This is a sub-page of our page on Geometry.
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Popular Science Monthly/Volume 66/March 1905/A Study of the Development of Geometric Methods during the 19th century, by M. Jean-Gaston Darboux, Secretaire Perpétuel de L’Academie des Science.
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To appreciate the progress geometry has made during the century just ended, it is of advantage to cast a rapid glance over the state of mathematical science at the beginning of the nineteenth century.
We know that, in the last period of his life, Lagrange, fatigued by the researches in analysis and mechanics, which assured him, however, an immortal glory, neglected mathematics for chemistry, which, according to him, was easy as algebra, for physics, for philosophic speculations.
This mood of Lagrange we almost always find at certain moments of the life of the greatest savants. The new ideas which came to them in the fecund period of youth and which they introduced into the common domain have given them all they could have expected; they have fulfilled their task and feel the need of turning their mental activity towards wholly new subjects. This need, as we recognize, manifested itself with particular force at the epoch of Lagrange.
At this moment, in fact, the program of researches opened to geometers by the discovery of the infinitesimal calculus appeared very nearly finished up. Some differential equations more or less complicated to integrate, some chapters to add to the integral calculus, and one seemed about to touch the very outmost bounds of science.
Laplace had achieved the explanation of the system of the world and laid the foundations of molecular physics. New ways opened before the experimental sciences and prepared the astonishing development they received in the course of the century just ended. Ampère, Poisson, Fourier and Cauchy himself, the creator of the theory of imaginaries, were occupied above all in studying the application of the analytic methods to mechanics, and seemed to believe that outside this new domain, which they hastened to cover, the outlines of theory and science were finally fixed.
Modern geometry, a glory we must claim for it, came, after the end of the eighteenth century, to contribute in large measure to the renewing of all mathematical science, by offering to research a way new and fertile, and above all in showing us, by brilliant successes, that general methods are not everything in science, and that even in the simplest subject there is much for an ingenious and inventive mind to do.
The beautiful geometric demonstrations of Huygens, of Newton and of Clairaut
were forgotten or neglected. The fine ideas introduced by Desargues and Pascal had remained without development and appeared to have fallen on sterile ground.
Carnot, by his ‘Essai sur les transversales‘ and his ‘Géometrie de position‘ above all Monge, by the creation of descriptive geometry and by his beautiful theories on the generation of surfaces, came to renew a chain which seemed broken. Thanks to them, the conceptions of the inventors of analytic geometry, Descartes and Fermat, retook alongside the infinitesimal calculus of Leibniz and Newton the place they had lost, yet should never have ceased to occupy. With his geometry, said Lagrange, speaking of Monge, ‘this demon of a man will make himself immortal’.
And, in fact, not only has descriptive geometry made it possible to coordinate and perfect the procedures employed in all the arts where precision of form is a condition of success and of excellence for the work and its products; but it appeared as the graphic translation of a geometry, general and purely rational, of which numerous and important researches have demonstrated the happy fertility.
Moreover, beside the ‘Géometrie descriptive‘ we must not forget to place that other master-piece, the ‘Application de l’analyse a la geometrie‘; nor should we forget that to Monge are due the notion of lines of curvature and the elegant integration of the differential equation of these lines for the case of the ellipsoid, which, it is said, Lagrange envied him. To be stressed is this character of unity of the work of Monge.
The renewer of modern geometry has shown us from the beginning, what his successors have perhaps forgotten, that the alliance of geometry and analysis is useful and fruitful, that this alliance is perhaps for each a condition of success.
II.
In the school of Monge were formed many geometers: Hachette, Brianchon, Chappuis, Binet, Lancret, Dupin, Malus, Gaultier de Tours, Poncelet, Chasles, etc. Among these Poncelet takes first rank. Neglecting, in the works of Monge, everything pertaining to the analysis of Descartes or concerning infinitesimal geometry, he devoted himself exclusively to developing the germs contained in the purely geometric researches of his illustrious predecessor.
Made prisoner by the Russians in 1813 at the passage of the Dnieper and incarcerated at Saratoff, Poncelet employed the leisure captivity left him in the demonstration of the principles which he has developed in the ‘Traité des propriétés projectives des figures‘ issued in 1822, and in the great memoirs on reciprocal polars and on harmonic means, which go back nearly to the same epoch. So we may say that the modern geometry was born at Saratoff.
Renewing the chain broken since Pascal and Desargues, Poncelet introduced at the same time homography and reciprocal polars, putting thus in evidence, from the beginning, the fruitful ideas on which the science has evolved during fifty years.
Presented in opposition to analytic geometry, the methods of Poncelet were not favorably received by the French analysts. But such were their importance and their novelty, that without delay they aroused, from diverse sides, the most profound researches.
Poncelet had been alone in discovering the principles; on the contrary, many geometers appeared almost simultaneously to study them on all sides and to deduce from them the essential results which they implicitly contained.
At this epoch, Gergonne was brilliantly editing a periodical which has to-day for the history of geometry an inestimable value. The Annales de Mathématiques Pures et Appliquées, published at Nimes from 1810 to 1831. was during more than fifteen years the only journal in the entire world devoted exclusively to mathematical researches.
Gergonne, who, in many regards, was a model editor for a scientific journal, had the defects of his qualities; he collaborated, often against their will, with the authors of the memoirs sent him, rewrote them, and sometimes made them say more or less than they would have wished. Be that as it may, he was greatly struck by the originality and range of Poncelet‘s discoveries.
In geometry some simple methods of transformation of figures were already known; homography even had been employed in the plane, but without extending it to space, as did Poncelet, and especially without recognizing its power and fruitfulness. Moreover all these transformations were punctual, that is to say they made correspond a point to a point.
In introducing polar reciprocals, Poncelet was in the highest degree creative, because he gave the first example of a transformation in which to a point corresponded something other than a point.
Every method of transformation enables it to multiply the number of theorems, but that of polar reciprocals had the advantage of making correspond to a proposition another proposition of wholly different aspect. This was a fact essentially new. To put it in evidence, Gergonne invented the system, which since has had so much success, of memoirs printed in double columns with correlative propositions in juxtaposition; and he had the idea of substituting for Poncelet‘s demonstrations, which required an intermediary curve or surface of the second degree, the famous ‘principle of duality,’ of which the signification, a little vague at first, was sufficiently cleared up by the discussions which took place on this subject between Gergonne, Poncelet and Pluecker.
Bobillier, Chasles, Steiner, Lamé, Sturm, and many others whose names escape me, were, at the same time as Pluecker and Poncelet, assiduous collaborators of the Annales de Mathématiques Pures et Appliquées.
Gergonne having become rector of the Acadamy of Montpellier, was forced to suspend in 1831 the publication of his journal. But the success it had obtained, the taste for research it had contributed to develop, had commenced to bear their fruit. Quetelet had established in Belgium the Correspondance mathématique et physique. Crelle, from 1826, brought out at Berlin the first sheets of his celebrated journal, where he published the memoirs of Abel, of Jacobi, of Steiner.
A great number of separate works began also to appear, wherein the principles of modern geometry were powerfully expounded and developed.
First came in 1827 the ‘barycentrische Calcul‘ of Moebius, a work truly original, remarkable for the profundity of its conceptions, the elegance and the rigor of its exposition; then in 1828 the ‘Analytisch-geometrische Entwickelungen‘ of Pluecker, of which the second part appeared in 1831 and which was soon followed by the ‘System der analytischen Geometrie‘ of the same author published at Berlin in 1835.
In 1832 Steiner brought out at Berlin his great work: ‘Systematische Entwickelung der Abhaengigkeit der geometrischen Gestalten von einander,’ and, the following year, ‘Die geometrischen Konstruktionen ausgefuehrt mittels der geraden Linie und eines festen Kreises,’ where was confirmed by the most elegant examples a proposition of Poncelet‘s relative to the employment of a single circle for the geometric constructions.
Finally, in 1830, Chasles sent to the Academy of Brussels, which happily inspired had offered a prize for a study of the principles of modern geometry, his celebrated ‘Aperçu historique sur l’origine et le développement des methodes en geometrie,’ followed by ‘Mémoire sur deux principes généraux de la science: la dualité et 1’homographie‘ which was published only in 1837.
