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Transformations

Transformations. Jehee Lee Seoul National University. Transformations. Local moving frame. Linear transformations Rigid transformations Affine transformations Projective transformations. T. Global reference frame. Linear Transformations.

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Transformations

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  1. Transformations Jehee Lee Seoul National University

  2. Transformations Local moving frame • Linear transformations • Rigid transformations • Affine transformations • Projective transformations T Global reference frame

  3. Linear Transformations • A linear transformationT is a mapping between vector spaces • T maps vectors to vectors • linear combination is invariant under T • In 3-space, T can be represented by a 3x3 matrix

  4. Examples of Linear Transformations • 2D rotation • 2D scaling

  5. Examples of Linear Transformations • 2D shear • Along X-axis • Along Y-axis

  6. Examples of Linear Transformations • 2D reflection • Along X-axis • Along Y-axis

  7. Properties of Linear Transformations • Any linear transformation between 3D spaces can be represented by a 3x3 matrix • Any linear transformation between 3D spaces can be represented as a combination of rotation, shear, and scaling • Rotation can be represented as a combination of scaling and shear • Is this combination unique ?

  8. Properties of Linear Transformations • Rotation can be computed as two-pass shear transformations

  9. Changing Bases • Linear transformations as a change of bases

  10. Changing Bases • Linear transformations as a change of bases

  11. Affine Transformations • An affine transformationT is a mapping between affine spaces • T maps vectors to vectors, and points to points • T is a linear transformation on vectors • affine combination is invariant under T • In 3-space,T can be represented by a 3x3 matrix together with a 3x1 translation vector

  12. Properties of Affine Transformations • Any affine transformation between 3D spaces can be represented by a 4x4 matrix • Any affine transformation between 3D spaces can be represented as a combination of a linear transformation followed by translation

  13. Properties of Affine Transformations • An affine transf. maps lines to lines • An affine transf. maps parallel lines to parallel lines • An affine transf. preserves ratios of distance along a line • An affine transf. does not preserve absolute distances and angles

  14. Examples of Affine Transformations • 2D rotation • 2D scaling

  15. Examples of Affine Transformations • 2D shear • 2D translation

  16. Examples of Affine Transformations • 2D transformation for vectors • Translation is simply ignored

  17. Examples of Affine Transformations • 3D rotation How can we rotate about an arbitrary line ?

  18. Pivot-Point Rotation • Rotation with respect to a pivot point (x,y)

  19. Fixed-Point Scaling • Scaling with respect to a fixed point (x,y)

  20. Scaling Direction • Scaling along an arbitrary axis

  21. Composite Transformations • Composite 2D Translation

  22. Composite Transformations • Composite 2D Scaling

  23. Composite Transformations • Composite 2D Rotation

  24. Composing Transformations • Suppose we want, • We have to compose two transformations

  25. Composing Transformations • Matrix multiplication is not commutative Translation followed by rotation Translation followed by rotation

  26. Composing Transformations • R-to-L : interpret operations w.r.t. fixed coordinates • L-to-R : interpret operations w.r.t local coordinates (Column major convention)

  27. Review of Affine Frames • A frame is defined as a set of vectors {vi | i=1, …, N} and a point o • Set of vectors {vi} are bases of the associate vector space • o is an origin of the frame • N is the dimension of the affine space • Any point p can be written as • Any vector v can be written as

  28. Changing Frames • Affine transformations as a change of frame

  29. Changing Frames • Affine transformations as a change of frame

  30. Rigid Transformations • A rigid transformationT is a mapping between affine spaces • T maps vectors to vectors, and points to points • T preserves distances between all points • T preserves cross product for all vectors (to avoid reflection) • In 3-spaces, T can be represented as

  31. Rigid Body Transformations • Rigid body transformations allow only rotation and translation • Rotation matrices form SO(3) • Special orthogonal group (Distance preserving) (No reflection)

  32. Summary • Linear transformations • 3x3 matrix • Rotation + scaling + shear • Rigid transformations • SO(3) for rotation • 3D vector for translation • Affine transformation • 3x3 matrix + 3D vector or 4x4 homogenous matrix • Linear transformation + translation • Projective transformation • 4x4 matrix • Affine transformation + perspective projection

  33. Questions • What is the composition of linear/affine/rigid transformations ? • What is the linear (or affine) combination of linear (or affine) transformations ? • What is the linear (or affine) combination of rigid transformations ?

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