A note on Nicolas-Auguste Tissot: At the origin of quasiconformal mappings (to appear in Vol. VII of the Handbook of Teichm{ü}ller Theory)
AA NOTE ON NICOLAS-AUGUSTE TISSOT: AT THE ORIGIN OFQUASICONFORMAL MAPPINGS
ATHANASE PAPADOPOULOS
Abstract.
Nicolas-Auguste Tissot (1824–1897) was a French mathematicianand cartographer. He introduced a tool which became known among geogra-phers under the name
Tissot indicatrix , and which was widely used during thefirst half of the twentieth century in cartography. This is a graphical represen-tation of a field of ellipses, indicating at each point of a geographical map thedistorsion of this map, both in direction and in magnitude. Each ellipse rep-resented at a given point is the image of an infinitesimal circle in the domainof the map (generally speaking, a sphere representing the surface of the earth)by the projection that realizes the geographical map.Tissot studied extensively, from a mathematical viewpoint, the distortionof mappings from the sphere onto the Euclidean plane, and he also developeda theory for the distorsion of mappings between general surfaces. His ideas areclose to those that are at the origin of the work on quasiconformal mappingsthat was developed several decades after him by Gr¨otzsch, Lavrentieff, Ahlforsand Teichm¨uller.Gr¨otzsch, in his papers, mentions the work of Tissot, and in some of thedrawings he made for his articles, the Tissot indicatrix is represented. Te-ichm¨uller mentions the name Tissot in a historical section in one of his fun-damental papers in which he points out that quasiconformal mappings wereinitially used by geographers.The name Tissot is missing from all the known historical reports on quasi-conformal mappings. In the present article, we report on this work of Tissot,showing that the theory of quasiconformal mappings has a practical origin.The final version of this article will appear in Vol. VII of the Handbook ofTeichm¨uller Theory (European Mathematical Society Publishing House, 2020).AMS Mathematics Subject Classification: 01A55, 30C20, 53A05, 53A30, 91D20.Keywords: Quasiconformal mapping, geographical map, sphere projection,Tissot indicatrix.
Contents
1. Introduction 12. Biographical note on Tissot 23. On the work of Tissot on geographical maps 5References 71.
Introduction
Darboux, starts his 1908 ICM talk whose title is
Les origines, les m´ethodeset les probl`emes de la g´eom´etrie infinit´esimale (The origins, methods andproblems of infinitesimal geometry) with the words: “Like many otherbranches of human knowledge, infinitesimal geometry was born in the studyof practical problems,” and he goes on explaining how problems that arise
Date : January 13, 2020. a r X i v : . [ m a t h . HO ] J a n ATHANASE PAPADOPOULOS in the drawing of geographical maps, that is, the representation of regionsof the surface of the Earth on a Euclidean piece of paper, led to the mostimportant developments in geometry made by Lagrange, Euler, Gauss andothers.The theory of quasiconformal mappings has its origin in the problemsof drawing geographical maps. Teichm¨uller, in the last part of his paper
Extremale quasikonforme Abbildungen und quadratische Differentiale (Ex-tremal quasiconformal mappings and quadratic differentials), published in1939 [35], which is the main paper in which he develops the theory thatbecame known as
Teichm¨uller theory , makes some comments on this origin,mentioning the work of the French mathematician and geographer Nicolas-Auguste Tissot (1824–1897). Gr¨otzsch, in his paper ¨Uber die Verzerrungbei nichtkonformen schlichten Abbildungen mehrfach zusammenh¨angenderschlichter Bereiche (On the distortion of non-conformal schlicht mappingsof multiply-connected schlicht regions), published in 1930 [19], mentionsseveral times the name Tissot, referring to the
Tissot indicatrix which herepresents in the pictures he drew for his article. The directions of the majorand minor and minor axes of this ellipse constitute are important elementin some of his results. A geographical map is the image of a mapping—henceforth called a projection—from the surface of the Earth, consideredas a sphere or spheroid, onto the Euclidean plane. The Tissot indicatrix isa device introduced by Tissot, who called it the indicating ellipse (ellipseindicatrice, which was used by geographers until the middle of the twentiethcentury. It is a field of ellipses drawn on the geographical map, each ellipserepresenting the image by the projection—assumed to be differentiable—ofan infinitesimal circle at the corresponding point on the sphere (or spher-oid) representing the surface of the Earth. Examples of Tissot indicatricesare given in Figure 1.In Figure 2, we have reproduced drawings from a paper of Gr¨otzsch inwhich he represents the Tissot indicatrix of the maps he uses.Although the work of Tissot is closely related to the theory of quasicon-formal mappings, his name is never mentioned in the historical surveys ofthis subject, and the references by Gr¨otzsch and by Teichm¨uller to his workremained unnoticed. In this note, I will give a few indications on this work.Before surveying the work of Tissot in §
3, I will give, in §
2, a shortbiographical note on him.2.