Time would fail us to give a worthy appreciation of these beautiful works and to apportion the share of each. Moreover, to what would such a study conduct us, but to a new verification of the general laws of the development of science. When the times are ripe, when the fundamental principles have been recognized and enunciated, nothing stops the march of ideas; the same discoveries, or discoveries almost equivalent, appear at nearly the same instant, and in places the most diverse. Without undertaking a discussion of this sort, which, besides, might appear useless or become irritating, it is, however, of importance to bring out a fundamental difference between the tendencies of the great geometers who, about 1830, gave to geometry a scope before unknown.
III.
Some, like Chasles and Steiner, who consecrated their entire life to research in pure geometry, opposed what they called synthesis to analysis and, adopting in the ensemble if not in detail the tendencies of Poncelet, proposed to constitute an independent doctrine, rival of Descartes‘ analysis.
Poncelet could not content himself with the insufficient resources furnished by the method of projections; to attain imaginaries he created that famous principle of continuity which gave birth to such long discussions between him and Cauchy.
Suitably enunciated, this principle is excellent and can render great service. Poncelet was wrong in refusing to present it as a simple consequence of analysis; and Cauchy, on the other hand, was not willing to recognize that his own objections, applicable without doubt to certain transcendent figures, were without force in the applications made by the author of the ‘Traité des propriétés projectives des figures.’
Whatever be the opinion of such a discussion, it showed at least in the clearest manner that the* geometric system of Poncelet rested on an analytic foundation, and besides we know, by the untoward publication of the manuscripts of Saratoff, that by the aid of Descartes‘ analysis were established the principles which serve as foundation for the ‘Traité des propriétés projectives des figures.’
Younger than Poncelet, who besides abandoned geometry for mechanics where his works had a preponderant influence, Chasles, for whom was created in 1847 a chair of Géométric supérieure in the Faculty of Science of Paris, endeavored to constitute a geometric doctrine entirely independent and autonomous. He has expounded it in two works of high importance, the ‘Traité de géométrie supérieure,’ which dates from 1852, and the ‘Traité des sections coniques,’ unhappily unfinished and of which the first part alone appeared in 1865.
In the preface of the first of these works he indicates very clearly the three fundamental points which permit the new doctrine to share the advantages of analysis and which to him appear to mark an advance in the cultivation of the science. These are:
(1) The introduction of the principle of signs, which simplifies at once the enunciations and the demonstrations, and gives to Carnot‘s analysis of transversals all the scope of which it is susceptible;
(2) the introduction of imaginaries, which supplies the place of the principle of continuity and furnishes demonstrations as general as-those of analytic geometry;
(3) the simultaneous demonstration of propositions which are correlative, that is to say, which correspond in virtue of the principle of duality.
Chasles studies indeed in his work homography and correlation; but he avoids systematically in his exposition the employment of transformations of figures, which, he thinks, can not take the place of direct demonstrations since they mask the origin and the true nature of the properties obtained by their means.
There is truth in this judgment, but the advance itself of the science permits us to declare it too severe. If it happens often that, employed without discernment, transformations multiply uselessly the number of theorems, it must be recognized that they often aid us to better understand the nature of the propositions even to which they have been applied. Is it not the employment of Poncelet‘s projective transformations which have led to the so fruitful distinction between projective properties and metric properties, which has taught us also the high importance of that cross-ratio whose essential property is found already in Pappus, and of which the fundamental role has begun to appear after fifteen centuries only in the researches of modern geometry?
The introduction of the principle of signs was not as new as Chasles supposed at the time he wrote his ‘Traité de Géometrie superiéure.’
Moebius, in his barycentrische Calcul, had already given issue to a desideratum of Carnot, and employed the signs in a way the largest and most precise, defining for the first time the sign of a segment and even that of an area.
Later he succeeded in extending the use of signs to lengths not laid off on the same straight and to angles not formed about the same point.
Besides Grassmann, whose mind has so much analogy to that of Moebius, had necessarily employed the principle of signs in the definitions which serve as basis for his methods, so original, of studying the properties of space.
The second characteristic which Chasles assigns to his system of geometry is the employment of imaginaries. Here, his method was really new and he illustrates it by examples of high interest. One will always admire the beautiful theories he has left us on [con]focal surfaces of the second degree, where all the known properties and others new, as varied as elegant, flow from the general principle that they are inscribed in the same developable circumscribed to the circle at infinity [which intersects each real projective plane in its circular points at infinity].
But Chasles introduced imaginaries only by their symmetric functions, and consequently would not have been able to define the cross-ratio of four elements when these ceased to be real in whole or in part. If Chasles had been able to establish the notion of the cross ratio of imaginary elements, a formula he gives in the ‘Géometrie supérieure‘ (p. 118 of the new edition) this would have immediately furnished him that beautiful definition of angle as the logarithm of a cross-ratio which enabled Laguerre, our regretted confrère, to give the complete solution, sought so long, of the problem of the transformation of relations which contain at the same time angles and segments in homography and correlation.
Like Chasles, Steiner, the great and profound geometer, followed the way of pure geometry; but he has neglected to give us a complete exposition of the methods upon which he depended. However, they may be characterized by saying that they rest upon the introduction of those elementary geometric forms which Desargues had already considered, on the development he was able to give to Bobillier‘s theory of polars, and finally on the construction of curves and surfaces of higher degrees by the aid of sheaves or nets of curves of lower orders. In default of recent researches, analysis would suffice to show that the field thus embraced has just the extent of that into which the analysis of Descartes introduces us without effort.
IV.
While Chasles, Steiner, and, later, as we shall see, von Staudt, were intent on constituting a rival doctrine to analysis and set in some sort altar against altar, Gergonne, Bobillier, Sturm, above all Pluecker, perfected the geometry of Descartes and constituted an analytic system in a manner adequate to the discoveries of the geometers.
It is to Bobillier and to Pluecker that we owe the method called abridged notation. Bobillier consecrated to it some pages truly new in the last volumes of the Annales de Mathématiques Pures et Appliquées of Gergonne.
Pluecker commenced to develop it in his first work, soon followed by a series of works where are established in a fully conscious manner the foundations of the modern analytic geometry. It is to him that we owe tangential coordinates, trilinear coordinates, employed with homogeneous polynomial equations, and finally the employment of canonical forms whose validity was recognized by the method, so deceptive sometimes, but so fruitful, called the enumeration of constants.
All these happy acquisitions infused new blood into Descartes‘ analysis and put it in condition to give their full signification to the conceptions of which the geometry called synthetic had been unable to make itself completely mistress.
Pluecker, to whom it is without doubt just to adjoin Bobillier, carried off by a premature death, should be regarded as the veritable initiator of those methods of modern analysis where the employment of homogeneous coordinates permits treating simultaneously and, so to say, without the reader perceiving it, together with one figure all those deducible from it by homography and correlation.
V.
Parting from this moment, a period opens brilliant for geometric researches of every nature.
The analysts interpret all their results and are occupied in translating them by constructions.
The geometers are intent on discovering in every question some general principle, usually undemonstrable without the aid of analysis, in order to make flow from it without effort a crowd of particular consequences, solidly bound to one another and to the principle whence they are derived. Otto Hesse, brilliant disciple of Jacobi, develops in an admirable manner that method of homogeneous coordinates to which Pluecker perhaps had not attached its full value. Boole discovers in the polars of Bobillier the first notion of a covariant; the theory of forms is created by the labors of Cayley, Sylvester, Hermite, Brioschi. Later Aronhold, Clebsch and Gordan and other geometers still living gave to it its final notation, established the fundamental theorem relative to the limitation of the number of covariant forms and so gave it all its amplitude.
The theory of surfaces of the second order, built up principally by the school of Monge, was enriched by a multitude of elegant properties, established principally by O. Hesse, who found later in Paul Serret a worthy emulator and continuer.
The properties of the polars of algebraic curves are developed by Pluecker and above all by Steiner. The study, already old, of curves of the third order is rejuvenated and enriched by a crowd of new elements. Steiner, the first, studies by pure geometry the double tangents of curves of the fourth order, and Hesse, after him, applies the methods of algebra to this beautiful question, as well as to that of points of inflection of curves of the third order.
