3D transformations in computer graphics are essentially tools to manipulate how we view and interact with 3D objects on a screen. They allow us to move, resize, rotate, and even distort these objects to create dynamic and realistic scenes. Here's a breakdown of the key types:
Translation: This involves shifting an object from one position to another in 3D space. Imagine moving a teacup across a table. We define the movement by specifying distances along the X, Y, and Z axes.
We can define translation using a vector with the desired offset in X (Tx), Y (Ty), and Z (Tz). The new coordinates after translation are:
- X' = X + Tx
- Y' = Y + Ty
- Z' = Z + Tz
This simply adds the translation vector to the original coordinates (X, Y, Z) to get the new position (X', Y', Z').
Rotation: This spins the object around a specific axis. You can rotate a teapot around its vertical axis for a pouring effect. Rotation is defined by an angle and the axis of rotation.
Rotation around X-axis (θ):
New coordinates: X' = X (no change along X) Y' = Y * cos(θ) - Z * sin(θ) Z' = Y * sin(θ) + Z * cos(θ)
Rotation around Y-axis (θ):
New coordinates: X' = X * cos(θ) + Z * sin(θ) Y' = Y (no change along Y) Z' = X * sin(θ) - Z * cos(θ)
Rotation around Z-axis (θ):
New coordinates: X' = X * cos(θ) - Y * sin(θ) Y' = X * sin(θ) + Y * cos(θ) Z' = Z (no change along Z)
Scaling: This modifies the size of the object. You can shrink a house model or stretch a spring animation. Scaling can be done uniformly (enlarging or shrinking all dimensions equally) or non-uniformly (stretching in one direction while shrinking another).
Scaling modifies the size of the object. We can define separate scaling factors for each axis: Sx for X-axis, Sy for Y-axis, and Sz for Z-axis. Here's how scaling is represented:
- X' = X * Sx
- Y' = Y * Sy
- Z' = Z * Sz
Each original coordinate is multiplied by its corresponding scaling factor to achieve the desired size change. Uniform scaling uses the same factor (S) for all axes, resulting in:
- X' = Y' = Z' = X * S
Shear: This one is a bit more complex. It skews the object along a particular axis, tilting its shape without changing its size. Imagine shearing a cube to turn it into a prism.
Shear is a more complex transformation that skews the object along a particular axis. It's defined by shear factors (Shxy, Shxz, Shyx, Shyz). Here are the equations for shear:
- X' = X + Shyx * Y + Shzx * Z
- Y' = Shxy * X + Y + Shzy * Z
- Z' = Shxz * X + Shyz * Y + Z
These transformations are often applied using transformation matrices, which are mathematical tools that efficiently handle the calculations involved in repositioning or modifying object coordinates.
By combining these transformations, you can achieve a vast range of effects in computer graphics, from animating characters walking to creating dynamic camera movements in a video game.