4. 3D Transformations

3D transformations use $4 \times 4$ matrices in homogeneous coordinates.

In 3D transformations, we represent points and transformations using 4x4 matrices, which allow for a more complex representation of geometric objects than 2D. Each point in 3D space is represented as a vector with four components, where the fourth component is used to facilitate transformations and ensures that linear transformations can be applied uniformly. This extra dimension is very helpful for translating, rotating, and scaling objects within a three-dimensional environment.

Think of a 3D virtual world, like the environment in a video game. Just like how you can move around in a room to see things from different angles, the 4x4 matrix helps you position and manipulate objects within that virtual space, allowing for actions like spinning a character or moving an object up and down.

a. Translation

Translation in 3D involves moving an object from one position to another in three-dimensional space. This is done by adding a specific value to the x, y, and z coordinates of the object. For example, if an object is located at (x, y, z) in space and we want to move it by a distance (dx, dy, dz), the new position will be (x + dx, y + dy, z + dz). The transformation matrix for this operation is a 4x4 matrix where the last column represents the translation offsets.

Imagine you're pushing a box across the floor. When you push it three steps forward, two steps to the right, and one step up onto a shelf, that entire action is akin to a 3D translation where the box's position updates based on your push. The matrix captures exactly how far the box moves in each direction.

b. Scaling

Scaling in 3D changes the size of an object along the x, y, and z dimensions. This type of transformation is accomplished by multiplying the original coordinates by scaling factors. For instance, if we scale an object by factors of sx, sy, and sz, the new coordinates of a point will be (sx * x, sy * y, sz * z). The transformation matrix reflects these scaling factors along its diagonal.

Think of scaling like inflating a balloon. The balloon expands equally in all directions, getting larger without changing its shape. If a balloon is blown up twice its size, every point on its surface moves away from the center, just like a 3D object scales up based on its scaling factors.

c. Rotation About x-axis: About y-axis: About z-axis:

Rotation in 3D can occur about any of the three axes: x, y, or z. Each axis has its own specific rotation matrix that defines how points in space will be rotated around it. For example, rotating a point around the x-axis affects its y and z coordinates, while the x-coordinate remains unchanged. By using these rotation matrices, we can quickly compute the new positions of points after rotation.

Imagine a spinning globe. When you spin the globe around the vertical axis (like the Earth's rotation), the locations of countries change in your view. Similarly, in 3D graphics, rotating an object around any of the three axes alters its orientation just as we visually perceive it changing when the globe turns.

d. Reflection (over principal planes) Over x-y plane (z = 0):

Reflection in 3D can occur over various planes—common ones include the x-y, y-z, and x-z planes. When reflecting a point over the x-y plane, for example, the z-coordinate is inverted while the x and y coordinates remain constant. This transformation effectively flips the position of the object across the specified plane.

Think of looking at your reflection in a perfectly still lake. The water acts as the reflective plane, mirroring your image upside down. Similarly, in 3D transformations, reflecting a shape over a plane makes it appear as if you are viewing its mirrored version.

e. General 3D Transformation Concatenation Multiple 3D transformations are combined by multiplying respective $4 \times 4$ matrices, following the order of operations required by the application.

When performing multiple transformations on an object in 3D, each transformation can be represented by its own 4x4 matrix. To apply all these transformations, you multiply the matrices together in a specific sequence. This process is important because the order in which transformations are applied affects the final result. For instance, translating an object after rotating it will yield different results than translating it before the rotation.

Picture a factory assembly line, where an item first gets painted, then packaged, and finally shipped. If the order is changed, the item may not be ready for shipment at all. Similarly, in 3D transformations, the sequence of operations matters as each subsequent transformation is affected by the previous one.

--