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Transformations in 3D

Transformations in 3D. Lecture 17 Mon, Oct 6, 2003. 3D Affine Transformations. In 3-dimensional scenes we have the same three kinds of transformation. Translations. Rotations. Scalings. Points have three coordinates (x, y, z), with a fourth coordinate w = 1 added: (x, y, z, 1).

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Transformations in 3D

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  1. Transformations in 3D Lecture 17 Mon, Oct 6, 2003

  2. 3D Affine Transformations • In 3-dimensional scenes we have the same three kinds of transformation. • Translations. • Rotations. • Scalings. • Points have three coordinates (x, y, z), with a fourth coordinate w = 1 added: (x, y, z, 1)

  3. 3D Translations • The translation T(dx, dy, dz). =

  4. 3D Scalings • The scaling S(sx, sy, sz). =

  5. 3D Rotations • Rotations in space are a little more complicated. • A rotation in space is about an axis, not a point. • Any line may be the axis of rotation. • Generally we rotate about the x, y, and z axes.

  6. 3D Rotations about the z-axis • The rotation Rz(). =

  7. Example: Rotation about the z-axis • Rotate 90 about the z-axis. • What is the image of (3, 4, 5)? Rz(90) = (3, 4, 5)  (-4, 3, 5)

  8. 3D Rotations about the x-axis • The rotation Rx(). =

  9. Example: Rotation about the x-axis • Rotate 90 about the x-axis. • What is the image of (3, 4, 5)? Rx(90) = (3, 4, 5)  (3, -5, 4)

  10. 3D Rotations about the y-axis • The rotation Ry(). =

  11. Example: Rotation about the y-axis • Rotate 90 about the z-axis. • What is the image of (3, 4, 5)? Ry(90) = (3, 4, 5)  (5, 4, -3)

  12. Direction of Rotation • For each axis (x, y, z), the positive direction of rotation is determined by the right-hand rule. • Point the thumb of your right hand in the positive direction of the axis. • Curl your fingers. • The direction of curl is the positive direction of rotation.

  13. 3D Transformations in OpenGL • OpenGL uses the functions • glTranslatef(dx, dy, dz) • glRotatef(angle, vx, vy, vz) • v = (vx, vy, vz) is a vector indicating the axis of rotation. • glScalef(sx, sy, sz)

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