##
1.1 MATHEMATICAL ELEMENTS OF COMPUTER GRAPHICS

###
1.1.1 Introduction

• To put geometries on the computer screen we depend on basic mathematical tools and methods.

• After a geometric model is constructed, it must be displayed (rendered).

• Rendering is mainly limited by computer hardware, and the geometric model.

• The main methods for doing computer graphics are,

- Wire Frame

- Wire frame with hidden line removal

- Polygon drawing (backface, and clipping)

- Shaded polygons

- Raytracing

• As the scene becomes more complicated, the computing time becomes longer, but the picture becomes more realistic.

• The basic history of research on geometric modeling can be summarized as,

- 2D computer drafting - Mid 60s

- 2.5 D - Late 60s

- 3D Wire Frame Systems - Early 70s

- 3D Surface Systems - Mid 70s

- 3D Primitive Solids - Early to Mid 70s

- 3D Arbitrary Solids - Mid to Late 70s

###
1.1.2 PIXELS

• The fundamental task is converting lines, points and surfaces in 3D space, to be depicted on a 2D screen using colored pixels, or printed on paper with dots, or plotted with pens.

• A computer screen is made up of an square array of points (pixels). The points can be lit up. When viewed as a whole these points make a picture.

• One major problem is making a map between a geometry model (a collection of points) and what we see on the screen. This is accomplished with the perspective transform.

###
1.1.3 The Perspective Transform

• A set of basic viewing parameters may be defined (variations are also common),

- The point the Eye is looking at, and from which direction

- The focal distance to the viewing plane

- The size of the viewing plane being focused on

- Which direction is up for the eye

• As seen above the viewing parameters can all be combined using simple matrix multiplication which will convert a point in 3D space to a point on the screen.

• The process of drawing an object is merely applying this transformation to each point in the 3D model, then using the resulting (x, y) point on the 2D screen. (Note: If this transformation is done properly then z = depth in the view plane.)

• The point mapped to the computer screen can then be converted to a single pixel using a simple scaling calculation. (Note: It is not shown, but if a point is off the screen, then it cannot be drawn.)

• Visual display can be done using,

- CRT monitors with Frame Buffer memory to store the image.

- plotters which draw one line at a time

- printers using special and proprietary graphics languages

• For the sake of simplicity, the remaining graphics methods ignore some trivial operations such as screen coordinates, line clipping at edge of screen, etc.

• The ‘z’ value after the perspective transform gives a relative depth of a point. This can be used later for depth sorting, or to set light intensity to cue the user to view depth.