Derive equation for window-viewport transformation.

Introduction

Window-viewport transformation is a crucial concept in computer graphics. It is a process that allows for the scaling of the picture or the graphical image, which is defined in a window in the world coordinate system, to fit into the viewport in the device coordinate system. This transformation is essential as it enables the proper display of graphical images on different output devices, regardless of their resolution or size.

Window and Viewport

In computer graphics, a window is a rectangular region in the world coordinate system where the picture or graphical image is defined. It is the area of interest in the world coordinate system, and it is where the actual picture is drawn.

On the other hand, a viewport is a rectangular region on the output device (like a computer screen) where the content of the window is to be displayed. It is defined in the device coordinate system. The viewport can be of any size, and it is not necessarily the same size as the window.

The relationship between a window and a viewport is that the window defines what is to be viewed (the picture), and the viewport defines where it is to be viewed (on the output device).

Window-Viewport Transformation

The process of window-viewport transformation involves scaling the picture defined in the window so that it fits into the viewport. This is necessary because the size of the window (in world coordinates) and the size of the viewport (in device coordinates) are usually different. Without this transformation, the picture would not fit properly into the viewport, leading to distortion or clipping of the image.

Derivation of the Window-Viewport Transformation Equation

The window-viewport transformation involves two steps: scaling and translation.

1. Scaling: The scaling transformation changes the size of the picture to match the size of the viewport. The scaling factors (Sx and Sy) are calculated as follows:

   Sx = (viewport width / window width)

   Sy = (viewport height / window height)

2. Translation: The translation transformation moves the scaled picture to the correct position in the viewport. The translation distances (Tx and Ty) are calculated as follows:

   Tx = viewport left - window left * Sx

   Ty = viewport bottom - window bottom * Sy

The window-viewport transformation equation can then be derived by combining the scaling and translation transformations:

Xv = Xw * Sx + Tx

Yv = Yw * Sy + Ty

Where:

• Xv and Yv are the coordinates in the viewport (device coordinates).
• Xw and Yw are the coordinates in the window (world coordinates).

Example of Window-Viewport Transformation

Let's consider an example where we have a window with coordinates (1, 1) and (4, 4) and a viewport with coordinates (2, 2) and (6, 6).

First, we calculate the scaling factors:

Sx = (6 - 2) / (4 - 1) = 4 / 3

Sy = (6 - 2) / (4 - 1) = 4 / 3

Then, we calculate the translation distances:

Tx = 2 - 1 * (4 / 3) = 2 / 3

Ty = 2 - 1 * (4 / 3) = 2 / 3

Finally, we apply the window-viewport transformation equation to transform a point (2, 2) in the window to the viewport:

Xv = 2 * (4 / 3) + (2 / 3) = 10 / 3

Yv = 2 * (4 / 3) + (2 / 3) = 10 / 3

So, the point (2, 2) in the window corresponds to the point (10 / 3, 10 / 3) in the viewport.

Conclusion

Understanding the window-viewport transformation equation is vital in computer graphics as it enables the proper display of graphical images on different output devices. It involves scaling the picture defined in the window to match the size of the viewport and then translating the scaled picture to the correct position in the viewport.

Diagram: Not necessary.

Summary

Window-viewport transformation is a process in computer graphics that allows for the scaling and positioning of a picture or graphical image defined in a window to fit into a viewport on an output device. This transformation is necessary to ensure proper display of the image on different devices with varying resolutions and sizes.

Analogy

Imagine you have a painting on a canvas (window) that you want to display on a picture frame (viewport) of a different size. You need to scale and position the painting so that it fits perfectly within the picture frame without any distortion or clipping.