The notion of class introduced by Gergonne, the study of a paradox in part elucidated, by Poncelet and relative to the respective degrees of two curves reciprocal polars one of the other, give birth to the researches of Pluecker relative to the singularities called ordinary of algebraic plane curves. The celebrated formulas to which Pluecker is thus conducted are later extended by Cayley and by other geometers to algebraic skew curves, by Cayley again and by Salmon to algebraic surfaces.
The singularities of higher order are in their turn taken up by the geometers; contrary to an opinion then very widespread, Halphen demonstrates that each of these singularities cannot be considered as equivalent to a certain group of ordinary singularities and his researches close for a time this difficult and important question.
Analysis and geometry, Steiner, Cayley, Salmon, Cremona, meet in the study of surfaces of the third order, and, in conformity with the anticipations of Steiner, this theory becomes as simple and as easy as that of surfaces of the second order.
The algebraic ruled surfaces, so important for applications, are studied by Chasles, by Cayley, of whom we find the influence and the mark in all mathematical researches, by Cremona, Salmon, La Gournerie; so they will be later by Pluecker in a work to which we must return.
The study of the general surface of the fourth order would seem to be still too difficult; but that of the particular surfaces of this order with multiple points or multiple lines is commenced, by Pluecker for the surface of waves, by Steiner, Kummer, Cayley, Moutard, Laguerre, Cremona and many other investigators.
As for the theory of algebraic skew curves, grown rich in its elementary parts, it receives finally, by the labors of Halphen and of Noether, whom it is impossible for us here to separate, the most notable extensions.
A new theory with a great future is born by the labors of Chasles, of Clebsch and of Cremona; it concerns the study of all the algebraic curves which can be traced on a determined (algebraic) surface.
Homography and correlation, those two methods of transformation which have been the distant origin of all the preceding researches, receive from them in their turn an unexpected extension; they are not the only methods which make a single element correspond to a single element, as might have shown a particular transformation briefly indicated by Poncelet in the ‘Traité des propriétés projectives des figures.’
Pluecker defines the transformation by reciprocal radii vectors or inversion, of which Sir William Thomson and Liouville hasten to show all the importance, as well for mathematical physics as for geometry.
A contemporary of Moebius and Pluecker, Magnus believed he had found the most general transformation which makes a point correspond to a point, but the researches of Cremona teach us that the transformation of Magnus is only the first term of a series of birational transformations which the great Italian geometer teaches us to determine methodically, at least for the figures of plane geometry.
The Cremona transformations long retained a great interest, though later researches have shown us that they reduce always to a series of successive applications of the transformation of Magnus.
VI.
All the works we have enumerated, others to which we shall return later, find their origin and, in some sort, their first motive in the conceptions of modern geometry; but the moment has come to indicate rapidly another source of great advances for geometric studies.
Legendre‘s theory of elliptic functions, too much neglected by the French geometers, is developed and extended by Abel and Jacobi. With these great geometers, soon followed by Riemann and Weierstrass, the theory of Abelian functions which, later, algebra would try to follow solely with its own resources, brought to the geometry of curves and surfaces a contribution whose importance will continue to grow.
Already, Jacobi had employed the analysis of elliptic functions in the demonstration of Poncelet‘s celebrated theorems on inscribed and circumscribed polygons, inaugurating thus a chapter since enriched by a multitude of elegant results; he had obtained also, by methods pertaining to geometry, the integration of Abelian equations.
But it was Clebsch who first showed in a long series of works all the importance of the notion of deficiency ([genus]) of a curve, due to Abel and Riemann, in developing a crowd of results and elegant solutions that the employment of Abelian integrals would seem, so simple was it, to connect with their veritable point of departure.
The study of points of inflection of curves of the third order, that of double tangents of curves of the fourth order and, in general, the theory of osculation on which the ancients and the moderns had so often practised, were connected with the beautiful problem of the division of elliptic functions and Abelian functions.
In one of his memoirs, Clebsch had studied the curves which are rational or of deficiency [= genus] zero; this led him, toward the end of his too short life, to envisage what may be called also rational surfaces, those which can be simply represented by a plane. This was a vast field for research, opened already for the elementary cases by Chasles, and in which Clebsch was followed by Cremona and many other savants. It was on this occasion that Cremona, generalizing his researches on plane geometry, made known not indeed the totality of birational transformations of space, but certain of the most interesting among these transformations.
The extension of the notion of deficiency [= genus] to algebraic surfaces is already commenced; already also works of high value have shown that the theory of integrals, simple or multiple, of algebraic differentials will find, in the study of surfaces as in that of curves, an ample field of important applications; but it is not proper for the reporter on geometry to dilate on this subject.
VII.
While thus were constituted the mixed methods whose principal applications we have just indicated, the pure geometers were not inactive. Poinsot, the creator of the theory of couples, developed, by a method purely geometric, ‘that, said he, where one never for a moment loses from view the object of the research,’ the theory of the rotation of a solid body that the researches of d’Alembert, Euler and Lagrange seemed to have exhausted: Chasles made a precious contribution to kinematic by his beautiful theorems on the displacement of a solid body, which have since been extended by other elegant methods to the case where the motion has divers degrees of freedom. He made known those beautiful propositions on attraction in general, which figure without disadvantage beside those of Green and Gauss.
Chasles and Steiner met in the study of the attraction of ellipsoids and showed thus once more that geometry has its designated place in the highest questions of the integral calculus.
Steiner did not disdain at the same time to occupy himself with the elementary parts of geometry. His researches on the contacts of circles and conics, on isoperimetric problems, on parallel surfaces, on the center of gravity of curvature, excited the admiration of all by their simplicity and their depth.
Chasles introduced his principle of correspondence between two variable objects which has given birth to so many applications; but here analysis re-took its place to study the principle in its essence, make it precise and generalize it.
It was the same concerning the famous theory of characteristics and the numerous researches of de Jonquieres, Chasles, Cremona and still others, which gave the foundations of a new branch of the science, Enumerative Geometry.
During many years, the celebrated postulate of Chasles was admitted without any objection: a crowd of geometers believed they had established it in a manner irrefutable.
But, as Zeuthen then said, it is very difficult to recognize whether, in demonstrations of this sort, there does not exist always some weak point that their author has not perceived; and, in fact, Halphen, after fruitless efforts, crowned finally all these researches by clearly indicating in what cases the postulate of Chasles may be admitted and in what cases it must be rejected.
VIII.
Such are the principal works which restored geometric synthesis to honor and assured to it, in the course of the last century, the place belonging to it in mathematical research. Numerous and illustrious workers took part in this great geometric movement, but we must recognize that its chiefs and leaders were Chasles and Steiner. So brilliant were their marvelous discoveries that they threw into the shade, at least momentarily, the publications of other modest geometers, less preoccupied perhaps in finding brilliant applications, fitted to evoke love for geometry, than to establish this science itself on an absolutely solid foundation.
Their works have received perhaps a recompense more tardy, but their influence grows each day; it will without doubt increase still more. To pass them over in silence would be without doubt to neglect one of the principal factors which will enter into future researches.
We allude at this moment above all to von Staudt. His geometric works were published in two books of grand interest: the ‘Geometrie der Lage,’ issued in 1847, and the ‘Beiträge zur Geometrie der Lage,’ published in 1856, that is to say, four years after the ‘Géométrie supérieure of Chasles.’
Chasles, as we have seen, had devoted himself to constituting a body of doctrine independent of Descartes‘ analysis and had not completely succeeded. We have already indicated one of the criticisms that can be made upon this system: the imaginary elements are there defined only by their symmetric functions, which necessarily excludes them from a multitude of researches. On the other hand, the constant employment of cross-ratio, of transversals and of involution, which requires frequent analytic transformations, gives to the ‘Géométrie supérieure‘ a character almost exclusively metric which removes it notably from the methods of Poncelet. Returning to these methods, von Staudt devoted himself to constituting a geometry freed from all metric relation and resting exclusively on relations of situation.
