The Universe of Discourse

Linogram: Declarative drawing
As we saw in yesterday's article, The definition of the EAS component is twenty lines of strange, mostly mathematical notation. I could have drawn the Etch-a-Sketch in a WYSIWYG diagram-drawing system like xfig. It might have been less trouble. Many people will prefer this. Why invent linogram?

Some of the arguments should be familiar to you. The world is full of operating systems with GUIs. Why use the Unix command line? The world is full of WYSIWYG word processors. Why use TeX?

Text descriptions of processes can be automatically generated, copied, and automatically modified. Common parts can be abstracted out. This is a powerful paradigm.

Collectively, the diagrams contained 19 "gears". Partway through, I decided that the black dot that represented the gear axle was too small, and made it bigger. Had I been using a WYSIWYG system, I would have had the pleasure of editing 19 black dots in 10 separate files. Then, if I didn't like the result, I would have had the pleasure of putting them back the way they were. With linogram, all that was required was to change the 0.02 to an 0.05 in eas.lino:

        define axle {
          param number r = 0.05;
          circle a(fill=1, r=r);
        }

The Etch-a-Sketch article contained seven similar diagrams with slight differences. Each one contained a require "eas"; directive to obtain the same definition of the EAS component. Partway through the process, I decided to alter the aspect ratio of the Etch-a-Sketch body. Had I been drawing these with a WYSIWYG system, that would have meant editing each of the seven diagrams in the same way. With linogram, it meant making a single trivial change to eas.lino.

A linogram diagram has a structure: it is made up of component parts with well-defined relationships. A line in a WYSIWYG diagram might be 4.6 inches long. A line in a linogram diagram might also be 4.6 inches long, but that is probably not all there is to it. The south edge of the body box in my diagrams is 4.6 inches long, but only because it has been inferred (from other relationships) to be 1.15 w, and because w was specified to be 4 inches. Change w, and everything else changes automatically to match. Each part moves appropriately, to maintain the specified relationships. The distance from the knob centers to the edge remains 3/40 of the distance between the knobs. The screen remains 70% as tall as the body. A WYSIWYG system might be able to scale everything down by 50%, but all it can do is to scale down everything by 50%; it doesn't know enough about the relationships between the elements to do any better. What will happen if I reduce the width but not the height by 50%? The gears are circles; will the WYSIWYG system keep them as circles? Will they shrink appropriately? Will their widths be adjusted to fit between the two knobs? Maybe, or maybe not. In linogram, the required relationships are all explicit. For example, I specified the size of the black axle dots in absolute numbers, so they do not grow or shrink when the rest of the diagram is scaled.

Finally, because the diagrams are mathematically specified, I can leave the definitions of some of the components implicit in the mathematics, and let linogram figure them out for me. For example, consider this diagram:

        gear3 gears(width=WIDTH, r1=1/4, r3=1/12);

I specified r1, the radius of the left gear, and r3, the radius of the right gear. Where is the middle gear? It's implicit in the definition of the gear3 type. The definition knows that the three gears must all touch, so it calculates the radius of the middle gear accordingly:

define gear3 {
  ...
  number r2 = (1 - r1 - r3) / 2;
  ...
}

linogram gives me the option of omitting r2 and having it be calculated for me from this formula, or of specifying r2 anyway, in which case linogram will check it against this formula and raise an error if the values don't match.

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More complete information about linogram is available in Chapter 9 of Higher-Order Perl; complete source code is available from the linogram web site.

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