COMPUTING SCIENCE

# Seeing between the Pixels

# Straightedge and Compass

Pixels were not always the universal building blocks of computer graphics. In the 1960s, many computer display screens worked on another principle altogether, called vector graphics. Instead of scanning a beam over the entire surface of the screen in closely spaced parallel lines (the *raster* pattern of modern displays), the beam was steered from point to point as needed to create a line drawing. For example, directing the beam from the *x,y* coordinates 0,0 to the point 6,0, then to 0,6 and finally back to 0,0 would draw an isosceles right triangle. The corresponding hard-copy device was the pen plotter, a kind of automated Etch-A-Sketch in which a motor-driven pen zipped from point to point over a sheet of paper.

Specialized hardware for vector graphics is now a rarity, and yet the underlying idea has not gone away. Even though modern displays and printers are raster devices, many computer programs adopt the vector format as an internal data structure for graphic objects. Computer-aided-design software works this way; an architect's rendering of a building is made up of lines, not pixels. Often, vector-based programs are referred to as "drawing" software, whereas pixel-based programs are for "painting." (To those who work in the graphic arts, the difference between vectors and pixels is the difference between Adobe Illustrator and Adobe Photoshop.)

Although the term "vector graphics" suggests drawing with straight lines, modern programs offer a compass as well as a straightedge, and even a French curve. Indeed, vector systems can depict the entire universe of planar Cartesian geometry—any form described by equations in *x* and *y*. Linear equations yield lines or line segments; quadratic equations describe circles and other conic sections; cubic equations define a particularly important class of curves are called splines, after the flexible strips of metal or wood that were once used to outline smooth curves in shipbuilding.

At least one common vector-graphics system evolved from a language for driving pen plotters; it is HP-GL, the Hewlett-Packard Graphics Language. But the most widespread notation for vector graphics is Postscript, developed by Adobe Systems. Every letter on this page was created as a sequence of line and arc and spline commands in Postscript.

Postscript is a versatile language that can accommodate pixel-based images as well as vectors. It is also a fully equipped programming language, with facilities for iteration, recursion and defining procedures. One consequence is that a finite expression in Postscript can describe a geometric object of unbounded complexity, such as the Hilbert curve, a one-dimensional line so contorted that it fills a two-dimensional area. (But is the Postscript encoding a representation of the image or merely a program that when executed generates a representation?)

An important advantage of vector formats is that geometric objects retain their identity and structure. In a vector-based drawing program you can select a line and alter its thickness or color. These operations are much harder with an image made up of nothing but pixels, where the concept of "line" does not exist. In many cases vector drawings are also more compact than pixel-based images. And a final virtue of vector formats is that the same drawing can be displayed or printed at any resolution. For pixel images, the only way to show finer detail is to squeeze in more pixels, but the equations that define vector forms are mathematical objects of potentially unlimited precision. The same Postscript letterforms can be rendered on a monitor screen at less than 100 dots per inch or on film at more than 1,000 dots per inch.

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