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Re: Circles

From: David Kastrup
Subject: Re: Circles
Date: Thu, 11 Aug 2022 14:14:10 +0200
User-agent: Gnus/5.13 (Gnus v5.13) Emacs/29.0.50 (gnu/linux)

Werner LEMBERG <> writes:

>>> Well, Metafont uses a third-order Bézier approximation to build up
>>> circles (from four quarter circles).  However, the Metafont book
>>> says that the difference to a real circle is less than 0.06%, which
>>> is thus imperceptible.
>>> The same holds for PostScript-based fonts.
>> PostScript uses second-order Béziers, so it makes little sense to
>> cite the mathematics of Metafont here.
> This is not correct.  PostScript-derived fonts (Type1, CFF/CFF2 in
> OpenType) *do* use third-order curves.  I think you are mixing this up
> with TrueType fonts, which indeed use second-order curves.

I was more talking about the curveto operator than fonts, but you are
right either way.  It was just Truetype that used only quadratic

Sort of embarrassing, since I distinctly remember putting explicit
quartercircle approximations into the curved rectangles PostScript code.
I did not find anything like that in the repository master though; it
must have been an ephemeral part of the work for better rasterized PDF
stems I did quite some time ago.

David Kastrup

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