This is the spirit in which was conceived his first work, the ‘Geometrie der Lage‘ of 1847. The author there takes as point of departure the harmonic properties of the complete quadrilateral and those of homologic triangles, demonstrated uniquely by considerations of geometry of three dimensions, analogous to those of which the School of Monge made such frequent use.
In this first part of his work, von Staudt neglected entirely imaginary elements. It is only in the Beiträge zur Geometrie der Lage, his second work, that he succeeds, by a very original extension of the method of Chasles, in defining geometrically an isolated imaginary element and distinguishing it from its conjugate.
This extension, although rigorous, is difficult and very abstract. It may be defined in substance as follows: Two conjugate imaginary points may always be considered as the double points of an involution on a real straight; and just as one passes from an imaginary to its conjugate by changing \, i \, into \,-i \, , so one may distinguish the two imaginary points by making correspond to each of them one of the two different senses which may be attributed to the straight. In this there is something a little artificial; the development of the theory erected on such foundations is necessarily complicated.
By methods purely projective, von Staudt establishes a calculus of cross-ratios of the most general imaginary elements. Like all geometry, the projective geometry employs the notion of order and order engenders number; we are not astonished therefore that von Staudt has been able to constitute his calculus; but we must admire the ingenuity displayed in attaining it. In spite of the efforts of distinguished geometers who have essayed to simplify its exposition, we fear that this part of the geometry of von Staudt, like the geometry otherwise so interesting of the profound thinker Grassmann, can not prevail against the analytic methods which to-day have won favor almost universal.
Life is short; geometers know and also practise the principle of least action. Despite these fears, which should discourage no one, it seems to us that under the first form given it by von Staudt, projective geometry must become the necessary companion of descriptive geometry, that it is called to renovate this geometry in its spirit, its procedures and its applications.
This has already been comprehended in many countries, and notably in Italy where the great geometer Cremona did not disdain to write, for the schools, an elementary treatise on projective geometry.
IX.
In the preceding articles, we have essayed to follow and bring out clearly the most remote consequences of the methods of Monge and Poncelet. In creating tangential coordinates and homogeneous coordinates, Pluecker seemed to have exhausted all that the method of projections and that of reciprocal polars could give to analysis.
It remained for him, toward the end of his life, to return to his first researches to give them an extension enlarging to an unexpected degree the domain of geometry.
Preceded by innumerable researches on systems of straight lines, due to Poinsot, Moebius, Chasles, Dupin, Malus, Hamilton, Kummer, Transon, above all to Cayley, who first introduced the notion of the coordinates of the straight, researches originating perhaps in statics and kinematics, perhaps in geometrical optics, Pluecker‘s geometry of the straight line will always be regarded as the part of his work where are met the newest and most interesting ideas.
That Pluecker first set up a methodic study of the straight line, that already is important, but that is nothing beside what he discovered. It is sometimes said that the principle of duality shows that the plane as well as the point may be considered as a space element. That is true; but in adding the straight line to the plane and point as possible space element, Pluecker was led to recognize that any curve, any surface, may also be considered as space element, and so was born a new geometry which already has inspired a great number of works, which will raise up still more in the future.
A beautiful discovery, of which we shall speak further on, has already connected the geometry of spheres with that of straight lines and permits the introduction of the notion of coordinates of a sphere.
The theory of systems of circles is already commenced; it will be developed without doubt when one wishes to study the representation, which we owe to Laguerre, of an imaginary point in space by an oriented circle.
But before expounding the development of these new ideas which have vivified the infinitesimal methods of Monge, it is necessary to go back to take up the history of branches of geometry that we have neglected until now.
X.
Among the works of the school of Monge, we have hitherto confined ourselves to the consideration of those connected with finite geometry; but certain of the disciples of Monge devoted themselves above all to developing the new notions of infinitesimal geometry” applied by their master to curves of double curvature, to lines of curvature, to the generation of surfaces, notions expounded at least in part in the ‘Application de l’Analyse a la Géométrie‘ Among these we must cite Lancret, author of beautiful works on skew curves, and above all Charles Dupin, the only one perhaps who followed all the paths opened by Monge
Among other works, we owe to Dupin two volumes Monge would not have hesitated to sign: the ‘Développements de Géométrie pure,’ issued in 1813 and the ‘Applications de Géométrie et de Méchanique‘ dating from 1822.
There we find the notion of indicatrix, which was to renovate, after Euler and Meusnier, all the theory of curvature, that of conjugate tangents, of asymptotic lines which have taken so important a place in recent researches.
Nor should we forget the determination of the surface of which all the lines of curvature are circles, nor above all the memoir on triple systems of orthogonal surfaces where is found, together with the discovery of the triple system formed by surfaces of the second degree, the celebrated theorem to which the name of Dupin will remain attached.
Under the influence of these works and of the renaissance of synthetic methods, the geometry of infinitesimals re-took in all researches the place Lagrange had wished to take away from it forever.
Singular thing, the geometric methods thus restored were to receive the most vivid impulse in consequence of the publication of a memoir which, at least at first blush, would appear connected with the purest analysis; we mean the celebrated paper of Gauss: ‘Disquisitiones generales circa superficies curvas‘ which was presented in 1827 to the Göttingen Society, and whose appearance marked, one may say, a decisive date in the history of infinitesimal geometry.
From this moment, the infinitesimal method took in France a free scope before unknown.
Frenet, Bertrand, Molins, J. A. Serret, Bouquet, Puiseux, Ossian Bonnet, Paul Serret, develop the theory of skew curves. Liouville, Chasles, Minding, join them to pursue the methodic study of the memoir of Gauss.
The integration made by Jacobi of the differential equation of the geodesic lines of the ellipsoid started a great number of researches. At the same time the problems studied in the ‘Application de l’Analyse‘ of Monge were greatly developed.
The determination of all the surfaces having their lines of curvature plane or spherical completed in the happiest manner certain partial results already obtained by Monge.
Gabriel Lamé, who according to the judgment of Jacobi, at this moment was one of the most penetrating of geometers, and who, like Charles Sturm, had commenced with pure geometry, had already made to this science contributions of a most interesting kind, in a little book published in 1817, by memoirs inserted in the Annales of Gergonne.
Lamé utilized the results obtained by Dupin and Binet on the system of confocal surfaces of the second degree and, rising to the idea of curvilinear coordinates in space, Lamé became the creator of a wholly new theory destined to receive in mathematical physics the most varied applications.
XI.
Here again, in this infinitesimal branch of geometry are found the two tendencies we have pointed out à propos of the geometry of finite quantities.
Some, among whom must be placed J. Bertrand and O. Bonnet, wish to constitute an independent method resting directly on the employment of infinitesimals. The grand ‘Traité de Calcul différentiel,’ of Bertrand, contains many chapters on the theory of curves and of surfaces, which are, in some sort, the illustration of this conception.
Others follow the usual analytic ways, being only intent to clearly recognize and put in evidence the elements which figure in the first plan. Thus did Gabriel Lamé in introducing his theory of differential parameters. Thus did Beltrami in extending with great ingenuity the employment of these differential invariants to the case of two independent variables, that is to say, to the study of surfaces.
It seems that to-day is accepted a mixed method whose origin is found in the works of Ribaucour, under the name périmorphie. The rectangular axes of analytic geometry are retained, but made mobile and attached as seems best to the system to be studied. Thus disappear most of the objections which have been made to the method of coordinates. The advantages of what is sometimes called intrinsic geometry are united to those resulting from the use of the regular analysis. Besides, this analysis is by no means abandoned; the complications of calculation which it almost always carries with it, in its applications to the study of surfaces and rectilinear coordinates, usually disappear if one employs the notion on the invariants and the covariants of quadratic powers of differentials which we owe to the researches of Lipschitz and Christoffel, inspired by Riemann‘s studies on the non-Euclidean geometry.
XII.
The results of so many labors were not long in coming. The notion of geodesic curvature which Gauss already possessed, but without having published it, was given by Bonnet and Liouville, the theory of surfaces of which the radii of curvature are functions one of the other, inaugurated in Germany by two propositions which would figure without disadvantage in the memoir of Gauss, was enriched by Ribaucour, Halphen, Sophus Lie and others, with a multitude of propositions, some concerning these surfaces envisaged in a general manner; others applying to particular cases where the relation between the radii of curvature takes a form particularly simple; to minimal surfaces for example, and also to surfaces of constant curvature, positive or negative.
