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1. Graphics & Visualization Chapter 3 2D and 3D Coordinate Systems and Transformations Graphics & Visualization: Principles & Algorithms

2. Introduction • In computer graphics is often necessary to change: • the form of the objects • the coordinate system • Examples: • In a model of a scene, the digitized form of a car may be used in several instances, positioned at various points & directions and in different sizes • In animation, an object may be transformed from frame to frame • As objects traverse the graphics pipeline, they change their coordinate system: object coordinates  world coordinates world coordinates  eye coordinates • Coordinates transformations: - tools of change - the most important & classic topic in computer graphics Graphics & Visualization: Principles & Algorithms Chapter 3

3. Introduction (2) • Points in 3D Euclidean space E3: 3×1 column vectors (here) • Linear transformations: - 3×3 matrices - they are post-multiplied by a point to produce another point Graphics & Visualization: Principles & Algorithms Chapter 3

4. Introduction (3) • If points were represented by 1×3 row vector the linear transformations would be pre-multiplied by the point • Right-handed three-dimensional coordinate systems are used: Graphics & Visualization: Principles & Algorithms Chapter 3

5. Affine Transformations - Combinations • Mathematics: Transformations: mappings whose domain & range are the same set (e.g. E3 to E3) • Computer graphics & visualization: Affine Transformations: transformations which preserve important geometric properties of the objects being transformed • Affine transformations preserve affine combinations • Examples of Affine combinations: • line segments, convex polygons, triangles, tetrahedra  the building blocks of our models Graphics & Visualization: Principles & Algorithms Chapter 3

6. Affine Combinations • An affine combination of points is a point : : affine coordinates of p with respect to where • Convex affine combination: • if all ai ≥ 0 , i = 0,1,…,n • The affine combination p is within the convex hull of the original points • E.g.1: Line segment between points p1and p2 is the set of points p: p = a1 ∙ p1 + a2 ∙ p2 with 0 ≤ a1 ≤ 1 and a2 = 1 – a1 • E.g. 2: Triangle with vertices p1,p2,p3 is the set of points p: p = a1 ∙ p1 + a2 ∙ p2 + a3 ∙ p3 with 0 ≤ a1, a2, a3≤ 1 and a1+ a2 + a3 = 1 Graphics & Visualization: Principles & Algorithms Chapter 3

7. Affine Transformations • Affine transformation: • transformation which preserves affine combinations • it retains the inter-relationship of the points • A transformation is affine if where : an affine combination • The result of the application of an affine transformation onto the result p of an affine combination should equal the affine combination of the result of performing the affine transformation on the defining points with the same weights ai • E.g. Graphics & Visualization: Principles & Algorithms Chapter 3

8. Affine Transformations (2) • Practical consequence: • Internal points need not be transformed • It suffices to transform the defining points • E.g. to perform an affine transformation on a triangle: • Transform its three vertices only, not its (infinite) interior points General affine transformation Mappings of the form (1) where A is a 3×3 matrix is a 3×1 matrix are affine transformations in E3 . Proof: we shall show that (1) preserves affine combinations Graphics & Visualization: Principles & Algorithms Chapter 3

9. 2D Affine Transformations • 2D results generalize to 3D • Four basic affine transformations: translation scaling rotation shearing Graphics & Visualization: Principles & Algorithms Chapter 3

10. 2D Translation • Defines a movement by a certain distance in a certain direction, both specified by the translation vector • The translation of a 2D point by a vector gives another point : • Instantiation of where (I is the 2×2 identity matrix) Graphics & Visualization: Principles & Algorithms Chapter 3

11. 2D Scaling • Changes the size of objects • Scaling factor: specifies the change in each direction • 2D: sx & sy are the scaling factors which are multiplied by the respective coordinates of p=[x, y]T • Scaling of 2D point p=[x, y]T to give p’=[x’, y’]T : • Instantiation of where Graphics & Visualization: Principles & Algorithms Chapter 3

12. 2D Scaling (2) • Not possible to observe the effect on a single point • If scaling factor < 1  reduce the object’s size scaling factor > 1  increase the object’s size • Translation side-effect (proportional to the scaling factor): • The object has moved toward the origin of the x-axis (sx < 1) • The object has moved away from the origin of the y-axis (sy > 1) • Isotropic scaling: • If all the scaling factors are equal (in 2D sx = sy) • Preserves the similarity of objects (angles) • Mirroring: • Special case, using a -1 scaling factor • About x-axis: S(1,-1) and y-axis: S(-1,1) Graphics & Visualization: Principles & Algorithms Chapter 3

13. 2D Rotation • Turns the objects about the origin • The distance from the origin does not change, only the orientation changes • Counterclockwise rotation is positive • We can estimate from : Thus: Graphics & Visualization: Principles & Algorithms Chapter 3

