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CAP4730: Computational Structures in Computer Graphics

CAP4730: Computational Structures in Computer Graphics. 2D Transformations. 2D Transformations. World Coordinates Translate Rotate Scale Viewport Transforms Putting it all together. Transformations. Rigid Body Transformations - transformations that do not change the object. Translate

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CAP4730: Computational Structures in Computer Graphics

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  1. CAP4730: Computational Structures in Computer Graphics 2D Transformations

  2. 2D Transformations • World Coordinates • Translate • Rotate • Scale • Viewport Transforms • Putting it all together

  3. Transformations • Rigid Body Transformations - transformations that do not change the object. • Translate • If you translate a rectangle, it is still a rectangle • Scale • If you scale a rectangle, it is still a rectangle • Rotate • If you rotate a rectangle, it is still a rectangle

  4. Vertices • We have always represented vertices as (x,y) • An alternate method is: • Example:

  5. Matrix * Vector

  6. Matrix * Matrix Does A*B = B*A? What does the identity do?

  7. Practice

  8. Translation • Translation - repositioning an object along a straight-line path (the translation distances) from one coordinate location to another. (x’,y’) (tx,ty) (x,y)

  9. Translation • Given: • We want: • Matrix form:

  10. Translation Examples • P=(2,4), T=(-1,14), P’=(?,?) • P=(8.6,-1), T=(0.4,-0.2), P’=(?,?) • P=(0,0), T=(1,0), P’=(?,?)

  11. Which one is it? (x’,y’) (tx,ty) (tx,ty) (x,y) (x,y)

  12. Recall • A point is a position specified with coordinate values in some reference frame. • We usually label a point in this reference point as the origin. • All points in the reference frame are given with respect to the origin.

  13. Applying to Triangles (tx,ty)

  14. What do we have here? • You know how to:

  15. Scale • Scale - Alters the size of an object. • Scales about a fixed point (x’,y’) (x,y)

  16. Scale • Given: • We want: • Matrix form:

  17. Non-Uniform/Differential Scalin’ (x’,y’) (x,y) S=(1,2)

  18. Rotation • Rotation - repositions an object along a circular path. • Rotation requires an  and a pivot point

  19. Rotation

  20. Example • P=(4,4) • =45 degrees

  21. What is the difference? Revisited V(-0.6,0) V(0,-0.6) V(0.6,0.6) Translate (1.2,0.3) V(0,0.6) V(0.3,0.9) V(0,1.2) Translate (1.2,0.3) V(0.6,0.3) V(1.2,-0.3) V(1.8,0.9) V(0,0.6) V(0.3,0.9) V(0,1.2)

  22. Rotations V(-0.6,0) V(0,-0.6) V(0.6,0.6) Rotate -30 degrees V(0,0.6) V(0.3,0.9) V(0,1.2)

  23. Combining Transformations Q: How do we specify each transformation?

  24. Specifying 2D Transformations • Translation • T(tx, ty) • Translation distances • Scale • S(sx,sy) • Scale factors • Rotation • R() • Rotation angle

  25. Combining Transformations • Using translate, rotation, and scale, how do we get:

  26. Combining Transformations • Note there are two ways to combine rotation and translation. Why?

  27. Let’s look at the equations

  28. Combining them • We must do each step in turn. First we rotate the points, then we translate, etc. • Since we can represent the transformations by matrices, why don’t we just combine them?

  29. 2x2 -> 3x3 Matrices • We can combine transformations by expanding from 2x2 to 3x3 matrices.

  30. Homogenous Coordinates • We need to do something to the vertices • By increasing the dimensionality of the problem we can transform the addition component of Translation into multiplication.

  31. Homogenous Coordinates • Homogenous Coordinates - term used in mathematics to refer to the effect of this representation on Cartesian equations. Converting a pt(x,y) and f(x,y)=0 -> (xh,yh,h) then in homogenous equations mean (v*xh,v*yh,v*h) can be factored out. • What you should get: By expressing the transformations with homogenous equations and coordinates, all transformations can be expressed as matrix multiplications.

  32. Final Transformations - Compare Equations

  33. Combining Transformations

  34. How would we get:

  35. How would we get:

  36. Coordinate Systems • Object Coordinates • World Coordinates • Eye Coordinates

  37. Object Coordinates

  38. World Coordinates

  39. Screen Coordinates

  40. Coordinate Hierarchy

  41. Let’s reexamine assignment 2b

  42. Transformation Hierarchies • For example:

  43. Transformation Hierarchies • Let’s examine:

  44. Transformation Hierarchies • What is a better way?

  45. Transformation Hierarchies • What is a better way?

  46. Transformation Hierarchies • We can have transformations be in relation to each other

  47. More Complex Models

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