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Chapter 5: Transforming Objects

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Different viewpoints of same scene (move camera) Computer animation ... (a,b,1)T = M(c,d,1)T. Successive changes in coordinate frame: ... – PowerPoint PPT presentation

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Title: Chapter 5: Transforming Objects


1
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

2
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.

3
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

4
Chapter 5 Transforming Objects
  • Rotation CCW
  • Shearing

5
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

6
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

7
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)

8
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)

9
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

10
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)

11
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.

12
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.

13
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.
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