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4.1.1 Rendering

As a graph is a visual object, we now preoccupy ourselves with rendering graphs. We will not attempt to fully render any graph; the influence we may exert on the world does not permit us to render either the intricate details or the vast stretches present in most graphs. We will content ourselves, for now, with discretely rendering small portions of graphs.

There is a wide variety of physical rendering devices which may be controlled by modern computers. Examples include: monitors, projection units, laser printers, ink-jet printers, dot-matrix printers, thermal printers, and plotters. We phrase our discussion in terms of an abstract rendering device, which approximates actual rendering devices. Our abstract rendering device outputs a perfectly rendered image R, which is a collection of pixels.

Usually, the pixels are rectangular in shape and form a tex2html_wrap_inline37557 grid. Each pixel   tex2html_wrap_inline37559 represents a region of the plane. The pixel tex2html_wrap_inline31435 represents the region   tex2html_wrap_inline37563 . If M is given by

math24903

then the portion of the graph lying within the unit square is rendered, where tex2html_wrap_inline37567 is the coordinates of pixel tex2html_wrap_inline31435 . Other portions of the graph may be rendered by appropriately modifying M.

There are many other possibilities. When rendering with polar coordinates, it may be convenient to have pixels which are wedge-shaped and form concentric rings about the origin.

figure24915

This type of pixel allows for a more natural mapping M, which will in turn enable better interval renderings. When rendering polar graphs, rectangular pixels may still be used, but with a more cumbersome mapping M.

Another possibility is for each pixel to be in the shape of a small cube. As with rectangular pixels, a simple axis-aligned affine mapping may be used, but would now allow portions of space to be presented. Three-dimensional rendering devices would allow the rendering to be presented directly; two-dimensional rendering devices may present projections and slices of the rendering.

In the remainder of this chapter, our illustrations will be of two-dimensional renderings using a grid of rectangular pixels. The accompanying descriptions are general, and may be applied to the other cases mentioned above. In higher (and lower) dimensions, references to tex2html_wrap_inline37577 and tex2html_wrap_inline34277 may be replaced with references to tex2html_wrap_inline37581 and tex2html_wrap_inline37583 , respectively.


next up previous notation contents
Next: 4.1.2 Batch Rendering Up: 4.1 Graphs Previous: 4.1 Graphs
Jeff TupperMarch 1996