Biographical note on Tissot
Nicolas-Auguste Tissot was born in 1824, in Nancy, which was to become,26 years later, the birthplace of Henri Poincar´e (whom we shall mentionsoon). Tissot entered the ´Ecole Polytechnique in 1841. He started by oc-cupying a career in the Army and defended a doctoral thesis on November The expression “infinitesimal circle” means here, as is usual in the theory of quasiconformalmappings, a circle on the tangent space at a point. In practice, it is a circle on the surface whichhas a “tiny radius.” In the art of geographical map drawing, these circles, on the domain surfaces,are all supposed to have the same small size, so that the collection of relative sizes of the imageellipses becomes also a meaningful quantity. The reader should note that the ´Ecole Polytechnique was, and is still, is a military school.
NOTE ON NICOLAS-AUGUSTE TISSOT 3
Figure 1.
Four geographical maps on which the field of ellipses (Tissotindicatrix) are drawn. The maps are extracted from the book
Album ofmap projections [33]. These are called, from left to right, top to bottom,the stereographic [33, p. 180],
Lagrange [33, p. 180], central cylindrical [33, p. 30] and equidistant conical projections [33, p. 92]. The firsttwo projections are conformal and not area-preserving. The last two areneither conformal nor area-preserving.
Figure 2.
Two figures from Gr¨otzsch’s paper ¨Uber die Verzerrung beinichtkonformen schlichten Abbildungen mehrfach zusammenh¨angenderschlichter Bereiche [19]. Gr¨otzsch drew the Tissot indicatrices of hisquasiconformal mappings. (In each drawing, the major and minor axesof the ellipses are shown.)
ATHANASE PAPADOPOULOS
17, 1851; cf. [36]. On the cover page of his thesis, he is described as “Ex-Capitaine du G´enie.” Tissot became later a professor at the famous Lyc´eeSaint-Louis in Paris, and at the same time examiner at the ´Ecole Poly-technique, in particular for the entrance exam. He eventually became anassistant professor ( r´ep´etiteur ) in geodesy at the ´Ecole Polytechnique.After having published, in the period 1856–1858, several papers and
Comptes Rendus notes on cartography, in which he analyzed the distor-tion of some known geographical maps (see [38], [39], [40]), Tissot starteddeveloping his own theory, on which he published three notes, in the years1859–1860, [41, 42, 43], and then a series of others in the years 1865–1880[45, 46, 47, 48, 49, 50, 51]. He then collected his results in the memoir [52],published in 1881, in which he gives detailed proofs. In a note on p. 2 ofthis memoir, Tissot declares that after he published his first
Comptes Ren-dus notes on the subject, the statements that he gave there without proofwere reproduced by A. Germain in his
Trait´e des projections des cartesg´eographiques [18] and by U. Dini in his memoir
Sopra alcuni punti dellateoria delle superfici [12]. He notes that Germain and Dini gave their ownproofs of these statements, which are nevertheless more complicated thanthose he had in mind and which he gives in the memoir [52]. He also writesthat Dini showed that the whole theory of curvature of surfaces may bededuced from the general theory that he had developed himself. In fact,Dini applied this theory to the representation of a surface on a sphere, usingGauss’s methods. Tissot also says that his ideas were used in astronomy, byHerv´e Faye, in his
Cours d’astronomie de l’ ´Ecole Polytechnique [16]. Thetexts of the two
Comptes rendus notes [43] and [40] of Tissot are reproducedin the Germain’s treatise [18].Besides his work on geographical maps, Tissot wrote several papers on ele-mentary geometry. We mention incidentally that several preeminent Frenchmathematicians of the nineteenth and the beginning of the twentieth cen-tury published papers on this topics. We mention Serret, Catalan, Laguerre,Darboux, Hadamard and Lebesgue; see e.g. [44, 37, 53].On the title page of Tissot’s memoir [52] (1881), the expression
Exam-inateur `a l’ ´Ecole Polytechnique follows his name, as he was in charge ofthe entrance examination. In his ´Eloge historique de Henri Poincar´e [8],Darboux relates the following episode about Tissot, examining Poincar´e: Before asking his questions to Poincar´e, Mr. Tissot suspended the examduring 45 minutes: we thought it was the time he needed to preparea sophisticated question. Mr. Tissot came back with a question of theSecond Book of Geometry. Poincar´e drew a formless circle, he marked thelines and the points indicated by the examiner, then, after wandering longenough in front of the blackboard, with his eyes fixed on the ground, heconcluded loudly: “It all comes down to proving the equality AB = CD .This is a consequence of the theory of mutual polars, applied to the twolines.” Mr. Tissot interrupted him: “Very good, Sir, but I want a moreelementary solution.” Poincar´e started wandering again, this time not Poincar´e entered the ´Ecole Polytechnique in 1873. In the French system of oral examinations,which is still in use, a student is given a question or a set of questions which he is asked toprepare while another student (who had already been given some time to prepare his questions)is explaining his solutions at the blackboard, in the same room. Thus, it is not unusual that atsuch an examination, some students listen to the examinations of others.
