MetaFog:Converting METAFONT ShapestoContours

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1 MetaFog:Converting METAFONT ShapestoContours Richard J. Kinch 6994 Pebble Beach Court Lake Worth FL USA URL: hhtp://www.emi.net/~kinch Abstract TheComputerModernTypefaceshavetheiroriginalspecificationintermsof the METAFONTlanguage. Theindividualglyphprogramsrelyonthesophisticatedalgebraandmarkingmethodsof METAFONT.Manyofthe METAFONT primitives, such as stroked pens and overlapping ink, are not directly expressable inoutlinetypefaceformatssuchastype1andtruetype,whichsupportonly topographiccontoursexpressedas non-overlappingbézier curves. We explainthe computationalgeometryinvolvedinthe conversionfrom METAFONTshapes to outlines, why this is a difficult problem, and why previouseffortshavefallenshort. WedescribeMetaFog,asetofprogramswritten to post-process METAPOSToutput to complete such conversions, and the algorithmsimplementedtosolvethemathematicalproblems.thetwomostsignificant problems are(1) finding the envelope of an elliptical pen stroked along abéziercurve(analgebraicproblem),and(2)reducingoverlappingpathsto anequivalent,non-overlappingcontour(atopologicalproblem). Weproposea schemetoembedtype1andtruetypehinttechnologyinto METAFONTsources to reduce the duplication of effort to produce well-hinted fonts. We compare theaccuracyofmetafog sanalyticconversionstoapproximationsbasedonautotracingof METAFONT sbit-mappedoutput,andshowexamplesoferrorsinthe Computer Modern Typefaces which are hidden in METAFONT proofs but visible in MetaFog proofs. TEXanditsFonts ModernimplementationsofTEXlikeTrueTEX r have eliminated bit-mapped meta-fonts in favor of outlineformatssuchastruetypeortype1. TEX did an admirable job of producing its own font bit-maps in the days before operating systems supportedfonts.buttodaythemostpopularoperating systems and print engines require outline fonts. These scalable formats facilitate previewing and printing TEX documents in a powerful, portable, andflexiblefashionwhichbit-mappedfontscannot achieve. While pure TEX is independent of any particular fonts, TEX is nevertheless just as dependent today on Computer Modern and other METAFONTbasedfontsasever.Thusarisestheneedforconversionof METAFONT programsintoequivalentoutline forms. While METAFONT programscandescribea glyph in terms of complex, overlapping paths, the outlineformatsrequirethatwespecifyglyphsasa setofcontours (non-overlappingoutlines). Herein lies the most difficult aspect of conversion: META- FONT s primitive shapes are built from third-degree parametriccurvesmodulatedbythird-degreepaths, andsuchshapescanoverlap,addandsubtractin arbitrarilydeviousways. Conversions: analytic versus approximate. MetaFog is a system for exact, analytic conversionof METAFONT shapestocontours. Thatis, MetaFog always store shapes in terms of their pure, parametriccurves.by analytic wemeanthatthe methods we use analyze and solve the underlying equations for the parametric curves. We use no intermediate approximations such as converting curvestopolygons,sothateveryresultcurveisa direct derivation of an input curve and every input point is unchanged in the output. By exact wemeanthattheresultcurvesfollow the METAFONTshapes to within one pixel in the1024or2048pixels/emgridusedintypicaloutlinefontformats.insomecases METAFONT design TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting 233

2 RichardJ.Kinch envelopes cannot be represented exactly by Bézier curves,andweusethismetrictodeterminethedegree of curve-fitting needed. We use a METAFONT mode_deffor a perfect output device needing no corrections for fill-in or overshoot. Automatingananalyticconversionof META- FONTshapesrequiresamajoreffortinbothmathematicsandsoftware.Itrequiressolutionstoproblems which Knuth managed to avoid in METAFONT by using numerical tricks and simplifications. Earlier projects have attempted the task, but either fall shortoforapproximatethefullsolution(yanaiand Berry,1990;Carr,1987;Henderson,1989). Outline conversions of meta-fonts have also been done before using approximation techniques, thus avoiding the difficulty of an exact, analytic conversion. For example, autotracing attempts to fit an outline to a high-resolution bit-map. With enough skilled labor, autotracing yields an aesthetically pleasing result, although the shapes will tend to have certain artifactual deviations from the precise METAFONT originals. The BlueSky- Y&YconversionsofComputerModernandother meta-fontsshowthatcarefulautotracingandhandtuning can produce a result equal to that of a conventionally-designedcommercialfont. More recently Malyshev(1994) has published thebakomafonts,whichcontainverypreciseoutlineconversionsofcomputermodern. Malyshev s publication is limited to the results(that is, the outlinefontsthemselves);hehasnotrevealedthedetailsofhistechnique,althoughheclaimsthatitis analyticandnotanautotracedorotherwiseadigitized approximation. We will show below examples of font details which an analytic conversion would preserve,butwhicharemissingfromthebakoma fonts. Malyshev s claim of analytic perfection could neverthelessbetrue,ifsucherrorswereintroduced, forexample,bybugsinhisconversionsoftware.on the other hand, if a hidden approximationis involved somewhere in the BaKoMa conversion process, the result would not meet our strict definition ofbeingboth exact and analytic. Thisisnot tosaythatthebakomafontsarepoorconversions; itisevidentthattheshapesareexcellentinevery wayimportanttofontdesignersandthattheyare generally faithful to the METAFONT originals. The Nature of the Conversion Letusconsiderthe natureofthe conversionsinvolved. METAFONTcan actually do more sophisticated things than we are about to describe, but we will restrict our consideration to those META- FONTfeaturesthatareactuallyusedintypefaces like Computer Modern. Béziercurves. We will consider Bézier(Glassner, 1990)contourstobeourtargetformat. ABézier curve(figure1)isaparametriccurvegovernedby the equation: z(t)=(1 t) 3 z 1 +3(1 t) 2 tz 2 +3(1 t)t 2 z 3 +t 3 z 4 Parametertiscalledthetimealongthecurveand rangesovertheinterval[0,1];thepointattimetis z(t). ABézierpathisasetofBéziercurveswhich connectinachainattheirendpointstoformamore complex curve. A closed path which does not overlap describes a complete circuit and encloses an area. A set of such paths make a Bézier contour,which candescribetheoutlinesofaglyph. Thepathsin thecontourofawell-formedglyphdonotintersect eachother,andaswelltheydonotintersectthemselves.thisistherepresentationusedinthetype1 fontformat(adobesystems,1990). Conversionof Type 1 glyphs to TrueType glyphs(which use lowerorderparametriccurves)isastraightforwardconversion. In METAFONT(as documented in the literate sourcecode),knuth calls the Bézier paths cubic splines(an equivalent mathematical term), and usesadatastructureconsistingofknotlocations andcontrolpointstospecifypaths.thisistheterminologyweuseinmetafog.infigure1,pointsz 1 andz 4 areknots,andz 2 andz 3 arecontrolpoints. The goal of MetaFog conversion is to produce Bézier outlines which accurately represent the METAFONT designs. This will be close to theminimalsetofknotsneededtofitthedesign, because both METAFONT andcomputer Modern are economical in their use of reference points, andthereferencepointsinametafontprogram generally expand into the minimal set of knots to implement a fitted curve. Because METAFONT divides curves into octants, METAFONT s curves tendtohavecontrolpointsevery45degreesorso, versus Type 1 fonts which often subtend curves of90or180degreespercontrolpoint. Sointhis sense METAFONT designs have more controlpoints than good Type 1 designs. On the other hand, thetype1formatmandatesrulesfortangentsand extrema points that tend to add redundant control points to designs, so in this sense METAFONTdesignshavefewer controlpointsthangoodtype1 designs. MetaFogpreservesthepure METAFONT design, suchastheadditionof45degreecontrol points and the absence of redundant extrema points versus a likely implementation in Type 1. The final conversion code will optionally add redundant pointstomeetthetype1mandates. 234 TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting

3 MetaFog: Converting METAFONT Shapes to Contours z 2 z 1 = z(0) z(t) z 3 z 4 = z(1) Figure1:Béziercurve,startingatz 1 andending atz 4.Theoutgoingcontrolpointisz 2,incoming isz 3. Before METAFONT digitizes a glyph into a bitmap,itrepresentstheglyphasacollectionof shapes. Each shape can be an outline determined by a set of Bézier curves or the envelope of an ellipse stroked along a path. Each shape also can addorsubtractink. Thisistheinternalrepresentation which we wish to reduce to an equivalent set of Bézier outlines, which are the shapes which atype1fontusesdirectlyorwhichcanbeeasily converted into the shapes for a TrueType font. METAFONT shapesalsohavecolor;inpractice thismeansthatwecanthinkofeachshapeaseither additive black ink or white ink that subtracts blackinkalreadydrawn.wecanseethattheorder ofdrawingshapesinaglyphmustthereforebepreserved. Within MetaFog we use the winding-number convention(like PostScript s) for controlling color (black versus white), while METAFONT stores an explicitcolorforeachshape. METAFONT shapes usually, but not always, follow a consistent winding directionfortheassociatedcolor.metafogiscareful tocheckshapesoninputsothatthewindingnumber andcolorareconsistent. WhenMetaFogdiscovers an inconsistency, it reverses the input path. The different models treat edges differently whenrenderingbitmaps.wehaveyettotakethis into account in our conversions. Bézier tools. MetaFog uses quite a few algebraic tools to manipulate curves. Some are reimplementedorgeneralizedalgorithmsfrom META- FONT,andsomeareentirelynewconcepts: Findthecoordinatesofagiventimevalueona curve. Findclosesttimeonacurvetoagivenpoint. Auditacurve,path,orcontourdatastructure for consistency. Test whether a path is degenerate(zero winding number). Test whether a path is redundant(contained within) with respect to another. Test whether a path(possibly pivoted) duplicates another. Testwhethertwopathsoverlap(thatis,havea common segment). Find all intersections between two curves, associating mutual intersecting locations. Find all intersections in a contour, associating themwitheachappropriatecurveintermsof time. Sort intersection times associated with curves in a contour. N-sectacurveintoNcurvesgivenasetoftimes. Givenanellipse,findthepointontheellipseat agivenanglefromthemajoraxis. Givenanellipseandanangleofrotation,find themaximumpoint(horizontaltangent)onthe ellipse. Testiftwolinesegmentscross. Given the parallel curves of a stroked path, stretchorshrinktheendpointstofitagiven ellipsewitharbitraryrotation. Givenacurveandarotatedellipse,returnthe 6to8curvesfittingtheenvelope. Given an ellipse, generate a four-curve approximation. Giventhepositionsandtangentsofacurveendpoints, and a midpoint position, locate the endpoints which fit the constraints. Test a curve for simpleness (that is, turning angle π 2 andnoinflections). Givenanopenpathandanellipse,returnthe envelope, reducing overlapping segments. Testwhetheragivenpointisonacurve(with giventolerance). Testwhetheragivenpointisinsideaclosed path. Testwhetheragivenpathisinteriortoanother path. Findallcircuitsinacontour. The above algorithms, plus syntactical and data-structurechores,makeupabout12,000lines ofcprogramcode. TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting 235

4 RichardJ.Kinch Loading Shape Information from METAPOST. METAPOST(Hobby,1989)relievesusofthedifficulttaskofrunning METAFONTandextractingthe Béziercurveinformationrelevanttoacharacter.We chosetohavemetafoginterpretthepostscriptoutputfrom METAPOSTandtoconstructtheMetaFog contour data structures during this interpretation, ratherthantryingtomodify METAPOSTto make outputinamoreconvenientform.thisallowsusto staycurrentwith METAPOST improvements. METAPOST outputs outline curves in Post- Script by first defining the path with newpath, moveto, curveto, lineto and closepath commands, followed by a zero-pen-width strokeand afill. For white ink METAPOSTusessetgray before stroking or filling. For elliptical pens and slantedcoordinateoutput transformations META- POSTuses dtransform sto applyaffinetransformations. MetaFog contains an input interpreter that converts METAPOSToutput to internal data structures. Renderingellipsesstrokingpaths. Oneofthe problems which Knuth sidestepped in METAFONT was computing the envelope of an ellipse stroking along a Bézier curve. Knuth here chose to use Hobby s method to compute the envelope in terms oftherasterinsteadofscalablecurves; thecomputational geometry then reduces to a matter of manipulatingline-segmentsandpolygonsinsteadof polynomial curves(the METAFONTbook, 524). We instead want to compute a Bézier curve outline for stroked-ellipse envelope. Algebra tells us thatstrokinga3rddegreepolynomialcurve(theellipse approximated by Bézier curves)along a 3rd degreepolynomialcurve(the Béziercurveofthe strokedpath)resultsina6thdegreeenvelopecurve. We will have to approximate these 6th degree exact envelopecurveswith3rddegree(bézier)curves. Figure 2 shows how an ellipse contour may be approximatedby acontourmadefrombézier curves. This is similar to the four-curve approximation to a circle cited by Knuth. The Bézier control points for a unit circle are located symmetrically 4 3 ( 2 1) 0.552unitsawayfromtheend points.(this quantity does not appear explicitly in METAFONT, but we can solve for it by substituting the known angles and locations at the ends and midpoints of the curves.) The affine properties of Bézier curves permit us to linearly distort the Béziercontrolpointsoftheunitcircleinproportion totheeccentricityofaunitellipsetofitabézier contour to that ellipse. We can also apply linear transformationsofrotation,scaling,andtranslation totilt,size,andplaceaunitellipseasdesired. Figure 2:Contour of four Bézier curves which approximate an ellipse. Toproceedtotheenvelopeproblem,letusassume that the situation looks like Figure 3, where (without loss of generality) we have rotated and translatedthecoordinatessuchthatthestartofthe strokingpathisattheoriginandhaszeroslope there. Untransformed Ellipse Transformed Ellipse Ellipse at Midpoint Ellipse at Endpoint Figure3:StrokinganEllipticalPenonaBézier Path. Wewillfitanenvelopeconsistingoftwoends and two sides. The sides are parallel to the 236 TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting

5 MetaFog: Converting METAFONT Shapes to Contours strokingpath,andtheendsaresubsetsoftheellipseatthestartandfinishofthestroke. Weuse asetofboundaryconditionsfortheapproximation which will be natural and visually appealing: The slopesofthesidecurvesstartatzeroandendwith the same angle at which the stroke curve ends. Wefixthemidpointsandanglesofthesidecurves basedonthelocationoftheellipseatthemidpoint ofthestroke,usingthetangentpointsoftheellipse matching the angle of the stroke at its midpoint. This approximation is quite good when curves are nottoo sharp ;thatis,theydonotturnthrough morethan90degrees,andarenottoo tight ;that is,theydonothaveahigh2ndspatialderivative. We can always bisect sharp and/or tight curves to improve the accuracy of the approximation as needed;inpracticethecurvesarealmostalwaysso gentle as to be well-fitted without bisection. To compute the envelope curves, we must find their endpoints and their controlpoint locations. Wefirsttranslateandrotatethecoordinatesofthe problem to the normalized coordinate system to fit the model. Using the boundary conditions namely the endpoint locations, endpoint tangent angles,andthemidpointlocations abitofpolynomial algebra and a solution of simultaneous equations yields a closed-form solution to where to puttheendpointsandcontrolpointsoftheenvelope curves. Given these two curves, we can compute the subset of the ellipse curves as a maximization probleminanothertransformedcoordinatesystem. Invertingtherotationandtranslationofcoordinates yields the desired solution. Figure 4 shows some examples of envelopes computed with this method. Careful attention to generality and numericaldomains yields a robust algorithm, which is crucial to the wild data characteristicofgraphicalshapes. METAFONT usually uses circles(of course, a circleisaspecialcaseofanellipse)tostrokepens.the exceptions where METAFONT uses elliptical pens are thecalligraphiccapitalsandafewmathsymbols. Knuth also used circular pens quite liberally in Computer Modern. For example, circular pens draw the rectangularstems,sincethetechniquemakesparameterization of stem widths and rounding of corners somewhateasier,andtheserifprogramstakecare ofsquaringofftheroundcornersforromanfaces. The logical shape primitives OR and NOT. Once MetaFog expands any stroked paths to envelopes, it can proceed to intersect overlapping paths. MetaFog must compute all possible multipleintersectionsofeachpairofcurvesinapath, Figure 4: Envelopes computed for various Bézier strokes of an elliptical pen. insteadofassumingonlyonepossibleintersection as in METAFONT.(Two Bézier curves can have 8 intersections tryto findan exampleusingyour favorite drawing program!). MetaFog computes all intersections using an exhaustive extension to the recursive, numerical solution Knuth used in METAFONT; a closed-form solution employing zeroes to cubic polynomials is also possible but not implemented. Computing all such intersections and reconstituting the shapes with new knots at all intersections in the general case is a difficult problem which consumes most of MetaFog s running time. Weeding. The MetaFog weeder is a visual tool which allows a human operator to examine and hand-correct the output from automatic conversions. Manual input to the conversion process is vital, because METAFONT output often has degenerateshapesandintersectionsthatdefyanautomatic solution. In such cases, MetaFog cannot determine which shapes are overlapping, and so outputs a partial solution to the topological problem; the weeder allows the human designer to choose the proper Bézier shapes from intermediate METAFONT elements. TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting 237

6 RichardJ.Kinch Figure5showstheweedingdisplayforcharacter m of cmti10. The display showseach of the Bézier curves of the input shapes, intersected andbrokenintoseparatepieces. TheuserhasinvokedMetaFogin minimal mode(whichisguaranteed to succeed), which means that all curves used incomputingenvelopesofstrokesareretained;an intermediate mode(which does not always succeed) reduces each envelope to the exterior curves. NotehowMetaFoghasstrokedacircularpenalong Bézier paths and produced curves for the envelope. MetaFoghasalsoinsertednewknotswherecurves intersect; this computationcanbe quitecomplex sinceagivencurvecanhavearbitrarilymanyintersection points, resulting in a repeated bisection of thecurve. Thehumanoperatorusesthemouseto observeand togglebézier segments which make up thecorrectenvelopeofthecharacter;eachsegment changescolorasittoggleson(blue)oroff(red). Toggling proceeds quickly because the mouse click needonlybenear(notnecessarilyon)thedesired curve, clicks are buffered when the operator outpacesthe CPU, andasecondclickwilltoggleoff anyinadvertentlyerroneousselections. operatorvisuallycheckstheglyphforproperconstruction,andfinishesbyexportingthecharacter. Duringexport,theweedertakescareofoptimizing the output curves by removing redundant control points. Keypad functions allow the user to flip quicklythroughalltheglyphsinafont.thisallows careful previewing and weeding of any glyphs that needtouch-upfrommetafog.acheckplotprogram producesaprintoutofalltheglyphsinafontfor checking and documentation. Intheworstcase,MetaFogcanalwaysproduce a fully-intersected set of shapes with elliptical pen envelopes(ifany)alreadyexpanded. Theoperator oftheweederthenhasamoredetailedpointingjob, buttheresultwillbejustasperfectasanautomatic solution. Figure 6 shows the MetaFog weeder view of cmsy10 s calligraphic A. This illustrates how a tilted elliptical pen strokes a Bézier path in slanted coordinates. Figure 6: Weeder display for cmsy10calligraphic A, illustrating envelopes of elliptical pen strokes Figure 5: Weeder display for cmti10 m The weeder s user interface is optimized for speed. The PC s numeric keypad provides convenience functions, so that the operator keeps one handonthemouseandtheotheronthefunction keys. The operator can quickly switch between displaying all curves and displaying selected curves only. Previewing selected curves only gives an accuratecheckthatnosegmentsaremissingandthatno extrasegmentsareselected.zoom-inandzoom-out allow the operator to pick through busy areas wheremanycurveslieveryclosetogether. After toggling all the exterior edges of the glyph, the A few cases of the calligraphic capitals contain tightly turning curves which require handcorrections using the weeder. Hinting.Rendering fonts on low-resolution devices like video displays and laser printers requires heuristic help to yield a pleasing result. Without such help, the bit maps will have unnatural bumps, stems will be of uneven width, and drop-outs will occur. METAFONT handles these matters in the chapter DiscretenessandDiscretion ofthethe META- FONTbookandvia mode_defitems like fillinand o_correction. TheType1fontlanguageallows the designer to add hinting for the same purpose. TrueType calls the same notion instructions. Sincemostoftheindustryeffortinthisregardhas 238 TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting

7 MetaFog: Converting METAFONT Shapes to Contours been expended on Type 1 font hinting, hinting hasbecomethegenerictermforthisaspectoffont design. METAFONT has significant modeling differences fromtheoutlinefonthintingmethods,sothereisno translationpossibletoautomaticallymakeahinted outlinefontfromametafont design. Hintingcanbeappliedafterthetranslation,either automatically by auto-hinting software(yieldingapoortomodestresult)orbyaskilledprogrammer(yielding the best results, given enough expert effort). The big problem with adding hints in this post facto manner, is that the hints become detachedfromtheoriginal METAFONT programs, and any change to the meta-fonts will require repeatingthemanualhintingeffort.thebiggestloss hereisthatthemeta-nessof METAFONT programs doesnotcarryovertopost-factohand-hinting;one METAFONTcharacterprogrammakesvariantslike bold, italic, sans serif, etc., but each variant must be independently hand-hinted. Another important exampleisthatmostofthe METAFONTprograms forthe DCfontsrepeatprogramsfromComputer Modern, but such redundancies would not be usable aftertranslationtooutlineformats.outline-format fontsdosufferfromaninabilitytoexploittheseredundancies, and serious font designers typically have in-housetoolstoovercomethisproblem. Since METAFONThinting does notcarryover to Type 1 or TrueType hinting, the ideal solution would be to enhance the Computer Modern META- FONTprogramstocontainnewhintinginformation suitable for translation to other forms. Type 1 and TrueType hinting employ a limited number of techniques, which depend on the exact coordinatesanddesignofeachparticularcharacter. A programmercould add eachhint and the associated coordinates to each character s METAFONT programintheformofpseudo-comments. Ahinttranslatorprogramwouldconvertthe METAFONT pseudo-comments into Type 1 hint programs or TrueTypeinstructions. Makingtheshapetranslationindependentofthehinttranslationwouldallow adjusting shapes or hints independently, without having to re-run both aspects of the translation. The pseudo-comment language would be designed to represent the various hinting technologies and to exploit any commonality between them. The METAFONTlanguage is well-suited to an extension ofthissort. Forexample, Type1 flex hintingneedsto know the size and position of what is called the dish concavity in the Computer Modern serifs. Additionofthis informationto thetype1fonts improves the rendering of serifs. While this informationispresentinthe METAFONTprograms,itis lostintheprocessoftranslationtooutputshapes. The proposed method of pseudo-comments and hinttranslatorwouldpreserveandtranslatethisinformation. Hints are typically applied to stems, bowls, bulbs,andothercharacterfeatures,and METAFONT isquiteawareofthepertinentcoordinatesofthese items. This is also a database problem. Oneofthe difficulttasksoftranslatingtexfontsisthesurfeit of them. Just between Computer Modern and the DCfonts,spreadacrossvariousopticalsizes,there areseveralhundredfontseachhaving128or256 glyphs. Giventhatthetypicalglyphoutlinecontainsdozensofendpoints, eachhaving3pairsof coordinates,onecanseethatthetranslationenterprise involves millions of coordinates. Organizing thisinformationintoglyphdata,characternames, fonts,charactermetrics,encodings,accentcompositionrules,versioncontrols,kerningpairs,ligature rules, font families, output formats, hinting data, andsoonisasubstantialdatabaseproblem. Since we want to exploit redundancies like common subsets between OT1andT1encodings, we especially needacapabledatabaseapproachtomanagingthis information. MetaFog uses more of a rapid-prototype approach. Shellscriptsmanagethevariousstepsin translating a given font: running METAPOST to get intermediate conversions; running MetaFog itself to convertallorpartofagivenfonttooutlines,assemblingvariousfilesforacprogrammakefontwhich assemblesindividualcharacterdataintocomplete Type 1 fonts, including insertion of extrema points, initialproductionofanafmfile,andatexvirtual font file. Tables keep track of redundancies betweencharactersandfontssothatagiven META- FONTglyph need only be translated once. Tables suchasencodingvectorsaretypicallykeptinascii formandlook-upsareperformedbyshellscripts. GlyphinformationiskeptinPostScriptorpseudo- PostScriptformandrapidlymanipulatedbyCprogramsbuiltfromcommonfunctionlibraries. Tofinishthefonts,weuseseveraloutsideutilities.TheprogramsofHetherington st1utilscollectiontakecareofthedetailsofconversionstoand from the encrypted Type 1 font format, so that MetaFogneedbeconcernedonlywithASCIIType1 output.wealsotestthefontswithallthecommercialfonteditorscurrentlyavailable: Fontographer, Fontmonger, Type Designer, andfontlab; weuse TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting 239

