3D_Geometric_Transformations_computer graphics.pptx

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1. 3D Geometric Transformations An Overview of Concepts and Techniques

2. Introduction 3D geometric transformations are used to manipulate objects in 3D space.

3. Types of Transformations 1. Translation 2. Scaling 3. Rotation 4. Reflection 5. Shearing

4. Translation Shifts an object from one location to another in 3D space. Formula: P'(x,y,z) = P(x+tx, y+ty, z+tz)

5. Scaling Resizes an object by scaling factors along x, y, z axes. Formula: P'(x,y,z) = P(x*sx, y*sy, z*sz)

6. Rotation Rotates an object about x, y, or z axis. Example: Rotation about z-axis: [x' y' z'] = [x cosθ - y sinθ, x sinθ + y cosθ, z]

7. Reflection Flips an object about a plane (XY, YZ, or XZ). Example: Reflection in XY plane: (x, y, z) → (x, y, -z)

8. Shearing Distorts the shape of an object. Formula involves shifting one coordinate proportional to another.

9. Homogeneous Coordinates Used to represent transformations in matrix form. Adds an extra coordinate (w=1).

10. Matrix Representation Transformation matrices are used for compact representation. Example: Translation Matrix.

11. Composite Transformations Multiple transformations combined into one using matrix multiplication.

12. 3D Transformation Pipeline Steps: Modeling → Viewing → Projection → Clipping → Rasterization

13. Applications Used in Computer Graphics, CAD, Animation, Virtual Reality, and Gaming.

14. Example Problem Rotate point (2,3,4) about z-axis by 90°. Solution: (x',y',z') = (-3,2,4)

15. Visualization 3D transformations are visualized using wireframes, solid models, or rendering techniques.

16. Conclusion 3D transformations are fundamental for graphics, simulations, and real-world modeling.