NOTE ON NICOLAS-AUGUSTE TISSOT 5 in front of the blackboard, but in front of the table of the examiner,facing him, almost unconscious of his acts; then suddenly he developed atrigonometric solution. Mr. Tissot objected: “I would like you to stay inElementary Geometry.” Almost immediately after that, the examiner ofElementary Geometry was given satisfaction. He warmly congratulatedthe examinee and announced that he deserves the highest grade. Poincar´e kept a positive momory of Tissot’s examinations. He expressesthis in a letter to his mother sent on May 6, 1874, opposing them to the10-minute examinations (known as “colles”) that he had to take regularly atthe ´Ecole Polytechnique and which he said are pitiful. He writes: “WhenI think about the exams of Tissot and others, I can not help but take pityof these 10 minutes little colles where one puts in danger his future with anexpression which is more or less exact or a sentence which is more or lesswell crafted, and where a person is judged upon infinitesimal differences.” On the work of Tissot on geographical maps
Tissot studied at the ´Ecole Polytechnique, an engineering school wherethe students had a high level of mathematical training and at a periodwhere the applications of the techniques of differential geometry to all thedomains of science were an integral part of the curriculum. His work is partof a well-established tradition where mathematical tools are applied to thecraft of map drawing. This tradition passes through the works of preeminentmathematicians such as Ptolemy [31, 3], Lambert [22, 23], Euler [13, 14, 15]Lagrange [20, 21], Gauss [17], Chebyshev [5, 6], Beltrami [2], Liouville (seethe appendices to [26]), Bonnet [4] Darboux [9, 10, 11], and there are others.It was known since antiquity that there exist conformal (that is, angle-preserving) projections from the sphere to the Euclidean plane. But it wasnoticed that these projections distort other quantities (length, area, etc.),and the question was to find projections that realize a compromise betweenthese various distorsions. For instance, one question was to find the closest-to-conformal projection among the maps that are area-preserving. Hence,the idea of “closest-to-conformal” projection came naturally. Among themathematicians who worked on such problems, Tissot came closest to thenotion of quasiconformality. Avant d’interroger Poincar´e, M. Tissot suspendit l’examen pendant trois quarts d’heure : letemps de pr´eparer une question raffin´ee, pensions-nous. M. Tissot revint avec une question dudeuxi`eme Livre de G´eom´etrie. Poincar´e dessina un cercle informe, il marqua les lignes et les pointsindiqu´es par l’examinateur; puis, apr`es s’ˆetre promen´e devant le tableau les yeux fix´es `a terre pen-dant assez longtemps, conclut `a haute voix: Tout revient `a d´emontrer l’´egalit´e AB = CD . Elleest la cons´equence de la th´eorie des polaires r´eciproques, appliqu´ee aux deux droites.“Fort bien, Monsieur, interrompit M. Tissot; mais je voudrais une solution plus ´el´ementaire.”Poincar´e se mit `a repasser, non plus devant le tableau, mais devant la table de l’examinateur, face`a lui, presque inconscient de ses actes, puis tout `a coup d´eveloppa une solution trigonom´etrique.“Je d´esire que vous ne sortiez pas de la G´eom´etrie ´el´ementaire,” objecta M. Tissot, et presque aus-sitˆot satisfaction fut donn´ee `a l’examinateur d’´el´ementaires, qui f´elicita chaleureusement l’examin´eet lui annon¸ca qu’il avait m´erit´e la note maxima. [30], letter No. 62. Quand je pense aux exams de Tissot et autres, [. . . ] je ne puis m’empˆecher de prendre enpiti´e ces petites colles de 10 minutes o`u on joue son avenir dans une expression plus ou moinsexacte ou sur une phrase plus ou moins bien tourn´ee et o`u on juge un individu sur des diff´erencesinfinit´esimales. Ptolemy, in his
Geography , works with the strereographic projection, see [3]. See alsoPtolemy’s work on the
Planisphere [32].