The minimal surfaces were the object of works which make of their study the most attractive chapter of infinitesimal geometry. The integration of their partial differential equation constitutes one of the most beautiful discoveries of Monge; but because of the imperfection of the theory of imaginaries, the great geometer could not get from its formulas any mode of generation of these surfaces, nor even any particular surface. We will not here retrace the detailed history which we have presented in our ‘Leçons sur la théorie des surfaces‘; but it is proper to recall the fundamental researches of Bonnet which have given us, in particular, the notion of surfaces associated with a given surface, the formulas of Weierstrass which establish a close bond between the minimal surfaces and the functions of a complex variable, the researches of Lie by which it was established that just the formulas of Monge can to-day serve as foundation for a fruitful study of minimal surfaces.
In seeking to determine the minimal surfaces of smallest classes or degrees, we were led to the notion of double minimal surfaces which is dependent on Analysis situs.
Three problems of unequal importance have been studied in this theory.
The first, relative to the determination of minimal surfaces inscribed along a given contour in a developable equally given, was solved by celebrated formulas which have led to a great number of propositions. For example, every straight traced on such a surface is an axis of symmetry.
The second, set by S. Lie, concerns the determination of all the algebraic minimal surfaces inscribed in an algebraic developable, without the curve of contact being given. It also has been entirely elucidated.
The third and the most difficult is what the physicists solve experimentally, by plunging a closed contour into a solution of glycerine. It concerns the determination of the minimal surface passing through a given contour.
The solution of this problem evidently surpasses the resources of geometry. Thanks to the resources of the highest analysis, it has been solved for particular contours in the celebrated memoir of Riemann and in the profound researches which have followed or accompanied this memoir.
For the most general contour, its study has been brilliantly begun, it will be continued by our successors.
After the minimal surfaces, the surfaces of constant curvature attracted the attention of geometers. An ingenious remark of Bonnet connects with each other the surfaces of which one or the other of the two curvatures, mean curvature or total curvature, is constant.
Bour announced that the partial differential equation of surfaces of constant curvature could be completely integrated. This result has not been recovered; it would seem even very doubtful if we consider a research where S. Lie has essayed in vain to apply a general method of integration of partial differential equations to the particular equation of surfaces of constant curvature.
But, if it is impossible to determine in finite terms all these surfaces, it has at least been possible to obtain certain of them, characterized by special properties, such as that of having their lines of curvature plane or spherical; and it has been shown, by employing a method which succeeds in many other problems, that from every surface of constant curvature may be derived an infinity of other surfaces of the same nature, by employing operations clearly defined which require only quadratures.
The theory of the deformation of surfaces in the sense of Gauss has been also much enriched. We owe to Minding and to Bour the detailed study of that special deformation of ruled surfaces which leaves the generators rectilineal. If we have not been able, as has been said, to determine the surfaces applicable on the sphere, other surfaces of the second degree have been attacked with more success, and, in particular, the paraboloid of revolution.
The systematic study of the deformation of general surfaces of the second degree is already entered upon; it is one of those which will give shortly the most important results.
The theory of infinitesimal deformation constitutes to-day one of the most finished chapters of geometry. It is the first somewhat extended application of a general method which seems to have a great future.
Being given a system of differential or partial differential equations, suitable to determine a certain number of unknowns, it is advantageous to associate with it a system of equations which we have called auxiliary system and which determines the systems of solutions infinitely near any given system of solutions. The auxiliary system being necessarily linear, its employment in all researches gives precious light on the properties of the proposed system and on the possibility of obtaining its integration.
The theory of lines of curvature and of asymptotic lines has been notably extended. Not only have been determined these two series of lines for particular surfaces such as the tetrahedral surfaces of Lamé; but also, in developing Moutard’s results relative to a particular class of linear partial differential equations of the second order, it proved possible to generalize all that had been obtained for surfaces with lines of curvature plane or spherical, in determining completely all the classes of surfaces for which could be solved the problem of spherical representation.
Just so has been solved the correlative problem relative to asymptotic lines in making known all the surfaces of which the infinitesimal deformation can be determined in finite terms. Here is a vast field for research whose exploration is scarcely begun.
The infinitesimal study of rectilinear congruences, already commenced long ago by Dupin, Bertrand, Hamilton, Kummer, has come to intermingle in all these researches. Bibaucour, who has taken in it a preponderant part, studied particular classes of rectilinear congruences and, in particular, the congruences called isotropes, which intervene in the happiest way in the study of minimal surfaces.
The triply orthogonal systems which Lamé used in mathematical physics have become the object of systematic researches. Cayley was the first to form the partial differential equation of the third order on which the general solution of this problem was made to depend.
The system of [con]focal surfaces of the second degree has been generalized and has given birth to that theory of general cyclides in which may be employed at the same time the resources of metric geometry, of projective geometry, of inversive geometry, and of infinitesimal geometry. Many other orthogonal systems have been made known. Among these it is proper to signalize the cyclic systems of Ribaucour, for which one of the three families admits circles as orthogonal trajectories, and the more general systems for which these orthogonal trajectories are simply plane curves.
The systematic employment of imaginaries, which we must be careful not to exclude from geometry, has permitted the connection of all these determinations with the study of the finite deformation of a particular surface.
Among the methods which have permitted the establishment of all these results it is proper to note the systematic employment of linear partial differential equations of the second order and of systems formed of such equations. The most recent researches show that this employment is destined to renovate most of the theories.
Infinitesimal geometry could not neglect the study of the two fundamental problems set it by the calculus of variations.
The problem of the shortest path on a surface was the object of masterly studies by Jacobi and by Ossian Bonnet. The study of geodesic lines has been followed up; we have learned to determine them for new surfaces. The theory of ensembles has come to permit the following of these lines in their course on a given surface.
The solution of a problem relative to the representation of two surfaces one on the other has greatly increased the interest of discoveries of Jacobi and of Liouville relative to a particular class of surfaces of which the geodesic lines could be determined. The results concerning this particular case led to the examination of a new question: to investigate all the problems of the calculus of variations of which the solution is given by curves satisfying a given differential equation.
Finally, the methods of Jacobi have been extended to space of three dimensions and applied to the solution of a question which presented the greatest difficulties: the study of properties of minimum appertaining to the minimal surface passing through a given contour.
XIII.
Among the inventors who have contributed to the development of infinitesimal geometry, Sophus Lie distinguishes himself by many capital discoveries which place him in the first rank.
He was not one of those who show from infancy the most characteristic aptitudes, and at the moment of quitting the University of Christiania in 1865, he still hesitated between philology and mathematics.
It was the works of Pluecker which gave him for the first time full consciousness of his veritable calling.
He published in 1869 a first work on the interpretation of imaginaries in geometry, and from 1870 he was in possession of the directing ideas of his whole career. I had at this epoch the pleasure of seeing him often, of entertaining him at Paris, where he had come with his friend Felix Klein.
A course by M. Sylow followed by Lie had revealed to him all the importance of the theory of substitutions; the two friends studied this theory in the great treatise of C. Jordan; they were fully conscious of the important role it was called on to play in so many branches of mathematical science where it had not yet been applied.
They have both had the good fortune to contribute by their works to impress upon mathematical studies the direction which to them appeared the best.
In 1870, Sophus Lie presented to the Academy of Sciences of Paris a discovery extremely interesting. Nothing bears less resemblance to a sphere than a straight line, and yet Lie had imagined a single transformation which made a sphere correspond to a straight, and permitted, consequently, the connecting of every proposition relative to straights with a proposition relating to spheres and vice versa.
In this so curious method of transformation, each property relative to the lines of curvature of a surface furnishes a proposition relative to the asymptotic lines of the surface attained.
The name of Lie will remain attached to these deep-lying relations which join to one another the straight line and the sphere, those two essential and fundamental elements of geometric research. He developed them in a memoir full of new ideas which appeared in 1872.
