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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 <wl@gnu.org> 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
Béziers.
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

**Circles**, *Andrew Bernard*, `2022/08/10`