ATHANASE PAPADOPOULOS
Let us summarize a few of his results on this subject.An important observation made by Tissot right at the beginning of hismemoir [52] (p. 1) is that finding the most appropriate mode of projectiondepends on the shape of the region—and not only its size, that is, on theproperties of its boundary,. Finding maps of small “distorsion” (where,as we mentioned, this word has several possible meanings) was the aimof theoretical cartography. Tissot discovered that in order for the map tominimize an appropriately defined distortion, a certain function λ , definedby setting dσ = (1 + λ ) ds , must be minimized in some appropriate sense, where ds and dσ are the lineelements at the source and the target surfaces respectively. The minimalityof λ may mean, for example, that the value of the gradient of its squaremust be the smallest possible.In fact, Tissot studied mappings between surfaces that are more generalthan those between subsets of the sphere and of the Euclidean plane. Hestarted by noting that for a given mapping between two surfaces, there is,at each point of the domain, a pair of orthogonal directions that are sent toa pair of orthogonal directions on the image surface. Unless the mapping isangle-preserving at the given point, these pairs of orthogonal directions areunique. The orthogonal directions at the various points on the two surfacesdefine a pair of orthogonal foliations preserved by the mapping. Tissot callsthe tangents to these foliations principal tangents at the given point. Theycorrespond to the directions where the ratio of lengths of the correspondinginfinitesimal line elements attains its greatest and smallest values.Using the foliations defined by the principal tangents, Tissot gave amethod for finding the image of an infinitely small figure drawn in the tan-gent plane of the first surface. In particular, for a differentiable mapping, theimages of infinitesimal circles are ellipses. In this case, he gave a practicalway of finding the major and minor axes of these ellipses, and he providedformulae for them. This is the theory of the Tissot indicatrix.From the differential geometric point of view, the Tissot indicatrix givesinformation on the metric tensor obtained by pushing forward the metric ofthe sphere (or the spheroid) by the projection mapping.We recall that in modern quasiconformal theory, an important parameterof a map is the quasiconformal dilatation at a point, defined as the ratioof the major axis to the minor axis of the infinitesimal ellipse which is theimage of an infinitesimal circle by the map (assumed to be differentiableat the give point, so that its derivative sends circles centered at the originin the tangent plane to ellipses). The Tissot indicatrix gives much moreinformation than this quasiconformal dilatation, since it keeps track of (1)the direction of the great and small axes of the infinitesimal ellipse, and (2)the size of this ellipse, compared to that of the infinitesimal circle of whichit is the image.Darboux got interested in the work of Tissot on geography, and in par-ticular, in a projection described in Chapter 2 of his memoir [52]. He wrotea paper on Tissot’s work [11] explaining more carefully some of his results.He writes: “[Tissot’s] exposition appeared to me a little bit confused, and NOTE ON NICOLAS-AUGUSTE TISSOT 7 it seems to me that while we can stay in the same vein, we can follow thefollowing method [. . . ]” Tissot showed then how to construct mappings that have minimal distor-tion.Tissot’s work was considered as very important by cartographers. TheAmerican cartographer, in his book
Flattening the earth: two thousand yearsof map projections [34], published in 1997 and which is a reference in the sub-ject, after presenting the existing books on cartography, writes: “Almost allof the detailed treatises presented one or two new projections, they basicallydiscussed those existing previously, albeit with very thorough analysis. Onescholar, however, proposed an analysis of distorsion that has had a majorimpact on the work of many twentieth-century writers on map projections.This was Tissot [. . . ].”Modern cartographers are still interested in the theoretical work of Tissot,see [24].We mentioned several preeminent mathematicians who before Tissot workedon the theory of geographical maps. From the more recent era, let memention Milnor’s paper titled
A problem in cartography [25], published in1969. The reader interested in the theory of geographical maps developedby mathematicians is referred to the papers [29], [27] and [28] which alsocontain more on the work of Tissot.
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Memorie della Societ`a Italiana delleScienze (XL), s. 3, I, 2 (1868), p. 17–92.8Son exposition m’a paru quelque peu confuse, et il m’a sembl´e qu’en restant dans le mˆemeordre d’id´ees on pourrait suivre avec avantage la m´ethode suivante.
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NOTE ON NICOLAS-AUGUSTE TISSOT 9 [36] N.-A. Tissot,
Mouvement d’un point mat´eriel pesant sur une sph`ere. Suivi de Sur lad´etermination des orbites des plan`etes et des com`etes . Th`eses pr´esent´ees ´a la facult´e dessciences de Paris, Paris, Bachelier, Imprimeur-libraire de l’´Ecole Polytechnique et du bureaudes longitudes, 1852.[37] N.-A. Tissot, Sur les h´elices.
Nouvelles annales de math´ematiques, journal des candidats aux´ecoles polytechnique et normale
S´er. 1, 11 (1852), p. 454–457.[38] N.-A. Tissot, Sur la d´etermination des latitudes au moyen de la m´ethode de M. Babinet.
Comptes Rendus de l’Acad´emie des Sciences (Paris) 42 (1856), p. 287–288.[39] N.-A. Tissot, Sur les alt´erations d’angles et de distances dans le d´eveloppement modifi´e deFlamsteed.
Journal de l’ ´Ecole Polytechnique . Paris. 21 (1858), p. 217–225.[40] N.-A. Tissot, Sur le d´eveloppement modifi´e de Flamsteed.
Comptes Rendus de l’Acad´emiedes Sciences (Paris) 46 (1858), 646-648.[41] N.-A. Tissot, Sur les cartes g´eographiques.
Comptes Rendus de l’Acad´emie des Sciences (Paris) 49 (1859), p. 673–676.[42] N.-A. Tissot, Sur les cartes g´eographiques.
Comptes Rendus de l’Acad´emie des Sciences (Paris) 50 (1860), p. 474–476.[43] N.-A. Tissot, Sur les cartes g´eographiques
Comptes Rendus de l’Acad´emie des Sciences (Paris)50 (1860), p. 964–968.[44] N.-A. Tissot, D´emonstration nouvelle du th´eor`eme de Legendre sur les triangles sph´eriquesdont les cˆot´es sont tr`es-petits relativement au rayon de la sph`ere.
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S´er. 2, 1 (1862),p. 5–11.[45] N.-A. Tissot, Sur la construction des cartes g´eographiques.
Comptes Rendus de l’Acad´emiedes Sciences (Paris) 60 (1865), p. 933–934.[46] N.-A. Tissot, M´emoire sur la repr´esentation des surfaces et les projections des cartesg´eographiques.
Nouvelles annales de math´ematiques, journal des candidats aux ´ecoles poly-technique et normale
S´er. 2, 17 (1878), p. 49–55.[47] N.-A. Tissot, M´emoire sur la repr´esentation des surfaces et les projections des cartesg´eographiques.
Nouvelles annales de math´ematiques, journal des candidats aux ´ecoles poly-technique et normale
S´er. 2, 17 (1878), p. 145–163.[48] N.-A. Tissot, M´emoire sur la repr´esentation des surfaces et les projections des cartesg´eographiques.
Nouvelles annales de math´ematiques, journal des candidats aux ´ecoles poly-technique et normale
S´er. 2, 17 (1878), p. 351–366.[49] N.-A. Tissot, M´emoire sur la repr´esentation des surfaces et les projections des cartesg´eographiques.
Nouvelles annales de math´ematiques, journal des candidats aux ´ecoles poly-technique et normale
S´er. 2, 18 (1879), p. 337–356.[50] N.-A. Tissot, M´emoire sur la repr´esentation des surfaces et les projections des cartesg´eographiques.
Nouvelles annales de math´ematiques, journal des candidats aux ´ecoles poly-technique et normale
S´er. 2, 18, p. 385–397.[51] N.-A. Tissot, M´emoire sur la repr´esentation des surfaces et les projections des cartesg´eographiques.
Nouvelles annales de math´ematiques, journal des candidats aux ´ecoles poly-technique et normale
S´er. 2, Suppl´ement au tome 19 (1880), p. S3–S40.[52] N.-A. Tissot,
M´emoire sur la repr´esentation des surfaces et les projections des cartesg´eographiques
Paris: Gauthier-Villars, 1881.[53] N.-A. Tissot, Formules relatives aux foyers des coniques.
Nouvelles annales de math´ematiques,journal des candidats aux ´ecoles polytechnique et normale
S´er. 3, 13 (1894), p. 97–98
Athanase Papadopoulos, Universit´e de Strasbourg and CNRS, 7 rue Ren´e Descartes,67084 Strasbourg Cedex, France
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