6.6 Three-Dimensional Transformation

Three-dimensional (3D) transformations are used to manipulate 3D objects in a 3D coordinate system. These transformations can change the position, orientation, size, or shape of objects in 3D space. Like 2D transformations, 3D transformations are critical in computer graphics for applications like 3D modeling, animation, and gaming.

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1. Three-Dimensional Translation

Translation in 3D is the process of moving an object from one location to another along the X, Y, and Z axes. The object is shifted by a specified distance in each direction without altering its shape or orientation.

• Mathematical Representation: If $(x, y, z)$ is the original point, and $(tx, ty, tz)$ is the translation vector, the new point $(x', y', z')$ after translation is:

  • $x' = x + tx$

  • $y' = y + ty$

  • $z' = z + tz$

• Transformation Matrix:

  $T = \begin{bmatrix} 1 & 0 & 0 & tx \\ 0 & 1 & 0 & ty \\ 0 & 0 & 1 & tz \\ 0 & 0 & 0 & 1 \end{bmatrix}$

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2. Three-Dimensional Rotation

Rotation in 3D involves rotating an object around one of the three axes (X, Y, or Z). A point in 3D space is rotated by a specified angle around the axis of rotation.

• Rotation about the X-axis:

  • $x' = x$

  • $y' = y \cdot \cos \theta - z \cdot \sin \theta$

  • $z' = y \cdot \sin \theta + z \cdot \cos \theta$

• Rotation about the Y-axis:

  • $x' = x \cdot \cos \theta + z \cdot \sin \theta$

  • $y' = y$

  • $z' = -x \cdot \sin \theta + z \cdot \cos \theta$

• Rotation about the Z-axis:

  • $x' = x \cdot \cos \theta - y \cdot \sin \theta$

  • $y' = x \cdot \sin \theta + y \cdot \cos \theta$

  • $z' = z$

• Rotation Matrices:

  • Rotation about the X-axis:

    $R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta & 0 \\ 0 & \sin \theta & \cos \theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

  • Rotation about the Y-axis:

    $R_y(\theta) = \begin{bmatrix} \cos \theta & 0 & \sin \theta & 0 \\ 0 & 1 & 0 & 0 \\ -\sin \theta & 0 & \cos \theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

  • Rotation about the Z-axis:

    $R_z(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta & 0 & 0 \\ \sin \theta & \cos \theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

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3. Three-Dimensional Scaling

Scaling in 3D modifies the size of an object along the X, Y, and Z axes. It can be uniform (the same factor for all axes) or non-uniform (different scaling factors for each axis).

• Mathematical Representation: If $(x, y, z)$ is the original point and $(sx, sy, sz)$ are the scaling factors along the X, Y, and Z axes, then the new point $(x', y', z')$ is:

  • $x' = x \cdot sx$

  • $y' = y \cdot sy$

  • $z' = z \cdot sz$

• Transformation Matrix:

  $S = \begin{bmatrix} sx & 0 & 0 & 0 \\ 0 & sy & 0 & 0 \\ 0 & 0 & sz & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

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4. Three-Dimensional Reflection

Reflection in 3D mirrors an object across a plane. The reflection can be done across the XY, YZ, or XZ plane, or across an arbitrary plane.

• Reflection across the XY plane:

  • Mathematical Expression:

    • $x' = x$

    • $y' = y$

    • $z' = -z$

  • Transformation matrix:

    $R_{xy} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

• Reflection across the YZ plane:

  • Mathematical Expression:

    • $x' = -x$

    • $y' = y$

    • $z' = z$

  • Transformation matrix:

    $R_{yz} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

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5. Three-Dimensional Shear Transformation

Shear transformation in 3D distorts an object by shifting points in one direction based on their position in another direction. This transformation can be applied to all three axes.

• Mathematical Representation:

  • Shear along the X-axis:

    • $x' = x + sh_x \cdot y + sh_xz \cdot z$

    • $y' = y$

    • $z' = z$

  • Shear along the Y-axis:

    • $x' = x$

    • $y' = y + sh_y \cdot x + sh_yz \cdot z$

    • $z' = z$

  • Shear along the Z-axis:

    • $x' = x$

    • $y' = y$

    • $z' = z + sh_z \cdot x + sh_zy \cdot y$

• Transformation Matrix:

  • Shear along the X-axis:

    $H_{sh_x} = \begin{bmatrix} 1 & sh_x & sh_xz & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

  • Shear along the Y-axis:

    $H_{sh_y} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ sh_y & 1 & sh_yz & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

  • Shear along the Z-axis:

    $H_{sh_z} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ sh_z & sh_zy & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

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6. Three-Dimensional Composite Transformation

Composite transformation in 3D is the combination of multiple transformations (translation, rotation, scaling, reflection, shear) applied sequentially to an object. The resulting transformation matrix is obtained by multiplying the individual transformation matrices together.

For example, applying scaling, then rotation, then translation:

$T_{composite} = T \cdot R \cdot S$

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7. 3D Viewing Pipeline

The 3D viewing pipeline transforms a 3D world into a 2D screen space, similar to the 2D pipeline, but with additional steps to account for the third dimension.

1. Modeling Transformation: Transforms the object from model coordinates to world coordinates.

2. View Transformation: Converts the world coordinates to camera (view) coordinates.

3. Projection Transformation: Projects the 3D world onto a 2D plane (screen space).

4. Clipping: Removes objects outside the view frustum.

5. Viewport Transformation: Converts the 2D coordinates to screen coordinates.

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8. Projection Concepts

• Orthographic Projection:

  • Objects are projected onto a 2D plane along parallel lines. There is no perspective distortion; objects remain the same size regardless of distance from the camera.

  • Used for architectural drawings, engineering, etc.

• Parallel Projection:

  • Similar to orthographic but can also involve objects being projected along non-orthogonal directions.

• Perspective Projection:

  • Mimics the human eye's perception, where objects appear smaller as they move farther from the camera, creating a sense of depth.

  • Defined by a vanishing point and viewing distance.

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Conclusion

Three-dimensional transformations are the cornerstone of 3D graphics, enabling the manipulation of objects in a 3D coordinate system for applications such as modeling, animation, and visualization.

• Translation, Rotation, Scaling, Reflection, and Shear allow for fundamental changes in an object's position, orientation, size, and shape.

• Composite transformations efficiently combine multiple operations into a single matrix, streamlining complex workflows.

• The 3D Viewing Pipeline ensures that the transformation from 3D space to a 2D screen accurately represents the intended scene.

• Projection concepts like orthographic and perspective projection play a vital role in rendering realistic or technical views.

Mastering these transformations is essential for creating immersive and visually compelling 3D environments, driving innovation in fields like gaming, simulation, and virtual reality.