3D Transformations are operations used to change the position, size, orientation, or shape of objects in three-dimensional space.
👉 These transformations work on objects defined in (x, y, z) coordinates.
A point in 3D space: [ P(x, y, z) ]
Using homogeneous coordinates: [ P(x, y, z, 1) ]
👉 This allows us to use matrix multiplication for all transformations.
Shifts an object from one position to another.
[ x' = x + t_x,\quad y' = y + t_y,\quad z' = z + t_z ]
[
\begin{bmatrix}
1 & 0 & 0 & t_x
0 & 1 & 0 & t_y
0 & 0 & 1 & t_z
0 & 0 & 0 & 1
\end{bmatrix}
]
Changes the size of an object.
[ x' = x \cdot s_x,\quad y' = y \cdot s_y,\quad z' = z \cdot s_z ]
[
\begin{bmatrix}
s_x & 0 & 0 & 0
0 & s_y & 0 & 0
0 & 0 & s_z & 0
0 & 0 & 0 & 1
\end{bmatrix}
]
Rotates an object around an axis.
[
\begin{aligned}
x' &= x
y' &= y \cos\theta - z \sin\theta
z' &= y \sin\theta + z \cos\theta
\end{aligned}
]
[
\begin{aligned}
x' &= x \cos\theta + z \sin\theta
y' &= y
z' &= -x \sin\theta + z \cos\theta
\end{aligned}
]
[
\begin{aligned}
x' &= x \cos\theta - y \sin\theta
y' &= x \sin\theta + y \cos\theta
z' &= z
\end{aligned}
]
[
\begin{bmatrix}
\cos\theta & -\sin\theta & 0 & 0
\sin\theta & \cos\theta & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1
\end{bmatrix}
]
Produces a mirror image of an object.
Reflection about XY-plane: [ (x, y, z) \rightarrow (x, y, -z) ]
Reflection about YZ-plane: [ (x, y, z) \rightarrow (-x, y, z) ]
Reflection about XZ-plane: [ (x, y, z) \rightarrow (x, -y, z) ]
Distorts the shape by shifting coordinates.
[ x' = x + sh_x \cdot y + sh_z \cdot z ]
Representation of 3D points as: [ (x, y, z, 1) ]
Combination of multiple transformations.
Scale → Rotate → Translate
👉 Represented as: [ T = T_3 \cdot T_2 \cdot T_1 ]
⚠️ Order matters Changing order gives different results.
3D transformations modify objects in (x, y, z) space
Types:
Use 4×4 matrices + homogeneous coordinates
Order is important in composite transformations
Widely used in 3D graphics, animation, and games
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