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CS U540 Computer Graphics

CS U540 Computer Graphics. Prof. Harriet Fell Spring 2007 Lecture 11 – February 1, 2007. Today’s Topics. Linear Algebra Review Matrices Transformations New Linear Algebra Homogeneous Coordinates. Matrices. We use 2x2, 3x3, and 4x4 matrices in computer graphics.

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CS U540 Computer Graphics

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  1. CS U540Computer Graphics Prof. Harriet Fell Spring 2007 Lecture 11 – February 1, 2007

  2. Today’s Topics • Linear Algebra Review • Matrices • Transformations • New Linear Algebra • Homogeneous Coordinates

  3. Matrices • We use 2x2, 3x3, and 4x4 matrices in computer graphics. • We’ll start with a review of 2D matrices and transformations.

  4. Basic 2D Linear Transforms

  5. (1, 0) (0.5, 0) (0, 1) (0, 0.5) Scale by .5

  6. y y x x Scaling by .5

  7. y y x x General Scaling

  8. y y 1 sy x x sx 1 General Scaling

  9. sin() cos()  Rotation -sin() cos()

  10. y y x x Rotation 

  11. Reflection in y-axis

  12. y x Reflection in y-axis y x

  13. Reflection in x-axis

  14. Reflection in x-axis y y x x

  15. Shear-x s

  16. y y x x Shear x

  17. Shear-y s

  18. y y x x Shear y

  19. Linear Transformations • Scale, Reflection, Rotation, and Shear are all linear transformations • They satisfy: T(au + bv) = aT(u) + bT(v) • u and v are vectors • a and b are scalars • If T is a linear transformation • T((0, 0)) = (0, 0)

  20. Composing Linear Transformations • If T1 and T2 are transformations • T2 T1(v) =def T2(T1(v)) • If T1 and T2 are linear and are represented by matrices M1 and M2 • T2 T1 is represented by M2 M1 • T2 T1(v) = T2(T1(v)) = (M2 M1)(v)

  21. y y x x Reflection About an Arbitrary Line (through the origin)

  22. y x Reflection as a Composition

  23. Decomposing Linear Transformations • Any 2D Linear Transformation can be decomposed into the product of a rotation, a scale, and a rotation if the scale can have negative numbers. • M = R1SR2

  24. y y  x x Rotation about an Arbitrary Point  This is not a linear transformation. The origin moves.

  25. Translation (x, y)(x+a,y+b) y y (a, b) x x This is not a linear transformation. The origin moves.

  26. Homogeneous Coordinates y Embed the xy-plane in R3 at z = 1. (x, y)  (x, y, 1) y x x z

  27. Any 2D linear transformation can be represented by a 2x2 matrix 2D Linear Transformations as 3D Matrices or a 3x3 matrix

  28. Any 2D translation can be represented by a 3x3 matrix. 2D Linear Translations as 3D Matrices This is a 3D shear that acts as a translation on the plane z = 1.

  29. Translation as a Shear y y x x z

  30. 2D Affine Transformations • An affine transformation is any transformation that preserves co-linearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). • With homogeneous coordinates, we can represent all 2D affine transformations as 3D linear transformations. • We can then use matrix multiplication to transform objects.

  31. y y x x Rotation about an Arbitrary Point  

  32. y     x Rotation about an Arbitrary Point R() T(-cx, -cy) T(cx, cy)

  33. translate (A-a,B-b) Windowing Transforms (A,B) (a,b) scale (C,D) (C-c,D-d) translate (c,d)

  34. 3D Transformations Remember: A 3D linear transformation can be represented by a 3x3 matrix.

  35. 3D Affine Transformations

  36. 3D Rotations

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