Q3Computer Graphics and Multimedia

In computer graphics, a 2D Transformation is the strict mathematical process of altering the physical coordinates of a geometric object (like a polygon) to change its position, orientation, or size. To maximize computational speed in the GPU architecture, all transformations are aggressively represented as matrix multiplications. To allow Translation to be represented as a matrix multiplication rather than an addition, graphics engines utilize a highly advanced mathematical system called Homogeneous Coordinates, converting standard $(x,y)$ vectors into $(x,y,1)$ vectors.

1. Translation (Positional Shift)

Translation mathematically shifts an object along a straight linear path from one physical location to another. We aggressively add a translation distance $T_x$ to the original $x$ coordinate, and $T_y$ to the $y$ coordinate. $x^{\prime }=x+T_x$ $y^{\prime }=y+T_y$

The strict Homogeneous Matrix representation is: $$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& T_x\\ 0& 1& T_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

2. Scaling (Size Alteration)

Scaling violently alters the physical dimensions of the object by multiplying its coordinates by a strict scaling factor $S_x$ and $S_y$. If $S>1$, the object expands; if $S<1$, it physically shrinks. $x^{\prime }=x\cdot S_x$ $y^{\prime }=y\cdot S_y$

The strict Matrix representation is: $$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}{S}_x& 0& 0\\ 0& S_y& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

3. Rotation (Angular Spin)

Rotation mathematically revolves the object around the origin $(0,0)$ by a specific angle $\theta$. This requires aggressive trigonometric derivations. $x^{\prime }=x\cos \theta -y\sin \theta$ $y^{\prime }=x\sin \theta +y\cos \theta$

The strict Matrix representation is: $$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

By physically multiplying these homogeneous matrices together (Composite Transformation), the GPU can execute millions of complex movements simultaneously in a single, lightning-fast hardware cycle.