True circles from round-cap dash?

Question

In the discussion of tricuspoids Jongware stated that "circles" in Illustrator are not circles at all, but approximations "from four quadratic Bézier curves." Is the round cap for a stroke generated the same way, or would it be possible to generate a true circle by outlining the stroke of a 0 pt. round-capped dashed line?
Thanks,
Peter

Answer 1

The former Output Resolution setting was just Illustrator's interface for providing a flatness contol.
Flatness is a value in PostScript that governs how far (in terms of printer spots) the output device is allowed to deviate from the mathematical curve. In effect, it determines how much the bitmap of a curve is allowed to be approximated by rendering the curve as a series of facets. It mainly has to do with the issue of rendering the printer bitmap for long subtle curves which could choke earlier RIPs.
In older versions of Illustrator, rather than provide a straightforward Flatness setting as other programs did, Illustrator provided its so-called Output Resolution setting, such that...
Actual printing device resolution / Output Resolution setting = flatness.
The default was a flatness of 3, and the default assumed an output device with a resolution of 2400 spi. So Illustrator's default Output Resolution setting was 800 (2400spi / 800=3).
Many (probably most) Illustrator users for decades completely misunderstood the setting as something akin to PPI in a raster image. They thought of it as something like Illustrator's Raster Effects Resolution setting, to determine the raster PPI at which everything on the page was rasterized at print time.
FreeHand, by comparison, provided a straightforward Flatness setting that the user could set at the object level, not just the whole document level. (Just like FreeHand allows Halftone settings to be defined at the object level.)
Nowadays, processing power is not as hampered by low flatness settings as in the earlier days of PostScript imagesetters. And PostScript can now automatically set the flatness appropriately for the target device. Illustrator's default setting is now Automatic. You can see it (and change it if you feel you must) in the File>Print>Graphics dialog. There you'll find a slider control for adjusting Flatness. To see, turn off the Automatic checkbox, move the slider. Then mouseover the slider arrow. A tooltip will appear with the actual flatness setting.
It really has nothing to do with the fact that a cubic Bezier curves doesn't (very strictly speaking) define a true quarter circle.
Programs like Illustrator draw cubic (degree 3 polynomial) Bezier curves, not quadratic (degree 2). That's why a segment in Illustrator has two curve handles, not just one. You can see examples of quadratic Beziers in things like the old Macintosh Quickdraw programs, cheezy drawing modules in "office" or "works" programs, and even in the drawing tools in early versions of Flash. Even today, Flash actually renders its curves as quadratic Beziers in its actual SWF output because on average it's less data for the Flash Player to process on-the-fly. That's why FreeHand provided a Flash Preview mode; it allowed a FreeHand user who was drawing elements destined for use in Flash to see the subtle shape differences that would occur when FreeHand's cubic Beziers get converted to Flash's quadratic Beziers.
Some drawing programs in Illustrator's category and some CAD programs provide tools for both cubic and quadratic Beziers.
Having only one "curve handle" a quadratic Bezier curve cannot cross itself to form a loop. There are higher-degree Bezier curves, too; but processing and interface practicality make cubic Beziers the most advantageous for drawing programs.
Apple's TrueType used quadratic Beziers, which are a royal pain to use for drawing exact shapes like type glyphs. Today's OpenType font format can contain either cubic ("PostScript") or quadratic ("TrueType") Bezier curves.
JET