14. 2D Rotation (2) • where: • Rotation is an instantiation of the general affine transformation where and Graphics & Visualization: Principles & Algorithms Chapter 3

15. 2D Shear • Increases one of the object’s coordinates by an amount equal to the other coordinate times a shearing factor • Physical example: cards placed flat on a table and then tilted by a hard book. Graphics & Visualization: Principles & Algorithms Chapter 3

16. 2D Shear (2) • We can estimate from : shear along x axis  shear along y axis  where α and b are the respective shear factors • In matrix form: shear on x axis  , where shear on y axis  , where • Shear is an instantiation of the general affine transformation where or and Graphics & Visualization: Principles & Algorithms Chapter 3

17. Composite Transformations • Useful transformations in computer graphics and visualization rarely consist of a single basic affine transformation • All transformations must be applied to all objects of a scene • Objects are defined by thousands or even millions of vertices • EXAMPLE: Rotate a 2D object by 45o and then isotropically scale it by a factor of 2 • Apply the rotation matrix: • Then apply the scaling matrix: Graphics & Visualization: Principles & Algorithms Chapter 3

18. Composite Transformations (2) • It is possible to apply the matrices sequentially to every vertex p: S(2, 2) (R(45o) p)  NOT EFFICIENT • Another way is to exploit the associative property of matrix multiplication and apply the pre-computed composite to the vertices: (S(2, 2) R(45o) )p MORE EFFICIENT • Thus the composite transformation is only computed once and the composite matrix is applied to the vertices • Matrix multiplication is not commutative  the order of multiplying the transformation matrices is important • Having chosen the column representation of points  transformation matrices are right-multiplied by the points write the matrix composition in the reverse of the order of application Graphics & Visualization: Principles & Algorithms Chapter 3

19. Composite Transformations (3) • To apply the sequence of transformations T1,T2, ...,Tm, we compute the composite matrix Tm ·... ·T2 ·T1 • Problem with the translation transformation: Translation cannot be described by a linear transformation matrix such as: • Thus translation cannot be included in a composite transformation • Solution to the problem  homogeneous coordinates Graphics & Visualization: Principles & Algorithms Chapter 3

20. Homogeneous Coordinates • Homogeneous Coordinates use one additional dimension than the space we want to represent • 2D space: , where w is the new coordinate that corresponds to the extra dimension; w ≠ 0 • Fixing w=1 maintains our original dimensionality by taking slice w=1 • In 2D we use the plane w=1 instead of the xy-plane Graphics & Visualization: Principles & Algorithms Chapter 3

21. Homogeneous Coordinates (2) • Points whose homogeneous coordinates are multiples of each other are equivalent: e.g., [1,2,3]T and [2,4,6]T represent the same point • The actual point that they represent is given by their unique basic representation, which has w = 1 and is obtained by dividing all coordinates by w: [x/w, y/w, w/w]T = [x/w, y/w, 1]T • In general, we use the basic representation for points, and we ensure that our transformations preserve this property Graphics & Visualization: Principles & Algorithms Chapter 3

22. Homogeneous Coordinates (3) • How do homogeneous coordinates treat the translation problem? • Exploit the fact that points have w = 1, in order to represent the translation of a point p = [x, y, w]T by a vector , as a linear transformation: • Transformation on the w-coordinate ensures that the resulting point has Graphics & Visualization: Principles & Algorithms Chapter 3

23. Homogeneous Coordinates (4) • The above linear expressions can be encapsulated in matrix form, thus treating translation in the same way as the other basic affine transformations • In non-homogeneous transformations, the origin [0, 0]T is a fixed point: • A positive effect of homogeneous coordinates is that there is no fixed point under homogeneous affine transformations Graphics & Visualization: Principles & Algorithms Chapter 3

24. Homogeneous Coordinates (4) • The 2D origin is now [0, 0, 1]T which is not a fixed point • The point [0, 0, 0]T is outside w=1 plane  disallowed since it has w=0 • From here on points will be represented by their homogeneous coordinates: 2D  [x, y, 1]T 3D  [x, y, z, 1]T Graphics & Visualization: Principles & Algorithms Chapter 3

25. 2D Homogeneous Affine Transformations • 2D homogeneous translation matrix: • Translation is treated like the other basic affine transformations: • The last row of a homogeneous transformation matrix is always [0, 0, 1] in order to preserve the unit value of the w-coordinate • 2D homogeneous scaling matrix: Graphics & Visualization: Principles & Algorithms Chapter 3

26. 2D Homogeneous Affine Transformations (2) • 2D homogeneous rotation matrix: • 2D homogeneous shear matrices: shear on x axis  shear on y axis  Graphics & Visualization: Principles & Algorithms Chapter 3

