A Note on Digitized Angles

When I looked at them, I was sure that there must be a bug either in IDEAL or in the TROFF processor that typeset the IDEAL output, because the long shafts of the arrows did not properly bisect the angle made by the two short lines of the arrowheads. The shafts seemed to be drawn one pixel too high or too low. Chris spent many hours together with Brian Kernighan trying to find out what was wrong, but no errors could be pinned down. Eventually his thesis was printed on a high-resolution phototypesetter, and the problem became much less noticeable than it had been on the laser-printed proofs. There still was a glitch, but I decided not to hold up Chris’s graduation for the sake of a misplaced pixel. I remembered this incident at the end of 1983, when I was getting ready to write a new version of the METAFONT system for digital art [3]. I didn’t want my system to have such a flaw. But to my surprise, I learned that the problem is actually unavoidable in raster output: It is almost impossible to bisect a digitized angle exactly, except in very special circumstances. The two “halves” of the angle will necessarily appear somewhat different from each other, unless the resolution is quite high. Therefore Van Wyk (and Kernighan) were vindicated. Similar problems are bound to occur in MacDraw and in any other drawing package. For example, one of the things I noticed was the following curious fact. Consider the 45 angle that is made when a straight line segment of slope 2 comes up to a point (x0, y0) and then goes down along another line of slope 3: