Alternatives to Computer Modern Mathematics
Alan Hoenig 17 Bay Avenue Huntington, NY 11743 [email protected]
TEX users early on hustled to do everything with been adjusted so they more closely match their TEX that was available to other publishing software. accompanying text fonts. It has long been possible to use all kinds of graphics Computer Modern math plus new text fonts and fancy fonts with TEX. One hole remains in this scenario, and that is the ability to use fonts other It may be typographically dangerous to willy-nilly than Computer Modern for scientific typesetting. change the text font of a document while retaining This is a real puzzle, when you consider that the Computer Modern math, but it is possible to choose whole reason for TEX in the first place was scientific a font that does blend well with Computer Modern. typesetting. In the accompanying discussion, we Computer Modern fonts were designed using Mono- attempt to fill in this hole by presenting several type Modern No. 8A as a model. The digital font strategies for non-CM mathematical typesetting. most resembling these fonts is Monotype’s Modern Na¨ıvely, you might wonder why we just can’t font, widely available from digital font vendors. replace the Computer Modern text fonts of a doc- You should install the fonts as per the usual ument by some other fonts for a brand-new look. procedures. Then, LATEX users should add the If we combine Computer Modern math with Times command Roman text (or something comparable), the vari- ables look too anemic and thin compared with the text letters, and in an extended document this in- compatible contrast between text and math grates \renewcommand{\rmdefault}{mmo} on the reader. You may find that some differences between math and text are acceptable—after all, math is different from prose—but these variations are somehow too disparate. where mmo is the Berry name for the Modern font This article, drawn from the more extended family. No changes need be made to the math discussion found in chapter 10 of my book TEX font declarations, as we shall continue to use the Unbound (1998, Oxford University Press), contains Computer Modern math fonts. the results of some experiments showing how to use New math raw fonts TEX to generate technical documents using many handsome fonts. We will be creating series of virtual In an ideal world, math fonts would be 100% com- fonts to do the typesetting for us. patible with text fonts. For the math fonts available We will discuss several strategies for mathemat- to TEX, this statement is true only when Computer ical typesetting: Modern fonts are selected, when Times Roman is • replacing Computer Modern Roman by Mono- used with MathTime or the Mathematica fonts, or type Modern; when Lucida Bright is used with Lucida New Math. However, if reasonable compromises are permitted, • the use of commercial and other math fonts a much wider selection of font matches is available. that can then be integrated with text fonts. The x-height is the dominant physical feature Although vendors may supply macro and style of a font, since so much of a document is lowercase. files to perform this integration, we will explore In our math fonts, we take pains to match the x- virtual font approaches. The four special sets height of the math fonts with that of the text types. of raw math fonts include MathTime, Euler, Moreover, it makes more sense to scale the math to Lucida New Math, and Mathematica fonts; and the text, rather than vice versa, since there is almost always more text than math in a paper. Presumably, • using variations of the usual Computer Mod- the 10-pt size for a text font is the optimal size for ern bitmap math fonts whose parameters have that font.
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The MathInst utility need fontinst, version 1.5 or greater. Also, make sure the Perl executable appears on one of your system’s One way to install math fonts is via MathInst, writ- path directories. ten by me. MathInst creates an entire font envi- The raw math fonts consist of a series of outline ronment for typesetting. At the moment, Math- font files plus the associated afm files. It’s easy to Inst knows about four math fonts, the MathTime, install these fonts. Here are the steps appropriate Lucida, Euler math, and Mathematica math fonts. for traditional systems. The same steps apply to MathInst consists of a Perl script, and several ad- TDS systems, but it will be necessary to be more ditional files needed by fontinst. The main purpose specific about the paths for the files. of these Perl scripts is to match a designated text family with a set of math fonts to create new math 1. Place the math font files with the other scalable virtual fonts. fonts. MathInst produces lots of output. First are 2. Place the afm files with your other afm files. the fonts themselves which combine the given math 3. Make sure a proper entry exists for each math fonts with a family of text fonts. In case an author font in the map file for your dvi postprocessor. has provided the names of other fonts, such as Only the last point requires additional comment. • a sans serif family of fonts; For example, for the dvips psfonts.map file for a • a typewriter font; traditional TEX system, we need entries like • a calligraphic font; %% MathTime fonts... • a fraktur font; mtsy MTSY
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file. These entries on a traditional TEX system whereby the font family name for the new math should look something like fonts uses the two-character math designations to- %% Euler fonts... gether with the text font family designation. Sup- euex10 euex10
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In a LATEX document, all the work is done by have been. The extension font is quite sparse, but the package file whose first name is precisely the z- we can add virtual flesh using Computer Modern name for the math fonts. As with all package files, glyphs to fill in the blanks of the font table. its extension is sty. If you add a line like \usepackage{zmtmbv} Lucida New Math following \documentclass{...} then all the usual The Lucida math fonts for A T X were designed by to follow normal T X type- LTEX font-switching mechanisms will now apply to E E the zmtmbv fonts rather than to the default Com- setting conventions and yet be compatible with the puter Modern fonts. extensive Lucida and Lucida Bright font families, To typeset mathematics in a plain document, and are available for purchase from Y&Y. simply place a statement like This Lucida New Math family consists of five fonts. Because each contains the full complement \input zmtmbv of 256 characters, these fonts are crammed with all very near the beginning of your file. Thereafter, all kinds of additional glyphs. These additions include the usual plain font commands like \it, \rm, \tt, all the special symbols that occur in the additional $, $$, and so on, refer to the new math fonts. The symbol fonts commissioned and made available by pdcfsel package itself provides more flexibility than the American Mathematical Society and include a plain users are used to, and it would be well worth Blackboard Bold font. The three fonts useful for any (plain) reader’s time to gain familiarity with this standard TEXnical typesetting are the symbol, math package. oblique, and math extension fonts. A math italic font doesn’t follow the original math italic font quite The MathTime fonts Michael Spivak developed as closely as the oblique font. Finally, a math arrow the MathTime math fonts to be used with the Times font contains many, many new symbols plus the family of text fonts in a T X document;Y&Y is the E Blackboard Bold alphabet. vendor. Many authors will find these fonts the most Lucida math extension differs from other ex- useful for math typesetting. Their “Times Roman- tension fonts in that it contains many new glyphs y” look goes well with many other Roman fonts. in its upper half. With this font properly in place, The package consists of three fonts—an exten- you can use a new extensible set of open brackets, sion font, a math italic font, and a symbol font. additional wide accents, newly sized integral sym- The extension font mtex is directly analogous to bols, and a fully extensible integral. The font also cmex10 (the same characters are in the same posi- contains the uppercase Greek alphabet, which we al- tions), but the remaining fonts have slightly different ready use to construct the math Roman font. Math- layouts from their Computer Modern counterparts. Inst style files contain commands (where necessary) These differences are largely due to the elimination to use these new features. of the oldstyle digits and the calligraphic alphabet The wide accent symbols are automatically from the italic and symbol fonts. The slots opened in place; simply continue to use \widetilde and up by these omissions have been filled with upper- \widehat as before. case Greek letters and redesigned operator symbols. There are new kinds of square brackets that (The documentation that accompanies the fonts dis- grow to enclose filler material. These brackets, are cusses the differences in greater detail.) amenable to the usual “growth” mechanisms that Users can also create MathTime math fonts in govern \left, \right, \bigl, and so on. a second way—by following the instructions that ac- The several new integral signs include new sur- company these fonts. This approach is not so heavily face integrals, a new size for the regular and con- dependent on virtual fonts as is the MathInst way tour integrals, and pieces for a generally extensible and relies on a well-written macro file accompanying integral. The new command \surfint and the ex- this package. isting integral commands \int and \oint work as The Euler fonts Euler fonts consist of math liter- expected. In addition, there are large variants, sum- als (neither Roman nor italic, but a unique upright moned into play by \lint, \loint, and \lsurfint. font which is a compromise between the two forms), These control sequences ensure that the various in- symbols (with a compatible uppercase calligraphic tegral signs change their size depending on a text or alphabet), Fraktur, and extension fonts. Because display context. they predate virtual fonts, and because the font ta- You might like to have TEX select the right size bles themselves follow slightly quirky layouts, they integral for you. For that reason, there are three have not been as useful heretofore as they might variant integral commands, \varint, \varoint, and
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\varsint (for regular, contour, and surface inte- Mathematica math fonts The Mathematica math grals) that try to do that for you. Each of these fonts were in development as this book was written, takes as argument the contents of the integrand. but Wolfram Research graciously made their interim Figure 1 shows that this mechanism works poorly fonts available to me. The fonts in their eventual re- for the surface and contour integrals when the total lease may have different names, different characters, height of the integrand is taller than the largest of and a different ordering. the available integrals. These fonts consist of five font series compris- MathInst automatically places the TEX-hackery ing all of the characters that TEX normally expects, necessary in the Lucida style files it writes. You a calligraphic and a Blackboard Bold alphabet, and could type many more additional characters. Each series con- sists of four variants, normal, bold, typewriter, and $$ typewriter bold. As far as TEX is concerned, the \overbrace{\vphantom{\lint} characters in these fonts are scrambled in a funny \hbox{$\int\lint\oint\loint\surfint order, so we first create raw fonts, each of which ap- \lsurfint$\ }}% pears more meaningfully ordered to T X. You can ^{\hbox{text}} E \overbrace{\int\lint\oint\loint\surfint do this with the script makemma.tex, part of Math- \lsurfint}% Inst. Running this script, and then creating virtual ^{\hbox{display}} fonts in the usual way, creates three fonts mmami, $$ mmasy, and mmaex (math italic, symbol, and exten- $$\varint_{-\infty}^{+\infty}{\setlimits \left\Lbrack sion), which are themselves suitable components for \vcenter{\halign{\strut\hfil${#}$\hfil\cr virtual font shenanigans. Although these three are \widehat 1\,\widehat{23}\,\widehat{456}\, in fact virtual fonts, we will treat them as raw fonts \widehat{7890}\,\widehat{12345} in the creation of additional virtual fonts. \cr \widetilde{67890}\,\widetilde{1234}\, Fine tuning the new math fonts \widetilde{567}\, \widetilde{89}\,\widetilde{0} Adding special-purpose fonts Authors may want \cr to add special fonts to their math style. Here’s what \varoint{\short}\,\varoint{\med}\, MathInst allows you to add: \varoint{\tall}\, \varoint{\Tall}\,\varoint{\Talll}\, • a sans serif font family, \varoint{\VTall} • a typewriter font, \cr • a blackboard bold font, \varsint{\VTall}\,\varsint{\Talll}\, \varsint{\Tall}\, • a Fraktur font, \varsint{\tall}\,\varsint{\med}\, • a calligraphic font, and \varsint_{\scriptscriptstyle\partial C}% • a bold Greek font (suitable for setting bold {\setlimits\omega} \cr math). \left\Lbrack x\right\Rbrack\ You may add any, all, or none of these. If any \left\Lbrack\med\right\Rbrack of these fonts are present on your system, Math- \left\Lbrack \tall \right\Rbrack\ Inst adds high-level font-switching commands to the \left\Lbrack \Tall \right\Rbrack\ \left\Lbrack \Talll\right\Rbrack\ style and macro files it creates that recognize the \left\Lbrack \ontop{42pt}\right\Rbrack presence of these fonts. \cr Where do these fonts come from? Many of them \varint_0^9{\Talll}\ are proprietary, but a large number of them reside in \varint_{-1}^{+1}{\Tall}\ the public domain, albeit in unlikely or unsuspected \varint{\tall}\ \varint^{+\infty}_{-\infty}{\med}\ places. \varint{x}\cr As far as typewriter type is concerned, I strongly }}\right\Rbrack\,dx recommend the freely available Computer Modern }$$ typewriter font cmtt10, which blends well with al- most every other digital face. There are alterna- to get figure 1, which combines the Lucida math tives. The Pandora typewriter font pntt10 is also and Lucida Bright Roman fonts. Here, \short, free from ctan, and of course the printer-resident \med, \tall and so on are simply temporary control Courier font is widely available. There are several sequences to generate arguments of various relative other variants of cmtt10 in the TEX suite that some heights. users may prefer, and proprietary typewriter fonts
TEXNorthEast Conference, March 22–24, 1998 TUGboat, Volume 19 (1998), No. 2 181
text display
z }| {z Z I}| { +∞ R H 1 23 456 7890 12345 ⌠ 67890 1234 567 89 0 b c d Æ Æx. x x. . Èx Èx .g .f e. x . . . . x . . . I . x x I . . I I x . H x x. . x. x . . . x x . . . . ω . . . x x dx . . x ∂C . x x x. x x. . x x. . . . x . . . . x x . . . x x . . x 9 x +1 . x x +∞ . . . x . . . x x ⌠ . . x −∞ x ⌠ −1 x Z 0 R −∞ ⌡ ⌡ ⌡ Figure 1: New Lucida math extension characters in action.
include offerings in the Lucida Bright families and of ctan, the latter being in its ams subdirectory. ITC American Typewriter. Other authors may use Commercial sources include the Lucida New Math a monowidth sans serif font (such as Letter Gothic) family (the “arrows” font contains the Blackboard or some other contrasting face entirely. Bold glyphs) and Adobe’s Math Pi fonts (the sixth A wide choice exists for sans serif families. of these contains the Blackboard Bold). Choices Common choices will be Computer Modern sans on ctan for fraktur include Euler fraktur eufm in serif, and the Helvetica fonts resident in all Post- fonts/ams (a scalable version is part of the BaKoMa Script printers. I am personally partial to Gill collection, also on ctan) and the yfrak fonts in Sans (from Monotype) and the Lucida Sans fonts fonts/gothic/yfrak. Commercial choices include (Bigelow & Holmes), but both of these are commer- the Math Pi package from Adobe (check out the sec- cial fonts. ond font in the series for Fraktur) and a font called A calligraphic uppercase alphabet is necessary Fraktur from Bitstream. to make the \mathcal or \cal commands work Mathematicians often want formulas in bold properly. MathInst can add this alphabet (in a type. MathInst will create bold math fonts for you, virtual way) to the math symbol font. Among the but the sticking point might be bold variants of widely available candidates are alphabets from the the uppercase Greek letters. Computer Modern, Computer Modern symbol and Euler symbol fonts, Euler, and Mathematica fonts contain bold Greek and the printer-resident Zapf Chancery font. Several alphabets, but neither Lucida nor MathTime do. bitmap script fonts in the fonts area of ctan (such If a bold version of the Greek letters is available, as Calligra, the RSFS fonts, script fonts, and twcal) MathInst would like to know about it. There seems may be appropriate. Many commercial fonts are to be nothing available that exactly matches the suitable, but authors should refrain from choosing Lucida Greek types, but bold Greek Times fonts can too fancy a script. be purchased. There is less choice for a Blackboard Bold To make these fonts visible to MathInst, you’ll and Fraktur font. In ctan, we find the bbold need to enter the names of the font files to the right fonts (by Alan Jeffrey) and the msbm fonts devel- oped and provided by the American Mathemati- cal Society; both of these are in the fonts area
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of the equal signs in the statements making assign- Times-Roman are so common, I prepared templates ments to $tt_, $sansserif_, and so on. Don’t for- for these fonts. For fun, I prepared a template for get to remove any comment characters from the be- Monotype Baskerville. Times comes in regular and ginning of the line! Thus, to use pmp6 as the source bold series, and Baskerville in regular and semibold; for Blackboard Bold, we need the line Palatino is regular only. MathKit itself produces a number of scripts and $bbold_ = "pmp6"; batch files. Once these are properly executed, you in the parameter par file. Note that font names in get the following: these statements need double quotes fore and aft. 1. Virtual fonts for math and text typesetting. Note too that these changes need to be part of all You will also get fonts for bold math if you the par files (or at least all the ones you’ll be using). have supplied a template containing bold pa- MathInst produces test files testmath.tex for rameters. LAT X and testmatp.tex for plain. These files A E 2. Style files for plain TEX and LTEX (NFSS) . show how to implement the fonts you’ve just cre- These files support bold math if bold parameter ated and exercises these fonts in some reasonably templates were present. complete manner. The files themselves are closely The main MathKit script requires three param- modeled after a similar test file originally designed eters. These are: by Alan Jeffrey. (The original of this file appears on ctan in the fonts/utilities/fontinst area.) It 1. The name of the parameter template. ‘tm’ is a good idea to compile these tests and print one refers to Times-like parameters, ‘pl’ to Palatino- out each time you create a new math font family. like, and ‘bv’ to Baskerville-like. 2. The name under which text fonts are installed. New math fonts via Metafont This is apt to be something like ptm or mnt Think of the reasons that Computer Modern math for Adobe Times or Monotype Times New Ro- fonts clash with other text fonts—they are somehow man, ppl for Palatino, and mbv for Monotype quite too skinny, the wrong height and depth, and their Baskerville (which is different from ITC shapes may not harmonize well with text fonts. Be- New Baskerville). ing that they are meta-fonts, can we not alter the 3. The encoding your fonts follow. Only OT1 or parameters to generate math fonts that more closely ot1 (original TEX encoding) are allowed. approximate text fonts we may be using? This strat- For example, I type egy lies behind the MathKit scripts I have devel- perl ../mathkit tm ptm OT1 oped. MathKit aids in the creation of math fonts in my work directory to create Times-like fonts that may be compatible with a text font family. It following the original TEX encoding. (If your system consists of a Perl script and some auxiliary files to supports the #! syntax for specifying the name of help an author—even one ignorant of virtual fonts or an interpreter, then put the proper path at the METAFONT of —to perform these tasks. This mate- very top of mathkit, make sure the execute bit is rial can be found in the fonts/utilities/mathkit set, and type the simpler injunction ../mathkit tm ctan area of . ptm OT1 from the work directory.) METAFONT MathKit takes parameters that are I’ve had success matching bv (Baskerville-like) appropriate to an outline font family and uses these parameters to other Roman fonts. For example, I to create math fonts. The symbols and other spe- typed cial characters look pretty good—and are compati- ../mathkit bv mjn OT1 ble with your outline fonts—but the italics and nu- to generate a nice-looking set of fonts combining merals look ghastly. Using TEX’s virtual font mech- anism, we create math fonts that combine the new Monotype Janson text with Baskerville-like math special symbols with letters and numerals from the fonts. outline fonts. MathKit does some of this work for The following steps complete the font creation. you, and provides scripts for the remaining steps Perform them all within the MathKit work direc- (described in the accompanying documentation). It tory. also provides style files for plain TEX and for the 1. Use the mkdirs script to create any missing NFSS of LATEX for you to use these fonts in your directories. documents. 2. Execute the file makegf.bat to have META- The current version of MathKit comes with FONT create the pixel fonts for your fonts. This three sets of font templates. Since Palatino and step will take some time.
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3. You’ll need to pack all the pixel files. The file command, this command should be placed within called makepk.bat that may be helpful in this grouping symbols. regard. Caution: before executing this script, In LATEX documents, you simply need to include it may be necessary to edit it. the style name as part of the list of packages that 4. Execute the script makepl.bat to create some you use in the document. Thus, a typical document property list files needed by the next step. would have a statement like 5. Run the file makevp.tex through TEX. That \usepackage{ztmptm,epsf,pstricks,...} is, execute the command tex makevp or some- at the outset. thing appropriate for your system. This step If MathKit has created bold math fonts, a will take some time. Along with lots of super- boldface environment will typeset everything in fluous files, this creates many “virtual property that environment as bold, including all mathemat- list” files with extension vpl. ics. 6. Create the actual virtual files by running every If your outline fonts have been installed using vpl file through the program vptovf; execute expert fonts, you may need to alter the \rmdefault the file makevf.bat which MathKit creates for command. It might be necessary, say, to type you. \rmdefault{ptmx} 7. Execute the file putfonts.bat to place the font instead of \rmdefault{ptm}. and other files where they belong. Preparing parameter files This sequence is summarized for you again on the It was surprisingly easy to prepare these parameter files. I prepared computer screen when you execute MathKit. a test document in which individual characters are Using the new fonts MathKit produces two style printed on a baseline at a size of 750 pt. It’s (rela- files, one for LATEX and one for plain. Their file tively) easy to measure the dimensions of such large names are formed according the naming scheme characters, and METAFONT can be asked to divide by 75 to compute the proper dimension for 10-pt zhmock-familyihfont-familyi fonts. It was particularly easy for me to make these Here, hmock-familyi is the two-character designation measurements, as I use Tom Rokicki’s superior im- for one of the font parameter templates (such as plementation of TEX for NextStep. This package tm, pl, or bv); the word “mock” refers to the fact contains on-screen calipers, which take all the work that these fonts imitate but don’t equal the actual out of this chore. fonts in this family. hfont-familyi is the Berry family If you plan to create your own parameter files designation. Thus, if I create a Times-like set of for other font families, please use the supplied files fonts for use with font family ptm, I would find as models (those files with extensions mkr, mks, or A files ztmptm.sty (LTEX) and ztmptm.tex (plain). mkb). Make sure all measurements are given in terms In the same way, the style files for mock-Palatino of “pt#”; MathKit looks for this string. And please and mock-Baskerville fonts are named zplppl and consider placing this information in ctan. zbvmbv (with the appropriate extensions). At the top of a plain file, include the statement Rogues’ gallery \input ztmmnt The following displays show the results of mixing (or whatever the style file name is). Then, standard and matching various math families to many text font nicknames like \bf and \it and math toggles fonts. vfinst installed all the text fonts, and Math- like $ and \( will thenceforward refer to these new Inst or MathKit generated all the math+text fonts. fonts. These displays should be regarded as experi- If bold fonts have been generated, a com- ments only. I showed these pages to several people, mand \boldface typesets everything in its way and all concluded that some of the experiments are in boldface—prose, mathematics, whatever. Bold successful and others are failures. However, no one math may be appropriate for bold captions, sections agreed which were the successes and which were the heads, and the like. Like any other font changing failures!
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Computer Modern math + Monotype Modern
Unbound Orbits: Deflection of Light by the Sun
Consider a particle or photon approaching the sun from very great dis- tances. At infinity the metric is Minkowskian, that is, A(∞)=B(∞)=1, and we expect motion on a straight line at constant velocity V
b ≃ r sin(ϕ − ϕ∞) ≃ r(ϕ − ϕ∞) d dr ∞ −V ≃ dt (r cos(ϕ − ϕ )) ≃ dt
where b is the “impact parameter” and ϕ∞ is the incident directions. We see that they do satisfy the equations of motion at infinity, where A = B = 1, and that the constants of motion are
J = bV 2 (1) E =1− V 2. (2)
(Of course a photon has V =1, and as we have already seen, this gives E =0.) It is often more convenient to express J in terms of the distance r0 of closest approach to the sun, rather than the impact parameter b.At r0, dr/dϕ vanishes, so our earlier equations give
1/2 1 2 J = r0 − 1+V B(r0) The orbit is then described by
∞ 1 A 2 (r) dr ϕ(r)=ϕ∞ + 1 . Zr −1 2 2 1 1 1 1 r 2 − 2 − 2 − 2 r0 B(r) 1+V B(r0 ) 1+V r h ih i The total change in ϕ as r decreases from infinity to its minimum value r0 and then increases again to infinity is just twice its change from ∞ to r0, ′ that is, 2|ϕ(r0) − ϕ∞|. If the trajectory were a straight line, this would equal just π; ∆ϕ =2|ϕ(r0) − ϕ∞|−π. If this is positive, then the angle ϕ changes by more than 180◦,thatis,the trajectory is bent toward the sun; if ∆ϕ is negative then the trajectory is bent away from the sun.
Reprinted by permission of John Wiley & Sons, Inc. from Weinberg, Gravitation and Cosmology c 1972, John Wiley & Sons, Inc.
TEXNorthEast Conference, March 22–24, 1998 TUGboat, Volume 19 (1998), No. 2 185
MathTime math + Times New Roman (Monotype)
Unbound Orbits: Deflection of Light by the Sun
Consider a particle or photon approaching the sun from very great distances. At infinity the metric is Minkowskian, that is, A(∞) = B(∞) = 1,andwe expect motion on a straight line at constant velocity V
b ≃ r sin(ϕ − ϕ∞) ≃ r(ϕ − ϕ∞) d dr −V ≃ dt (r cos(ϕ − ϕ∞)) ≃ dt
where b is the “impact parameter” and ϕ∞ is the incident directions. We see that they do satisfy the equations of motion at infinity, where A = B = 1, and that the constants of motion are
J = bV 2 (1) E = 1 − V 2. (2)
(Of course a photon has V = 1, and as we have already seen, this gives E = 0.) It is often more convenient to express J in terms of the distance r0 of closest approach to the sun, rather than the impact parameter b.Atr0, dr/dϕ vanishes, so our earlier equations give
1/2 1 2 J = r0 − 1 + V B(r0 ) The orbit is then described by
∞ 1 A 2 (r) dr ϕ(r) = ϕ + . ∞ 1 Zr −1 2 r2 1 1 1 1 r2 B(r)−1+V 2 B(r )−1+V 2 − r2 0 h ih 0 i The total change in ϕ as r decreases from infinity to its minimum value r0 and then increases again to infinity is just twice its change from ∞ to r0,thatis, r ′ 2|ϕ( 0 ) − ϕ∞|. If the trajectory were a straight line, this would equal just π;
1ϕ = 2|ϕ(r0 ) − ϕ∞|−π.
If this is positive, then the angle ϕ changes by more than 180◦, that is, the trajectory is bent toward the sun; if 1ϕ is negative then the trajectory is bent away from the sun.
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Euler + Palatino (Adobe)
Unbound Orbits: Deflection of Light by the Sun
Consider a particle or photon approaching the sun from very great dis- tances. At infinity the metric is Minkowskian, that is, A(∞) = B(∞) = 1, and we expect motion on a straight line at constant velocity V
b ≃ r sin(ϕ − ϕ∞ ) ≃ r(ϕ − ϕ∞) ≃ d ≃ dr −V dt (r cos(ϕ − ϕ∞)) dt
where b is the “impact parameter” and ϕ∞ is the incident directions. We see that they do satisfy the equations of motion at infinity, where A = B = 1, and that the constants of motion are
J = bV2 (1) E = 1 − V2. (2)
(Of course a photon has V = 1, and as we have already seen, this gives E = 0.) It is often more convenient to express J in terms of the distance r0 of closest approach to the sun, rather than the impact parameter b.At r0 , dr/dϕ vanishes, so our earlier equations give
1/2 1 2 J = r0 − 1 + V B(r0 )
The orbit is then described by
∞ 1 2 = + A (r) dr ϕ(r) ϕ∞ 1 . Zr −1 2 2 1 1 1 1 r 2 − 2 − 2 − 2 r0 B (r) 1+V B (r0 ) 1+V r h ih i The total change in ϕ as r decreases from infinity to its minimum value r0 and then increases again to infinity is just twice its change from ∞ to ′ r0 ,thatis,2|ϕ(r0 ) − ϕ∞|. If the trajectory were a straight line, this would equal just π; ∆ϕ = 2|ϕ(r0 ) − ϕ∞|−π. ◦ If this is positive, then the angle ϕ changes by more than 180 ,thatis,the trajectory is bent toward the sun; if ∆ϕ is negative then the trajectory is bent away from the sun.
TEXNorthEast Conference, March 22–24, 1998 TUGboat, Volume 19 (1998), No. 2 187
Lucida New Math + Lucida Sans (Bigelow & Holmes; 8/10)
Unbound Orbits: Deflection of Light by the Sun
Consider a particle or photon approaching the sun from very great distances. At infinity the metric is Minkowskian, that is, A(∞) = B(∞) = 1, and we expect motion on a straight line at constant velocity V
b ≃ r sin(ϕ − ϕ∞ ) ≃ r(ϕ − ϕ∞ ) d dr −V ≃ dt (r cos(ϕ − ϕ∞ )) ≃ dt
where b is the “impact parameter” and ϕ∞ is the incident directions. We see that they do satisfy the equations of motion at infinity, where A = B = 1, and that the constants of motion are
J = bV 2 (1) E = 1 − V 2. (2)
(Of course a photon has V = 1, and as we have already seen, this gives E = 0.) It is often more convenient to express J in terms of the distance r0 of closest approach to the sun, rather than the impact parameter b.Atr0, dr/dϕ vanishes, so our earlier equations give 1/2 1 2 J = r0 − 1 + V B(r0) The orbit is then described by
∞ 1 A 2 (r) dr ϕ(r) = ϕ∞ + 1 . r Z −1 2 r2 1 1 1 − 1 r2 B(r)−1+V 2 B(r )−1+V 2 r2 0 0 ! h i The total change in ϕ as r decreases from infinity to its minimum value r0 and then increases again to infinity is just twice its change from ∞ to r0,thatis,2|ϕ(r0 ) − ′ ϕ∞|. If the trajectory were a straight line, this would equal just π;
ϕ = 2|ϕ(r0) − ϕ∞|−π.
If this is positive, then the angle∆ ϕ changes by more than 180◦,thatis,thetrajec- tory is bent toward the sun; if ϕ is negative then the trajectory is bent away from the sun. ∆
TEXNorthEast Conference, March 22–24, 1998