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Understanding Three-Dimensional Geometric and Modeling Transformations

This document provides an overview of essential concepts in three-dimensional geometric transformations, crucial in computer graphics. It covers topics such as 3D translations, scaling, rotations, reflections, and shearing. Each transformation type is explained with the use of mathematical matrices, illustrating how objects can be manipulated in 3D space. The text also discusses important geometric types including points, vectors, and scalars, as well as practical applications like perspective projection. This foundational knowledge is valuable for students and professionals in computer graphics.

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Understanding Three-Dimensional Geometric and Modeling Transformations

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  1. Prepared By Niddalabuswereh Mahmoudelqedra Supervised By Dr. Sana’a Wafa Al-Sayegh Computer Graphics University of Palestine ITGD3107

  2. ITGD3107Computer Graphics Chapter 11 Three-Dimensional Geometric and Modeling Transformations

  3. Three-Dimensional Geometric and Modeling Transformations • Some Basics • 3D Translations. • 3D Scaling. • 3D Rotation. • 3D Reflections. • Transformations.

  4. Some Basics • Basic geometric types. • Scalars s • Vectors v • Points p • Transformations • Types of transformation: • rotation, translation, scale,Reflections, shears. • Matrix representation • Order • P=T(P)

  5. 3DPoint • We will consider points as column vectors. Thus, a typical point with coordinates (x, y, z) is represented as:

  6. P is translated to P' by T: 3D Translations. Called the translation matrix T =

  7. 3D Translations.

  8. 3D Translations. • An object is translated in 3D dimensional by transforming each of the defining points of the objects.

  9. 3D Translations.

  10. P is scaled to P' by S: 3D Scaling Called the Scaling matrix S =

  11. 3D Scaling • Scaling with respect to the coordinate origin

  12. 3D Scaling • Scaling with respect to a selected fixed position (xf, yf, zf) • Translate the fixed point to origin • Scale the object relative to the coordinate origin • Translate the fixed point back to its original position

  13. 3D Scaling

  14. 3D Reflections • About an axis:equivalent to 180˚rotation about that axis

  15. 3D Reflections

  16. 3D Shearing • Modify object shapes • Useful for perspective projections: • E.g. draw a cube (3D) on a screen (2D) • Alter the values for xand y by an amount proportional to the distance from zref

  17. 3D Shearing

  18. Shears

  19. Rotation Positive rotation angles produce counterclockwise rotations about a coordinate axis mshe1990@hotmail.com &&

  20. Rotation mshe1990@hotmail.com &&

  21. Coordinate-Axes Rotations mshe1990@hotmail.com &&

  22. Coordinate-Axes Rotations mshe1990@hotmail.com &&

  23. Coordinate-Axes Rotations mshe1990@hotmail.com &&

  24. Coordinate-Axes Rotations mshe1990@hotmail.com &&

  25. General Three-Dimensional Rotations • An object is to be rotated about an axis that is parallel to one of the coordinate axes • Translate the object so that the rotation axis coincides with the parallel coordinate axis • Perform the specified rotation about that axis • Translate the object so that the rotation axis is moved back to its original position mshe1990@hotmail.com &&

  26. General Three-Dimensional Rotations • An object is to be rotated about an axis that is not parallel to one of the coordinate axes • Translate the object so that the rotation axis passes through the coordinate origin. • Rotate the object so that the axis of rotation coincide with one of the coordinate axes. • Perform the specified rotation about that coordinate axis. • Apply inverse rotations to bring the rotation axis back to its original orientation. • Apply the inverse Translation to bring the rotation axis back to its original position. mshe1990@hotmail.com &&

  27. Quiz Draw any shape, then moving translation matrix. Good Luck

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