8 RichardJ.Kinch Fontmonger toconvertthetype1fontstotrue- Typeform. Ifweweretorepeattheimplementation,one mightconsiderusingarelationaldatabasetostore theinformation,withqueryscriptsandcprograms doing the detailed work. OpticalOverkill. Fontsastheyareusedinoperating systems today do not favor the optical scaling which TEX is adept at exploiting. For example, TEXuseseightopticalsizesoftheComputerModern Romanfont(5 10,12,and17points). Thisistoo manyopticalsizes dowereallyneedeverystep from5to10points?nodoubtthiswasencouraged by the METAFONT facility at optically scaling with a simpleparameterchange.butwiththevariousembellishments of bold, italic, and so on, a minimally complete Computer Modern font set yields over 100 discretefonts. Users today are not accustomed to seeing so many fonts associated with an application. TEX has adistinctlyarchaicatmosphereinthisregard.operating systems that manage fonts are taxed to handle theplethoraoftinyvariationsintexfonts. Lately this overkill of optical sizes has worsened withthenfss,whichdoesagoodjobofhidingopticalsizesfromtheuser,butencouragesthestyle designer to multiply them. The painis excruciatingwithregardtooutline translation, where essentially identical problems withslightvariationsarerepeatedmanytimes.we wouldurgerestraintontexexpertswhenitcomes to selecting optical sizes. Comparisons of Various Approaches LetuscompareatypicalComputerModernglyphas translatedtooutlineformbyvariousmethods.figures7through12showtheoutputfor R ofcmr10 fromvariousconversions. NotethatFigure7andFigure8showextra, relocated,missing,orartifactcontrolpointswhich have lost the symmetry of the METAFONTcontrol points. The autotracing method used is evident in these examples. Figure9hasretainedmostofthe METAFONT controlpointsbutalsoinsertedartifacts.figures10 and11showthesetoftrue METAFONTpiecesfrom anintermediatestep,wheremetafoghasexpanded the circular pen strokes into their Bézier envelopes. Figure12showshowMetaFogretainsthe META- FONTcontrol points exactly, including all the octantsandallthesymmetry;therearenoextraor artifactualcontrolpoints. IncomparingFigures9 and12,notethatthetipoftheleginthebakoma conversion develops an asymmetry, that the flat top of the tip has narrowed, and that the 45 degree controlpointsaremissingfromthebowlandserif curves(which will underspecify these curves). The MetaFog conversion retains the proper symmetry, flatness,andprecision,whichareallaspectsofthis characterreadilyobservedinthe METAFONT proof in Computer Modern Typefaces. Missing flat Figure 7:BlueSkyResearchautotraced conversion. Missing symmetry Missing flat X-Rays reveal bugs in Computer Modern. MetaFog allows more detailed visualization of character designs than METAFONT proofs. While the proofs show reference points and marked areas, they cannotshowmostofthe relevantgeometric information. Indeed,fewoftheknots,notallthe outlines,andnoneofthestrokedpenenvelopesare accessible in METAFONT. Since MetaFog converts and manipulates all these items, it can also plot theminaconvenientform. Thisyieldsanewand sometimes surprising x-ray view of a character aviewunavailablein METAFONT.MetaFog s output files use a PostScript format so that proof pictures plot on any PostScript output device. The weeder is also a convenient visual tool for such views. Surprising aspects to some of the Computer Moderndesignsshowupinthe x-rays.theparametric nature of meta-designed features becomes visually apparent, and bugs in the design are clear wheretheywerenotbefore. Figure13showshow the serif subroutine has introduced an unexpected 240 TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting

