Chapter 5: Transforming Objects

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Chapter 5 Transforming Objects

• 5.2 Introduction to Transformations
• Affine transformations are useful
• Compose scene from instances
• Exploit and repeat symmetries
• Different viewpoints of same scene (move camera)
• Computer animation
• Graphics pipeline and current transformation (CT)
• Object transformation vs coordinate
  transformation

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Chapter 5 Transforming Objects

• 5.2.1 Transforming Points and Objects
• Map point P to image Q
• Most mappings continuous
• Restrict ourselves to affine (linear)
  transformations.
• 5.2.2 The Affine Transformations
• and similarly for vectors.

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Chapter 5 Transforming Objects

• 5.2.3 Geometric Effects of Elementary 2D Affine
  Transformations
• Combinations of Translation, scaling, rotation,
  shear.
• Translation
• Or, Q P d
• Scaling
• scaling about the origin
• negative reflection
• uniform vs differential scaling

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Chapter 5 Transforming Objects

• Rotation CCW
• Shearing

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Chapter 5 Transforming Objects

• 5.2.4 Inverse of an Affine Transformation
• Most affine transformations are nonsingular (ie
  det(M) is nonzero)
• To undo transformation Q MP, use P M-1Q.
• Scaling
• Rotation
• Shearing
• Translation

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Chapter 5 Transforming Objects

• 5.2.5 Composing Affine Transformations
• For homogeneous coordinates Affine
  transformations composed by matrix multiplication
  in reverse order.
• 5.2.6 Examples Composing 2D Transformations
• Rotate about an arbitrary point translate,
  rotate, translate
• Reflections about a tilted line

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Chapter 5 Transforming Objects

• 5.2.7 Useful Properties of Affine Transformations
• AT preserve affine combinations of
  pointsT(a1P1a2P2) a1T(P1) a2T(P2)
• AT preserve lines and planes If L(t)(1-t)AtB,
  thenQ(t) (1-t)T(A) tT(B)
• Parallelism of lines and planes is preserved
  Given Abt, we have M(Abt)MA (Mb)t.
  Independent of A, with same direction b.
• Columns of matrix reveal transformed coordinate
  frame
• m1Mi, m2Mj
• Frame (i,j,?) transforms into frame (m1,m2,m3)

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Chapter 5 Transforming Objects

• Relative ratios are preserved
• Effects of transformations on areasdet M
• Every AT is composed of elementary operations
• 2Dany M can be written as (translation)(shear)(sc
  ale)(rotation)
• 3Dany M as (transl)(scale)(rotation)(shear1)(shea
  r2)

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Chapter 5 Transforming Objects

• 5.3 3D Affine Transformations
• 5.3.1 Elementary 3D Transformations
• As for 2D. Selfstudy pp. 234-238.Note
  rotations x-roll, y-roll, z-roll.
• 5.3.2 Composing 3D Affine Transformations
• As for 2D. Selfstudy p. 238.
• 5.3.3 Combining rotations
• 3D rotation matrices do not commute!
• M Rz(ß3)Ry(ß2)Rx(ß1) Eulers angles

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Chapter 5 Transforming Objects

• Rotations about arbitrary axis
• Any rotation about a point is equivalent to a
  single rotation about some axis through the point
  (Eulers theorem).
• Ru(ß) Ry(-?)Rz(?)Rx(ß)Rz(?)Ry(?)
• OpenGL glRotated (angle, ux, uy, uz)

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Chapter 5 Transforming Objects

• Finding axis and angle of rotation Read.
• 5.4 Changing Coordinate Systems
• (a,b,1)T M(c,d,1)T
• Successive changes in coordinate frame (a,b,1)T
  M1(c,d,1)T M1M2(e,f,1)T
• Note to transform points, premultiply.To
  transform coordinate system, postmultiply.
• OpenGL postmultiply by default.

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Chapter 5 Transforming Objects

• Finding axis and angle of rotation Read.
• 5.5 Affine Transformations in a Program
• Selfstudy.
• 5.6 Drawing 3D Scenes with OpenGL
• Selfstudy. Note modelview matrix,
• projection matrix,
• viewport matrix.

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Chapter 5 Transforming Objects

• Homework Task 4
• Practice Exercise 5.2.6, p. 223.
• Practice Exercise 5.2.21, pp. 228.
• Practice Exercise 5.3.9, p. 243.
• Practice Exercise 5.5.3, p. 258.
• Practice Exercise 5.6.1, p. 264.
• Practice Exercise 5.8.10, p. 283.