Part 8: 2D Viewing Pipeline: Window-to-Viewport Transformation Coordinates

1. The 2D Viewing Pipeline Hierarchy

• Model Coordinates (MC): Local coordinate system where an individual 3D or 2D asset is designed (e.g. car wheel centered at $[0,0]$).
• World Coordinates (WC): Global Cartesian space where multiple assets are assembled into an environment.
• Viewing Window: A rectangular boundary in World Coordinates defining what is to be viewed: $[xw_{\min}, xw_{\max}] \times [yw_{\min}, yw_{\max}]$.
• Viewport: A designated rectangular sub-area of the physical display screen defining where the image is rendered: $[xv_{\min}, xv_{\max}] \times [yv_{\min}, yv_{\max}]$.

2. Derivation of Window-to-Viewport Transformation Equations

To preserve relative spatial positions, a point $(xw, yw)$ inside the world window must occupy the same relative proportional position within the viewport $(xv, yv)$:

Solving directly for viewport coordinates $(xv, yv)$ yields:

Where the horizontal and vertical scaling factors $s_x$ and $s_y$ are:

3. Aspect Ratio Preservation & Distortion

The aspect ratio of a rectangle is defined as $\text{Aspect Ratio} = \frac{\text{Width}}{\text{Height}}$.

• Distortion ($s_x \ne s_y$): If the aspect ratio of the world window differs from the viewport, circles will be squashed or stretched into ellipses, and squares will distort into rectangles.
• Aspect Ratio Preservation ($s_x = s_y$): Graphics engines constrain scaling to $s = \min(s_x, s_y)$, centering the rendered region and leaving horizontal or vertical blank margins—known as letterboxing and pillarboxing.

4. Matrix Formulation of Normalization ($M_{W\to V}$)

The complete window-to-viewport transformation can be executed in a single composite homogeneous matrix operation:

1. Translate window minimum to origin: $T(-xw_{\min}, -yw_{\min})$.
2. Scale by $(s_x, s_y)$: $S(s_x, s_y)$.
3. Translate from origin to viewport minimum: $T(xv_{\min}, yv_{\min})$.