The works which followed this brilliant début of Lie fully confirmed the hopes it had aroused. Pluecker‘s conception relative to the generation of space by straight lines, by curves or surfaces arbitrarily chosen, opens to the theory of algebraic forms a field which has not yet been explored, that Clebsch scarcely began to recognize and settle the boundaries of. But, from the side of infinitesimal geometry, this conception has been given its full value by Sophus Lie. The great Norwegian geometer was able to find in it first the notion of congruences and complexes of curves, and afterward that of contact transformations of which he had found, for the case of the plane, the first germ in Pluecker. The study of these transformations led him to perfect, at the same time with M. Mayer, the methods of integration which Jacobi had instituted for partial differential equations of the first order; but above all it threw the most brilliant light on the most difficult and the most obscure parts of the theories relative to partial differential equations of higher order. It permitted Lie, in particular, to indicate all the cases in which the method of characteristics of Monge is fully applicable to equations of the second order with two independent variables.
In continuing the study of these special transformations, Lie was led to construct progressively his masterly theory of continuous groups of transformations and to put in evidence the very important role that the notion of group plays in geometry. Among the essential elements of his researches, it is proper to signalize the infinitesimal transformations, of which the idea belongs exclusively to him.
Three great books published under his direction by able and devoted collaborators contain the essential part of his works and their applications to the theory of integration, to that of complex units and to the non-Euclidean geometry.
XIV.
By an indirect way I have arrived at that non-Euclidean geometry of which the study takes in the researches of geometers a place which grows greater each day.
If I were the only one to talk with you about geometry, I would take pleasure in recalling to you all that has been done on this subject since Euclid or at least from Legendre to our days.
Envisaged successively by the greatest geometers of the last century, the question has progressively enlarged.
It commenced with the celebrated postulatum relative to parallels; it ends with the totality of geometric axioms.
The ‘Elements’ of Euclid, which have withstood the action of so many centuries, will have at least the honor before ending of arousing a long series of works admirably enchained which will contribute, in the most effective way, to the progress of mathematics, at the same time that they furnish to the philosophers the points of departure the most precise and the most solid for the study of the origin and of the formation of our cognitions.
I am assured in advance that my distinguished collaborator will not forget, among the problems of the present time, this one, which is perhaps the most important, and with which he has occupied himself with so much success; and I leave to him the task of developing it with all the amplitude which it assuredly merits.
I have just spoken of the elements of geometry. They have received in the last hundred years extensions which must not be forgotten. The theory of polyhedrons has been enriched by the beautiful discoveries of Poinsot on the star polyhedrons and those of Moebius on polyhedrons with a single face. The methods of transformation have enlarged the exposition. We may say to-day that the first book contains the theory of translation and of symmetry, that the second amounts to the theory of rotation and of displacement, that the third rests on homothety and inversion. But it must be recognized that it is thanks to analysis that the ‘Elements’ have been enriched by their most beautiful propositions.
It is to the highest analysis we owe the inscription of regular polygons of 17 sides and analogous polygons. It is to it we owe the demonstrations so long sought, of the impossibility of the quadrature of the circle, of the impossibility of certain geometric constructions with the aid of a straightedge and a compass. It is to it finally that we owe the first rigorous demonstrations of the properties of maximum and of minimum of the sphere. It will appertain to geometry to enter upon this ground where analysis has preceded it.
What will be the elements of geometry in the course of the century which has just commenced? Will there be a single elementary book of geometry? It is perhaps America, with its schools free from all program and from all tradition, which will give us the best solution of this important and difficult question.
Von Staudt has sometimes been called the Euclid of the nineteenth century; I would prefer to call him the Euclid of projective geometry: but that geometry, however interesting it may be, is it destined to furnish the unique foundation of the future elements?
XV.
The moment has come to close this over-long recital, and yet there is a crowd of interesting researches that I have been, so to say, forced to neglect.
I should have loved to talk with you about those geometries of any number of dimensions of which the notion goes back to the first days of algebra, but of which the systematic study was commenced only sixty years ago by Cayley and by Cauchy. This kind of researches has found favor in your country and I need not recall that our illustrious president, after having shown himself the worthy successor of Laplace and Le Verrier, in a space which he considers with us as being endowed with three dimensions, has not disdained to publish, in the American Journal, considerations of great interest on the geometries of n dimensions.
A single objection can be made to studies of this sort, and was already formulated by Poisson: the absence of all real foundation, of all substratum permitting the presentation, under aspects visible and in some sort palpable, of the results obtained.
The extension of the methods of descriptive geometry, and above all the employment of Pluecker‘s conceptions on the generation of space, will contribute to take away from this objection much of its force.
I would have liked to speak to you also of the method of equipollences, of which we find the germ in the posthumous works of Gauss, of Hamilton‘s quaternions, of Grassmann‘s exterior algebra and in general of systems of complex units, of the Analysis situs, so intimately connected with the theory of functions, of kinematic geometries, of the theory of abaci, of geometrography, of the applications of geometry to natural philosophy to architecture and to the arts. But I fear, if I branched out beyond measure, some analyst, as has happened before, would accuse geometry of wishing to monopolize everything.
My admiration for analysis, grown so fruitful and so powerful in our time, would not permit me to conceive such a thought. But, if some reproach of this sort could be formulated to-day, it is not to geometry, it is to analysis it would be proper, I believe, to address it. The circle in which the mathematical studies appeared to be enclosed at the beginning of the nineteenth century has been broken on all sides.
The old problems present themselves to us under a new form, new problems offer themselves, whose study occupies legions of workers.
The number of those who cultivate pure geometry has become prodigiously restricted. Therein is a danger against which it is important to provide. We must not forget that, if analysis has acquired means of investigation which it lacked heretofore, it owes them in great part to the conceptions introduced by the geometers.
geometry must not remain in some sort entombed in its triumph. It is in its school we have learned; our successors must learn never to be blindly proud, of methods too general, to envisage the questions in themselves and to find, in the conditions particular to each problem, perhaps a direct way towards a solution, perhaps the means of applying in an appropriate manner the general procedures which every science should gather.
As Chasles said at the beginning of the ‘Aperçu historique‘:
‘The doctrines of pure geometry offer often, and in a multitude of questions, that way simple and natural which, penetrating to the very source of the truths, lays bare the mysterious chain which binds them to each other and makes us know them individually in the way most luminous and most complete.’
Cultivate therefore geometry, which has its own advantages, without wishing, on all points, to make it equal to its rival.
For the rest, if we were tempted to neglect it, it would soon find in the applications of mathematics, as it did once before, means to rise up again and develop itself anew. It is like the giant Antaeus who recovered his strength in touching the earth.
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Read September 24, 1904, at the Congress of Arts and Science at St. Louis.
Translated by Professor George Bruce Halsted.
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Geometries for CAGD
Helmut Pottmann, Stefan Leopoldseder, in Handbook of Computer Aided Geometric Design, 2002
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Beyond Flatland: The Geometric Forms due to Self-Assembly
STEPHEN HYDE, … SVEN LIDIN, in The Language of Shape, 1997
4.3 The composition of surfactant mixtures: the global constraint
A large number of different surface shapes satisfy eq. (4.3). These shapes are distinguishable by their topologies, characteristic of the degree of inter-connectivity of the structure. A feature of high topology surfaces is their high surface area. Indeed, it follows from the Gauss-Bonnet theorem (described in section 1.7) that the surface area per unit volume increases with the surface genus, provided the average value of the Gaussian curvature,
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Figure 4.3. (Left:) Two-dimensional view of a focal surface to an interface. This focal surface describes the location of centres of curvature of the interface. (Right:) Three-dimensional view of the focal surfaces F1 and F2 to a hyperbolic surface, S.
(Adapted from [5].)
The geometry of focal surfaces depends on the variation of curvatures along the interface. If the curvatures of the interface do not vary, e.g. spherical and cylindrical interfaces, the focal surface degenerates to a surface of vanishing area and the surface is “homogeneous”. Recall from section 1.12 that homogeneous surfaces are characterised by constant Gaussian curvature. For example, the focal surface of a sphere is just the point at the centre of the sphere; the focal surface of a cylinder is the axis of symmetry along the centre of the cylinder. By analogy, the focal surfaces of a hyperbolic surface of constant Gaussian curvature are two curves of vanishing area on either side of the surface.