27. 2D Homogeneous Inverse Transformations • Often necessary to reverse a transformation • 2D inverse homogeneous translation matrix: • 2D inverse homogeneous scaling matrix: Graphics & Visualization: Principles & Algorithms Chapter 3

28. 2D Homogeneous Inverse Transformations (2) • 2D inverse homogeneous rotation matrix: • 2D inverse homogeneous shear matrix: shear on x axis  shear on y axis  Graphics & Visualization: Principles & Algorithms Chapter 3

29. 2D Homogeneous Inverse Transformations • Applying an object transformation on an object is equivalent to the application of the inverse transformation on the coordinate system (axis transformation) EXAMPLE: • Isotropically scaling an object by a factor of 2 is equivalent to isotropically scaling the coordinate system axis by a factor of 1/2 (shrinking) Graphics & Visualization: Principles & Algorithms Chapter 3

30. Properties of Homogeneous Matrices Some properties of homogeneous affine transformation matrices: •  •  •  •  for isotropic scaling only (sx=sy) Graphics & Visualization: Principles & Algorithms Chapter 3

31. 2D Transformation Example 1 EXAMPLE 1: Rotation about an arbitrary point Determine the transformation matrix R(θ, p) required to perform rotation about an arbitrary point p by an angle θ SOLUTION • Step 1: Translate by • Step 2: Rotate by θ, R(θ) • Translate by Graphics & Visualization: Principles & Algorithms Chapter 3

32. 2D Transformation Example 2 EXAMPLE 2: Rotation of a triangle about a point Rotate the triangle by 45o about the point p=[-1,-1]T, where a=[0,0]T, b=[1,1]T and c=[5,2]T SOLUTION • The triangle can by represented by the matrix • We shall apply R(θ,p) [Ex. 1] to the triangle: • The rotated triangle is where a’=[-1, ]T, b’=[-1, ]T and c’=[ ]T Graphics & Visualization: Principles & Algorithms Chapter 3

33. 2D Transformation Example 3 EXAMPLE 3: Scaling about an arbitrary point Determine the transformation matrix S(sx,sy,p) required to perform scaling by sx and sy about an arbitrary point p SOLUTION • Step 1: Translate by • Step 2: Scale by sx and sy , S(sx,sy) • Step 3: Translate by to undo the initial translation Graphics & Visualization: Principles & Algorithms Chapter 3

34. 2D Transformation Example 4 EXAMPLE 4: Scaling of a triangle about a point Double the lengths of the sides of triangle keeping its vertex c fixed . The coordinates of its vertices are a=[0,0]T, b=[1,1]T , c=[5,2]T SOLUTION • The triangle can by represented by the matrix T [Ex. 2] • We shall apply the matrix S(sx,sy,p) [Ex. 3] to the triangle, setting the scaling factor equal to 2 and p=c • The scaled triangle is where a’=[-5,-2]T, b’=[-3,0]T and c’=[5,2]T Graphics & Visualization: Principles & Algorithms Chapter 3

35. 2D Transformation Example 5 EXAMPLE 5: Axis transformation Suppose that the coordinate system is translated by the vector . Determine the matrix that describes this effect SOLUTION • The required transformation matrix must produce the coordinates of the objects with respect to the new coordinate system. This is achieved by applying the inverse translation to the objects:  • Similar argument holds for any other axis transformation. Its effect is encapsulated by applying the inverse transformation to the objects Graphics & Visualization: Principles & Algorithms Chapter 3

36. 2D Transformation Example 6 EXAMPLE 6: Mirroring about an arbitrary axis Determine the transformation matrix required to perform mirroring about an axis specified by a point p=[px,py]T and a direction vector SOLUTION • Step 1: Translate by • Step 2: Rotate by –θ (negative as it is clockwise), R(-θ) θ forms between x-axis and vector and: The two previous steps make the general axis coincide with the x-axis • Step 3: Perform mirroring about the x-axis, S(1, -1) • Step 4: Rotate by θ, R(θ) Graphics & Visualization: Principles & Algorithms Chapter 3

37. 2D Transformation Example 6 (2) • Step 5: Translate by Summarizing the previous steps we have: Graphics & Visualization: Principles & Algorithms Chapter 3

38. 2D Transformation Example 7 EXAMPLE 7: Mirror polygon Given a polygon, determine its mirror polygon with respect to (a) the line y=2 and (b) the axis specified by the point p=[0,2]T and the vector The polygon is given by its vertices a=[-1,0]T, b=[0,-2]T, c=[1,0]T and d=[0,2]T SOLUTION • The polygon can be represented by matrix • In case (a) p=[0,2]T and thus θ=0ο, sinθ=0, cosθ=1 and we have: Graphics & Visualization: Principles & Algorithms Chapter 3