9 MetaFog: Converting METAFONT Shapes to Contours Missing knot Missing dish Asymmetry, missing flat, too busy Asymmetry Asymmetry Too wide Figure 8:PCTEX autotraced conversion. Figure 9:BaKoMaconversion. inflection in cmbx5letter x. Since the loss removes justafewpixelslikelytobefilledinphysicallyby most marking engines and optically by the human eye,theerrorisnotobviousinnormalusageoron METAFONT proofs. It becomes clear, however, during the MetaFog weeding. Figure14showshowjoiningofthe beak to thearmondigit 7 becomesdistortedatsmaller sizes. Thiserroriseasilymissedonproofsbutis visible under magnification. If you magnify and carefully examine the actual-size proofs in the Computer Modern Typefaces(VolumeEofComputers & Typesetting),thiserrorisvisibleasarowortwo ofextrapixelsatthetopofthecharacter. A frank error in dcr10 s thorn was easily discoveredinthisway,althoughithadescapedallthe proofchecksandactualusageforseveralyears(figure15). Thebottomserifshaveanextra step, which on bit-mapped proofs looks like a purposeful fillet.onthemetafogconversionitappearsclearly asanerror. (Thiserrorhasbeencorrectedinthe autographspursuanttothisdiscovery,anddoesnot appearonmorerecentversions.) Is There an Exact Translation? Is an exact translation possible? We used the standardof1pixelona2048pixel/emgrid. Nodoubt the noise of digitization and hinting creates many more varying pixels than this standard of error. Thereisnooutlinethatwillrenderthesamebit mapinatype1ortruetypeengineas METAFONT would render for a METAFONTprogram. The font format itself requires that we must approximate sixth-degree pen strokes with third-degree pieces. METAFONT cannot draw proof picture of the underlying curves, it can only produce a high-resolution bit map. And finally, device-specific mode definitionsin METAFONTresultinsignificantchangesto the proof mode device. Sothereisnosuchthing,inapracticalsense,as an exact translation, because there is no exact shape towhatametafontprogramdescribes! Perhaps weshouldinsteadspeakintermsofan ideal translation. Sample Output AsampleofMetaFogconversion,namelycmr10in Type1format,isavailableat: FTP://ftp.netcom.com/pub/Tr/TrueTeX Thisisanunhintedfontsuitableforviewinginafont editor, but not suited for textual use. MetaFog itself isaproprietaryproduct,andisnotinthepublic domain. Colophon We drafted this paper using the TrueTEXimplementationofL A TEX2εforWindows,whichallowed WYSIWYGpreviewing and printing, including all graphic images. We used three kinds of figures, and processedthemallthroughcoreldraw5:ordinary drawings,metafogimports,andscreensnapshots. TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting 241

10 RichardJ.Kinch Figure 10:MetaFogintermediatestep(overlaid), showing both explicit METAFONT shapes and Bézier envelopes of circular pen strokes. We created figures like the Bézier curve and ellipse examples using Corel Draw s drawing tools; MetaFog-output figures by importing MetaFog s PostScript-like output into Corel Draw; and screen snapshots by capturing with Corel Capture and pasting into Corel Draw via the Windows clipboard.weusedthefiguresincoreldrawtoexport Encapsulated PostScript(EPSF) files and inserted thefilesastex figures using the epsfig package for L A TEX. The TrueTEX dvips-compatible special-handlers allowed both screen previews and printing of the EPSF figures, including printing on anon-postscriptlaserprinter.weusedcoreldraw to print overhead transparencies of the figures. Draftcopiesandtransparencieswereimagedonan HP 4M Plus laser printer. The Proceedings editors usel A TEXanddvips,sothatnoconversionswere necessarybetweentheauthor ssubmissionandthe final production. Figure 11:MetaFogintermediatestep (exploded). The dish atthebottomisa white shape which subtracts from the serifs. J.D.Hobby. A METAFONT-like system with Post- Script output. TUGboat 10(4), ,1989. B.Malyshev. Automaticconversionof METAFONT fonts to Type1 PostScript. TUGboat 15(3), ,1994. S.YanaiandBerry, DanielM. Environmentfor translating METAFONT to PostScript. TUGboat 11(4), ,1990. References Adobe Systems. Adobe Type 1 Font Format, version 1.1.AddisonWesley,1990. L. Carr. Of METAFONT and PostScript. TEXniques 5, TEX UsersGroup,1987. A. Glassner, editor.graphics Gems. Academic Press,Cambridge,MA,1990. D. Henderson. Outline fonts with METAFONT. TUGboat 10(1),36 38, TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting

11 MetaFog: Converting METAFONT Shapes to Contours 45 degree control points Flat Symmetric, narrow flat Symmetric Figure 12: TrueTEXconversionviaMetaFog. Figure 14:ErrorinCMdigit7(cmbx5) 2.5x 2.5x Step Figure 13:ErrorinCMserifs(cmbx5) Figure 15:ErrorinDCthorn(dcr10) TUGboat,Volume16(1995),No.3 Proceedingsofthe1995AnnualMeeting 243

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