This situation is never realised, since a hyperbolic surface of constant Gaussian curvature cannot be immersed in three-dimensional euclidean space without singularities [6]. All hyperbolic surfaces in euclidean space have variations of Gaussian curvature over the surface. However, as long as the variations of curvature along a hyperbolic interface are small the foliation of space by parallel surfaces does nearly tile the volume (Fig. 4.4(a),(b)). We shall see that this approximation is a good one for three-periodic minimal surfaces (IPMS).
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Figure 4.4(a). A series of ellipses of increasing “homogeneity” (variation of curvature), and a circle, together with their respective focal curves (also called “evolutes”). The focal curve for the homogeneous case, the circle, degenerates to a single point at the circle’s centre.
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Figure 4.4(b). Three parallel surfaces, all hyperbolic. The focal surfaces to homogeneous hyperbolic surfaces degenerate to two curves (AB and CD) on either side of the surface.
In terms of the constituent aggregated molecules, the “quasi-homogeneity” condition – viz. small curvature variations – imposes the proviso that the variations in molecular dimensions are small. This condition implies that the surfactant parameter, which is dependent on the molecular shape, varies little throughout the aggregate. This translates naturally into chemical language: quasi-homogeneous interfaces are expected in aggregates containing an approximately monodisperse distribution of surfactant molecules.
Under this assumption, the (global) surface to volume ratio can be estimated from the (local) intrinsic geometry alone using the relations derived already for parallel surfaces. The volume is tiled without overlap or gaps by a dense foliation of parallel surfaces from the original (homogeneous) surface. For example, the (internal) volume, V, of a sphere is related to the area of the sphere, A, by the local relation (4.3), valid everywhere on the sphere,
V=Al(1−Hl+Kl23)
Since the scaled mean curvature, Hl, and the scaled Gaussian curvature, Kl2=1,
VA=R3
This is an exact value for the sphere (V=43πR3, A = 4πR2) due to its homogeneity. Recall that the surfactant parameter for a spherical interface is equal to 1/3 (eq. 4.4). This number is a measure of the average block shape in the interior of a spherical aggregate, yet it is also related to a global measure, the surface to volume ratio. This example is trivial, however it is immediately able to be generalised to more complex geometries. The same technique will be used in the next sections to derive similar data for hyperbolic surfaces.
Advances in Imaging and Electron Physics
Xingwei Yang, … Longin Jan Latecki, in Advances in Imaging and Electron Physics, 2011
2 Related Work
We mainly discuss closely related graph-based semi-supervised learning methods. A detailed survey of semi-supervised learning methods is available in Zhu (2008). Graph transduction methods have achieved state-of-the-art results in many applications. Two widely used classical methods are Gaussian fields and harmonic functions (Zhu et al., 2003) and local and global consistency (Zhou et al., 2003). In these methods, the label information is propagated to unlabeled data following the intrinsic geometry of the data manifold, which is described by the smoothness over the weighted graph connecting the data samples. With similar motivation, graph Laplacian regularization terms are combined with regularized least squares (RLS) or support vector machine (SVM) methods. These methods are denoted as Laplacian RLS (LapRLS) and Laplacian SVM (LapSVM) (Belkin et al., 2006; Sindhwani et al., 2005). The above methods can be viewed as balancing between label consistency and smoothness over the graph. Many methods use the same intuition. Chapelle and Zien (2005) use a density-sensitive connectivity distance between nodes to reveal the intrinsic relation between data. Blum and Chawla (2001) treat semi-supervised learning as a graph mincut problem. One problem with mincut is that it provides only hard classification without soft probability. Joachims (2003) proposes a novel algorithm called spectral graph transducer, which can be viewed as a loss function with a regularizer. To solve the problem of unstable label information, Wang et al. (2008) propose minimizing a novel cost function over both a function on the graph and a binary label matrix. They provide an alternating minimization scheme that incrementally adjusts the function and the labels toward a reliable local minimum. They solve the imbalanced label problem by adding a node regularizer for labeled data.
Some other works focus on different aspects of graph-based semi-supervised learning. A transductive algorithm on a directed graph is introduced in Zhou et al. (2005). Zhou et al. (2006) propose formulating relational objects using hypergraphs, where an edge can connect more than two vertices and extend spectral clustering, classification, and embedding to such hypergraphs. Nadler et al. (2009) discuss the limit behavior of semi-supervised learning methods based on the graph Laplacian.
Our proposed method is categorized as a diffusion-based semi-supervised learning method. Szummer and Jaakkola (2001) introduce a graph transduction algorithm based on the diffusion process. Szlam et al. (2008) improved the algorithm by considering the geometry of the data manifold with label distribution. However, neither approach solves the common problems with the diffusion process (see Section 1).
The most closely related work to our proposed approach on a tensor product graph is the diffusion kernel defined by Kondor and Lafferty (2002) and Vishwanathan et al. (2010). However, their construction of diffusions over the tensor product graph is completely different from the one proposed here. Moreover, Kondor and Lafferty (2002) and Vishwanathan et al. (2010) focus on defining new kernels, whereas we derive a novel semi-supervised learning framework on the tensor product graph.
Geometric Function Theory
Kenneth Stephenson, in Handbook of Complex Analysis, 2002
2.7 Classical analysis issues
We have seen several instances now of the close parallels between the discrete objects of circle packing and the familiar objects of classical analysis and conformal geometry. I would like to conclude this survey by emphasizing that the connections are a two-way street.
Circle packing brings to the classical theory a significant experimental capability, new methods of approximation, and a flexible visualization tool. It also has the potential to suggest new ideas, as it apparently did for He and Schramm, who in [55] took a major step towards solving Koebe’s Uniformization Conjecture using key ideas from their circle packing work.
In the other direction, circle packing theory has obviously benefited by following the rich classical model. That in turn has allowed it to carry notions from complex analysis into new topics. Graph embedding is a good example: any locally planar graph can be endowed with an intrinsic geometry via circle packing. In addition to ease of computation and visual appeal, these embeddings are often useful precisely because of their discrete conformal natures, for example in obtaining estimates for graph “separators” in [73], in estimating graph “resolution” in [65,4], and in proving existence of finite Dirichlet functions in [13].
Let us wrap up with a final example illustrating these mutual influences. The topic is “conformal tiling”, a notion introduced by Bowers and me in [21]. The original motivation came from the abstract combinatorial patterns of Jim Cannon, Bill Floyd, and Walter Parry (see [27]) arising in their investigation of Thurston’s Geometrization Conjecture. The pentagonal tiling of Figure 24(a) is one example.
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Fig. 24. Structure in a conformal tiling.
Circle packing began simply as a convenient embedding device for intricate graphs. Once concrete images were in hand, however, notions of discrete conformality quickly pointed to a classical formulation. The requisite theory was in fact available a century ago, but now the visual and experimental capabilities of circle packing kicked in – only with circle packing could one study specific examples in depth and develop essential insights. Unexpected structural features were observed and conjectures made; some have been confirmed using classical theory, others remain tantalizingly open.
Consider, for instance, the tiling of Figure 24(a). This is associated with a “twisted pentagonal” subdivision rule and though the circles themselves are not shown, this embedding is obtained from a circle packing. Relevant “aggregate” tiles arising in the combinatorics have been outlined in (b). Close observation suggests an underlying selfscaling, which seems to be confirmed when the outlines are rotated, scaled by λ ≈ 1.82, and overlayed in (c) – the corners of each aggregate tile fall directly on corners at the next aggregation level. Motivated by these experimental images, Cannon, Floyd, and Parry were lead first to a proof of scaling and then, with Rick Kenyon, to connections with the dynamics of rational functions leading to the exact scaling factor
λ=|25/16(5+3−15)|1/5≈1.8162516.
This example concludes our survey and perhaps shows best the potential for synergy between the familiar classical notions of analytic function theory and this new discrete realization in terms of circle packing.