39. 2D Transformation Example 7 (2) • In case (b) p=[0,2]T and thus sinθ = cosθ = and we have: where MSYM is the matrix of [Ex. 6] Graphics & Visualization: Principles & Algorithms Chapter 3

40. 2D Transformation Example 8 EXAMPLE 8: Window-to-Viewport transformation The contents of a 2D window must be transferred to a 2D “viewport”. Both the window and “viewport” are rectangular parallelograms with sides parallel to the x- and y-axis. Determine the window to viewport transformation matrix. Also determine how objects are deformed by this transformation. SOLUTION Suppose that the window and the viewport are defined by two opposite vertices and • Step 1: Translate to the origin, using where Graphics & Visualization: Principles & Algorithms Chapter 3

41. 2D Transformation Example 8 (2) • Step 2: Scale the window to the size of the viewport, using S(sx, sy) where • Step 3: Translate to the minimum viewport vertex , using where Summarizing the previous steps we have: Graphics & Visualization: Principles & Algorithms Chapter 3

42. 2D Transformation Example 8 (3) Finally : Graphics & Visualization: Principles & Algorithms Chapter 3

43. 2D Transformation Example 8 (4) • MWV contains non-isotropic scaling (sx≠sy)  objects will be deformed. A circle will become an ellipse and a square will become a rectangular parallelogram. • The aspect ratios of the window and the viewport are defined as the ratios of their x- to their y-sizes: • If aw≠av then objects are deformed. To avoid this, is to use the largest part of the viewport with the same aspect ratio as the window. For example we can change the vxmax or the vymax boundary of the viewport in the following manner: if (av>aw) then vxmax = vxmin+ aw*(vymax – vymin) else if (av<aw) then Graphics & Visualization: Principles & Algorithms Chapter 3

44. 2D Transformation Example 9 EXAMPLE 9: Window-to-Viewport transformation instances Determine the window-to-viewport transformation from the window: to the viewport : If there is deformation, how can it be corrected? SOLUTION • Direct application of the MWV [Ex. 8] For the window and viewport pair gives • Now and , so there is distortion since . It can be corrected by reducing the size of the viewport by setting: Graphics & Visualization: Principles & Algorithms Chapter 3

45. 2D Transformation Example 10 EXAMPLE 10: Tilted window-to-viewport transformation Suppose that the window is tilted and given by its four vertices a=[1,1]T, b=[5,3]T, c=[4,5]T and d=[0,3]T. Determine the transformation that maps it to the viewport SOLUTION • Step 1: Rotate the window by angle –θ about a point a. For this we shall use matrix R(θ, p) [Ex. 1], instantiating it as R(-θ, a) where • Step 2: Apply the window to viewport transformation MWV [Ex. 8] Graphics & Visualization: Principles & Algorithms Chapter 3

46. 2D Transformation Example 10 (2) Before Step 2 we must determine the maximum x- and y- coordinates of the rotated window by computing: Thus , and we have: Graphics & Visualization: Principles & Algorithms Chapter 3

47. 3D Homogeneous Affine Transformations • 3D homogeneous coordinates are similar to 2D • Add an extra coordinate to create [x, y, z, w]T where w corresponds to the extra dimension • Points whose homogeneous coordinates are multiples are equivalent e.g. [1, 2, 3, 2]T and [2, 4, 6, 4]T • Basic representation of a point • is unique • has w=1 • is obtained by dividing by w : [x/w, y/w, z/w, w/w]T = [x/w, y/w, z/w, 1]T , w≠0 Example: • Obtain a 3D projection of 4D space by setting w=1 • Points: 4×1 vectors. Transformations: 4×4 matrices Graphics & Visualization: Principles & Algorithms Chapter 3

48. 3D Homogeneous Translation • Specified by a 3-dimensional vector • Matrix form: • can be combined with other affine transformation matrices by matrix multiplication • Inverse translation: Graphics & Visualization: Principles & Algorithms Chapter 3

49. 3D Homogeneous Scaling • Three scaling factors: sx, sy, sz • If scaling factor < 1  the object’s size is reduced in the respective dimension scaling factor > 1  the object’s size is increased • Matrix form: • Scaling has a translation side-effect, proportional to the scaling factor Graphics & Visualization: Principles & Algorithms Chapter 3

50. 3D Homogeneous Scaling (2) • Isotropic scaling: • if sx = sy = sz • preserves the similarity of objects (angles) • Mirroring: • about one of the major planes (xy, xz, yz) • using a -1 scaling factor • e.g. mirroring about the xy-plane is S(1, 1, -1) • Inverse scaling: Graphics & Visualization: Principles & Algorithms Chapter 3