 100 0)cos()sin( 0)sin()cos( 100 0cossin 0sincos 100 0cossin 0sincos 2121 2121 11 11 22 22
- 28. 2D Composite Transformations Composite 2D Scaling ⋅ ⋅ = ⋅ 100 00 00 100 00 00 100 00 00 21 21 1 1 2 2 yy xx y x y x ss ss s s s s
- 30. −− +−− = − − ⋅ − ⋅ 100 sin)cos1(cossin sin)cos1(sincos 100 10 01 100 0cossin 0sincos 100 10 01 θθθθ θθθθ θθ θθ rr rr r r r r xy yx y x y x ( ) ( ) ( ) ( )θθ ,,,, rrrrrr yxRyxTRyxT =−−⋅⋅ Translate Rotate Translate (xr,yr ) (xr,yr ) (xr,yr ) (xr,yr )
- 32. − − = − − ⋅ ⋅ 100 )1(0 )1(0 100 10 01 100 00 00 100 10 01 yfy xfx f fx f f sys sxs y x s s y x y Translate Scale Translate (xr,yr ) (xr,yr ) (xr,yr ) (xr,yr ) ( ) ( ) ( ) ( )yxffffyxff ssyxSyxTssSyxT ,,,, ,, =−−⋅⋅
- 34. Example I sat in the car, and find the side mirror is 0.4m on my right and 0.3m in my front • I started my car and drove 5m forward, turned 30 degrees to right, moved 5m forward again, and turned 45 degrees to the right, and stopped • What is the position of the side mirror now, relative to where I was sitting in the beginning?
- 35. Other Two Dimensional Transformations Reflection Transformation that produces a mirror image of an object Image is generated relative to an axis of reflection by rotating the object 180° about the reflection axis
- 36. Reflection about the line y=0 (the x axis) − 100 010 001
- 37. Reflection about the line x=0 (the y axis) − 100 010 001
- 38. Reflection about the origin − − 100 010 001
- 39. Reflection when x = y
- 40. Example Consider the triangle ABC with co- ordinates x(4,1), y(5,2), z(4,3). Reflect the triangle about the x axis and then about the line y = -x
- 41. Shear Transformation that distorts the shape of an object is called shear transformation. Two shearing transformation used: Shift X co-ordinates values Shift Y co-ordinates values
- 42. X shear y x (0,1) (1,1) (1,0)(0,0) y x (2,1) (3,1) (1,0)(0,0) shx=2 100 010 01 xsh yy yshxx x = ⋅+= ' ' Preserve Y coordinates but change the X coordinates values
- 43. Y shear Preserve X coordinates but change the Y coordinates values x’ = x y’ = y + Shy . x y x (0,1) (1,1) (1,0)(0,0) y x (0,1) (1,2) (1,1)(0,0) 100 01 001 ysh
- 44. Example Perform x shear and y shear along on a triangle A(2,1), B(4,3), C(2,3) sh = 2
- 45. Shear relative to other axis X shear with reference to Y axis x y 1 1 yref = -1 x shx = ½, yref = -1 1 1 2 3 yref = -1 ( )x refx x sh y y y y ′ = + − ′ = 1 0 1 0 1 0 0 1 1 x x refx sh sh y x y y ′ − × ÷ ÷ ÷′ = ÷ ÷ ÷ ÷ ÷ ÷
- 46. Shear relative to other axis Y shear with reference to X axis ( )y ref x x y y sh x x ′ = ′ = + − 1 0 0 1 1 0 0 1 1 x y ref x x y sh sh x y ′ ÷ ÷ ÷′ = − × ÷ ÷ ÷ ÷ ÷ ÷ x y 1 1xref = -1 y x 1 1 2 xref = -1
- 47. Example Apply shearing transformation to square with A(0,0), B(1,0), C(1,1), D(0,1). Shear parameter value is 0.5 relative to line Yref = - 1 and Xref = - 1
- 48. 321321321 M)MM()MM(MMMM ⋅⋅=⋅⋅=⋅⋅ Associative properties Transformation is not commutative (CopyCD!) Order of transformation may affect transformation position Matrix Concatenation Properties
- 49. Transformations between 2DTransformations between 2D Coordinate SystemsCoordinate Systems To translate object descriptions from xy coordinates to x’y’ coordinates, we set up a transformation that superimposes the x’y’ axes onto the xy axes. This is done in two steps: 1. Translate so that the origin (x0, y0) of the x’y’ system is moved to the origin (0, 0) of the xy system. 2. Rotate the x’ axis onto the x axis. x y x0 x’ y’ y0 θ
- 50. Translation (x’,y’)=Tv(x,y) x’= x – tx y’=y-ty Rotation about origin (x’,y’)=Rϴ(x,y) x’= xcosϴ + ysinϴ y’= -xsinϴ + ycosϴ Scaling with origin (x’, y’)=Ssx, sy(x,y) x’= (1/sx)x y’= (1/sy)y Reflection about X axis (x’,y’)= Mx(x,y) x’= x y’= -y Reflection about Y axis (x’,y’)= My(x,y) x’= -x y’= y
- 51. Example Find the x’y’-coordinates of the xy points (10, 20) and (35, 20), as shown in the figure below: x y 30 x’ y’ 10 30º (10, 20) (35, 20)
- 52. Example Find the x’y’-coordinates of the rectangle shown in the figure below: x y 10 x’ y’ 10 60º 20