Mathematical context
In Transcendental Curves in the Leibnizian Calculus, 2017
4.4.7 Conclusion
Considering Johann Bernoulli’s lectures as a whole, we may observe an overarching conflict between general, abstract methods and concrete, specific geometry. Bernoulli shows a clear preference for obtaining solutions with a direct geometrical meaning as opposed to mere abstract formulas. This is perfectly understandable since, to Bernoulli and his contemporaries, the calculus is a method for advancing geometry rather than a self-contained analytical game. It is not enough that the calculus can incorporate geometrical and physical problems and generate answers in a language internal to itself, such as differential equations and complicated integrals. It is judged rather by the geometrical end product it can generate from these expressions. And the conditions under which the desired end product is produced are quite unmistakable: abstract method, abstract answer; geometry in, geometry out.
Thus, for instance, general analytic methods are often vastly inferior in specific cases to ad hoc methods respecting the intrinsic geometry of the curve. Examples include the general formulas for arc length, involute rectification, and caustics, as well as brute-force integration generally, all of which are abandoned more often than not by Bernoulli in specific cases. Indeed, curves characterised in geometrical terms (cycloids, spirals, involutes, envelopes, etc.) tend to be eminently treatable by more tailored methods and yield simple, geometrically interpretable answers, whereas when curves are treated analytically the calculations often become unfeasible (as often happens with arc length integrals, for example) or lead to unilluminating results (such as in the case of the tangent of the cissoid).
Further strengthening this dichotomy is the tendency for the set of geometrically elegant curves (such as cycloids, spirals, etc.) to be closed under geometrically elegant operations (such as evolutes, involutes, caustics, etc.). Jacob Bernoulli (1692) observed much the same thing and was so impressed with it that he later had the logarithmic spiral—the “spira mirabilis,” as he called it—engraved on his tombstone with the inscription “eadem mutata resurgo” (“[although] changed, I appear again the same”), referring to this curve’s property of being its own caustic, its own evolute, etc. (cf. Section 4.4.4.4).
Even more remarkably, simple and elegant geometry tends to be admirably applicable; one might call it an “unreasonable effectiveness,” to borrow the famous phrase of Wigner (1960). For instance, the simple geometrical idea of the involute is remarkably powerful for rectifying curves (again, often superior to the brute-force analytical method). And curves with simple geometrical definitions again and again turn out to have physical applications, such as the cycloid as the isochrone, or the epicycloid as the caustic.
Altogether, these kinds of examples suggest that geometry possesses a sort of self-sufficiency. They provide ample reason for one to dream of a utopia in which geometrically elegant problems have geometrically elegant solutions. Indeed, the most undisputed geometrical masters, such as Archimedes and Huygens, always seemed to pluck their finest fruits from this utopia. It is understandable, then, for this to be considered the mark of a truly great geometer. And consequently it is understandable also that abstract and general analytical methods and results will have a hard time finding justification in such a setting.
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3.5 NON-EUCLIDEAN GEOMETRIES
3.5.1 Hyperbolic geometry and geometric topology
Although we are usually designing in Euclidean space, there are various examples for applications of non-Euclidean geometries in geometric modeling.
A remarkable application is the following. Consider the hyperbolic plane H2, a model of which can be realized as follows. Take a circular disk with bounding circle u. The points in the open disk are the points of the hyperbolic plane. Collinear points in hyperbolic geometry lie on circles (or straight lines) which intersect u orthogonally. Such hyperbolic straight line segments are seen in Figure 3.16, left. Hyperbolic congruences are seen in this special model as Möbius transformations which preserve u as a whole.
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Figure 3.16. Tesselation of the hyperbolic plane (left); a function which is invariant under the associated discrete group is suitable for parametrizing a closed orientable surface of genus two (right).
There are other models of the hyperbolic plane, which are more appropriate for computations. One of these is the projective model, where points and lines appear as points and line segments inside a circle u and congruence transformations are given by projective maps which preserve u as a whole.
In the hyperbolic plane, there exist remarkable discrete groups G of congruences. They possess a domain F bounded by 4g-gon (g being an integer > 2) as fundamental domain. This means that application of the elements of the group G to F generates a tiling of the hyperbolic plane. Figure 3.16, left, shows such a tiling for g = 2. It illustrates a slightly more complicated fundamental domain, which is, however, equivalent to an octogon as the group in the sense that its value f(x) at a point x ∈ H2 and at all images of x under the elements of G are the same. Then, three such functions, evaluated at the fundamental domain F, may be seen as coordinate functions of a parametric surface in 3-space. It is well-known that this surface is a closed orientable surface of genus g and that all closed orientable surfaces of genus g > 2 may be obtained via hyperbolic geometry in this way [95],[115].
This hyperbolic approach to the design of closed surfaces of arbitrary genus and smoothness has first been taken by Ferguson and Rockwood [32]. [110] have further investigated this direction and shown, for example, how to design piecewise rational surfaces with arbitrarily high geometric continuity. Although theoretically very elegant, the practical use for complicated shapes seems to be limited. Most likely, subdivision based schemes will be preferred for applications.
3.5.2 Elliptic geometry and kinematics
The intrinsic geometry of the n-dimensional Euclidean sphere Sn ⊂ En+1, with identification of antipodal points, is called elliptic geometry. Three-dimensional elliptic geometry is very closely related to spherical kinematics and has important applications in the design and analysis of motions on the sphere and in Euclidean 3-space [69]. This relation as well as applications in computer animation and robot motion planning are discussed in chapter 29.
3.5.3 Isotropic geometry and analysis of functions and images
In order to visualize the function f: D ⊂ ℝ2 → ℝ, defined on a region D of the Euclidean plane E2 = ℝ2, we usually embed this plane as (x1, x2)-plane into 3-space ℝ3 and consider the graph surface Γ(f):= {(x1, x2, f(x1, x2)) ∈ ℝ3: (x1, x2) ∈ D}. This natural procedure is sometimes followed by the seemingly natural assumption to interpret ℝ3 as Euclidean space. However, it is much more appropriate for many applications to introduce a so-space. However, it is much more appropriate for many applications to introduce a so-called isotropic metric in ℝ3. In isotropic geometry, one investigates properties which are invariant under the following group of affine mappings,
(3.17)x1’=a1+x1cosφ−x2sinφ,x2’=a2+x1sinφ+x2cosφ,x3’=a3+a4x1+a5x2+x3.
Like the Euclidean motion group in ℝ3, this group of so-called isotropic motions depends on six real parameters Φ, α1,…, α5. As seen from the first two equations in (3.17), an isotropic motion appears as Euclidean motion in the projection onto the plane x3 = 0. A careful study of isotropic geometry in two and three dimensions is found in the monographs by H. Sachs [97],[98].
The application to the analysis and visualization of functions defined on Euclidean spaces is studied in [86]. For example, the standard thin plate spline functional in two dimensions,
J(f):=∫((∂2f∂x12)2+2(∂2f∂x1∂x2)2+(∂2f∂x22)2)dx,
has a purely geometric interpretation for the graph surface of f within isotropic geometry. It is the surface integral over the sum of squares of isotropic principal curvatures x1, x2,
J(f)=∫(x12+x22)dx.
The use of isotropic geometry has been extended to functions defined on surfaces (chapter 9) rather than flat Euclidean spaces [86]. Currently, it is investigated by Jan Koenderink for understanding images of surfaces along the lines described in [49].
Isotropic geometry also appears in the context of Laguerre geometry, namely in the so-called isotropic model. For example, the oriented tangent planes of a right circular cone appear as an isotropic circle in the isotropic model. This is in general a conic, whose projection onto x3 = 0 is a Euclidean circle. Smooth spline curves formed by such conic segments could be called “isotropic arc splines”. Their construction is completely analogous to arc splines in Euclidean 3-space. The transformation back to the standard model of Laguerre geometry gives developable surfaces, which consist of smoothly joined pieces of right circular cones [55]. Geometric computing with these cone spline surfaces rather than general developables has a variety of advantages: The computation of bending sequences and the planar development can be performed in an elementary way. The degree, namely two for both the implicit and parametric representation of the segments, is the lowest possible for generating smooth surfaces, and the offsets are of the same type [54],[56].
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