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Modeling Transformation

Modeling Transformation. Overview. 2D Transformation Basic 2D transformations Matrix representation Matrix Composition 3D Transformation Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ray casting. Modeling Transformation.

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Modeling Transformation

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  1. Modeling Transformation

  2. Overview • 2D Transformation • Basic 2D transformations • Matrix representation • Matrix Composition • 3D Transformation • Basic 3D transformations • Same as 2D • Transformation Hierarchies • Scene graphs • Ray casting

  3. Modeling Transformation • Specify transformations for objects • Allow definitions of objects in own coordinate systems • Allow use of object definition multiple times in a scene Ref] Hearn & Baker

  4. 2D Modeling Transformations

  5. 2D Modeling Transformations

  6. Basic 2D Transformations • Translation • x’ = x + tx • y’ = y + ty • Scale • x’ = x * sx • y’ = y * sy • Shear • x’ = x + hx * y • y’ = y + hy * x • Rotation • x’ = x * cosq - y * sinq • y’ = x * sinq + y * cosq

  7. Matrix Representation • Represent 2D transformation by a matrix • Multiply matrix by column vector • Apply transformation to point

  8. Matrix Representation • Transformations combined by multiplication • Matrices are a convenient and efficient way to represent a sequence of transformations

  9. 2 x 2 Matrices • What types of transformations can be represented with a 2 x 2 matrix? • 2D Identity • 2D Scale around (0, 0)

  10. 2 x 2 Matrices • What types of transformations can be represented with a 2x 2 matrix? • 2D Rotate around (0, 0) • 2D Shear shx = 2

  11. 2 x 2 Matrices • What types of transformations can be represented with a 2x 2 matrix? • 2D Mirror (=Reflection) over Y axis • 2D Mirror over (0, 0)

  12. 2 x 2 Matrices • What types of transformations can be represented with a 2x 2 matrix? • 2D Translation ? NO! Only linear 2D transformation can be presented with a 2 x 2 matrix

  13. Linear Transformations • Linear transformations are combinations of … • Scale • Rotation • Shear • Mirror (=Reflection) • Properties of linear transformations • Satisfies: • Origin maps to origin • Lines map to lines • Parallel line remain parallel • Ratios are preserved • Closed under composition

  14. 2D Translation • 2D Translation represented by a 3 x 3 matrix • Point represented with homogeneous coordinate

  15. Homogeneous Coordinates • Add a 3rd coordinate to every 2D point • (x, y, w) represents a point at location (x/w, y/w) • (x, y, 0) represents a point at infinity • (0, 0, 0) is not allowed

  16. Basic 2D Transformations • Basic 2D Transformations as 3 x 3 matrices

  17. Affine Transformations • Affine Transformations are combinations of … • Linear transformation • Translation • Properties of affine transformations • Origin does not necessarily map to origin • Lines map to lines • Parallel line remain parallel • Ratios are preserved • Closed under composition

  18. Transformation : Rigid-Body / Euclidean Transforms • Rigid Body Transformation (=rigid motion) • A transform made up of only translation and rotation • Preserve the shape of the objects that they act on • Preserves distances • Preserves angles Rigid / Euclidean Identity Translation Rotation

  19. Transformation : Similitudes / Similarity Transforms • Similarity Transformations • Preserves angles • Translation, Rotation, Isotropic scaling Similitudes Rigid / Euclidean Identity Translation Isotropic Scaling Rotation

  20. Transformation : Linear Transformations Similitudes Linear Rigid / Euclidean Scaling Identity Translation Isotropic Scaling Reflection Rotation Shear Translation is not linear

  21. Transformation : Affine Transformations • preserves parallel lines Affine Similitudes Linear Rigid / Euclidean Scaling Identity Translation Isotropic Scaling Reflection Rotation Shear

  22. Transformation : Projective Transformations • preserves lines Projective Affine Similitudes Linear Rigid / Euclidean Scaling Identity Translation Isotropic Scaling Reflection Rotation Shear Perspective

  23. Projective Transformations • Projective Transformations … • Affine transformation • Projective warps • Properties of projective transformations • Origin does not necessarily map to origin • Lines map to lines • Parallel line do not necessarily remain parallel • Ratios are not preserved (but “cross-ratios” are) • Closed under composition

  24. Cross-ratio • Given any four points A, B, C, and D, taken in order, define their cross ratio as • Cross Ratio (ABCD) = cross ratio (ABCD) = cross ratio (A'B'C'D')

  25. Matrix Composition • Transformations can be combined by matrix multiplication

  26. Matrix Composition • Matrices are a convenient and efficient way to represent a sequence of transformations • General purpose representations • Hardware matrix multiply • Efficiency with premultiplication

  27. Matrix Composition • Be aware: order of transformations matter • Matrix multiplication is not commutative “Global” “Local”

  28. Non-commutative Composition Scale then Translate: p' = T ( S p ) = TS p (5,3) (2,2) Scale(2,2) Translate(3,1) (1,1) (3,1) (0,0) (0,0) Translate then Scale: p' = S ( T p ) = ST p (8,4) (4,2) Translate(3,1) Scale(2,2) (6,2) (1,1) (3,1) (0,0)

  29. Matrix Composition • Rotate by q around arbitrary point (a, b) • M = T(a, b)* R(q) * T(-a, -b) • Translate (a, b) to the origin • do the rotation about origin • translate back • Scale by sx, sy around arbitrary point (a, b) • M = T(a, b)* S(sx, sy) * T(-a, -b) • Translate (a, b) to the origin • do the scale about origin • translate back

  30. Transformation Game

  31. Overview • 2D Transformation • Basic 2D transformations • Matrix representation • Matrix Composition • 3D Transformation • Basic 3D transformations • Same as 2D • Transformation Hierarchies • Scene graphs • Ray casting

  32. 3D Transformations • Same idea as 2D transformations • Homogeneous coordinates: (x, y, z, w) • 4 x 4 transformation matrices

  33. Basic 3D transformations

  34. Basic 3D transformations

  35. Non-commutative Composition: Order of Rotations • Order of Rotation Affects Final Position • X-axis  Z-axis • Z-axis  X-axis

  36. Matrix Composition : Rotations in space • Rotation about an Arbitrary Axis • Rotate about some arbitrary a through angle • Rotate about the z axis through an angle • The axis a lies in the yz plane • Rotate about the x axis through an angle • The axis a coincident with the z axis • Rotate about the a axis through • Same as a rotation about the now coincident z axis • Reverse the second rotation about the x axis • Reverse the first rotation about the z axis • Returning the axis a to its original position z a y x

  37. Transformation Hierarchies • Scene may have hierarchy of coordinate systems • Each level stores matrix representing transformation from parent’s coordinate system

  38. Transformation Example 1 • Well-suited for humanoid characters

  39. Transformation Example 1 Ref] Princeton University

  40. Transformation Example 2 • An object may appear in a scene multiple times

  41. Transformation Example 2

  42. Summary • Coordinate systems • World coordinates • Modeling coordinates • Representations of 3D modeling transformations • 4 x 4 Matrices • Scale, rotate, translate, shear, projections, etc. • Not arbitrary warps • Composition of 3D transformations • Matrix multiplication (order matters) • Transformation hierarchies

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