CONFORMAL REPRESENTATION. Conformal geom etry had its origin in the practical problem of so mapping the earth's curved surface upon a flat leaf of paper that differences of directions at any point of the surface shall be indicated by equal differences of direction at the corresponding point on the map. By the term conformal representation (also called isog onal, orthomorphic) is under stood any continuous mapping of one surface or region upon an other, with a one-to-one correspondence of their points, and in such manner that corresponding angles in the two surfaces or regions are equal. The term conform is due to Gauss, who referred thereby to the equivalent property that correspond ing infinitesimally small triangles on the two surfaces tend to conform, i.e., approach similarity, when their dimensions are in definitely diminished. Thus suppose ABC to be a small triangle on one surface formed by three intersecting curves, and A'B'C' the corresponding triangle on a second surface. As B and C approach A along BA and CA, correspondingly B' and C' will approach A' along B' A' and C' A' . When the transformation is conformal, the sides of the rectilinear triangles ABC and A'B'C' tend to become proportional, and the three angles of the two triangles to be respectively equal. Consequently the angle between the two curves meeting at A must be equal to the angle between the two corresponding curves at A'. A net of infinitesimal triangles closing around A so as to make up a curvilinear polygon enclosing A, is transformed into a similar polygon around A'. The ratio of mag nification in passing from the triangles at A to those at A' varies in general with the position of A.
The earliest known conformal representation is the stereogra phic projection of Hipparchus (c. 14o B.c.) and Ptolemy (C. A.D. 150). Another early conformal representation was afforded by the familiar Mercator's projection (1568). (See MAP: Projec tions.) Lambert (1772) was the first to seek the general conformal representation of the earth's surface on a plane, and Lagrange first noted its connection with functions of the complex variable. In 1822 the Society of Sciences at Copenhagen proposed as prize subject "the general solution of the problem to so build the parts of on.e given surface upon another given surface that the image shall be similar in its smallest parts to the (surface) imaged." The solution of this problem was obtained by Gauss and pub lished in 1825 in a memoir (Collected works, vol. 4) which marks the beginning of a general theory of conformal representation. The next great step was due to Riemann (1859), and among the important contributors in the last so years Schwarz and Klein should be particularly mentioned.
In his famous Erlanger Program, Klein revealed how the dif ferent kinds of geometry are associated each with a different collection or group of transfor mations. Each particular geom etry is a study of the properties and theorems invariant under ap plication of all transformations of the group. Ordinary Euclidean geometry rests on the so-called principa/ group, which results in combining in all possible ways translations, rotations, magnifica tions and reflections, and contains the theorems which are unaltered by these transformations. If with this group we combine all inversions of the plane with re spect to its circles, we obtain a special conformal group, known as the group of reciprocal radii.
Consider a continuous reversible point-to-point transformation of the interior of a region of the ordinary (x,y)-plane into the interior of a region of the (u,v)-plane, by means of the functions u=u(x,y), v=v(x,y) provided with continuous first derivatives.
It is easily established (see W. F. Osgood, Lehrbuch der Funk tionentheorie) that the necessary and sufficient condition that the two regions shall thereby be built conformally one on the other, without change of angle-sense, is that u and v shall satisfy the au Cauchy-Riemann differential equations, au = , — = — ax ay ay ax the conformality ceasing only in the points where these derivatives are all zero. In other words, w=u(x,y)-F-iv(x,y) must be an analytic function of z=x+iy in the region considered. Conformal geometry enters, therefore, into the very warp and woof of the theory of analytic functions. To obtain a conformal transforma tion we have only to select any analytic function, w=f(z)—for instance, zn, ez, or sin z,—with restricted or unrestricted domain, and split the function into its real and imaginary parts, f(z) _ u(x,y)-+iv(x,y). The equations u=u(x,y), v=v(x,y) then map any portion of the z-domain, in which f(z) is analytic, into a cor responding domain of the w-plane. The conformality ceases only at the isolated points for which f'(z) =o. The w-region may over lap itself if too large a portion of the z-plane is taken, giving rise then to an overlapping Riemann surface.
By the analytic transformation w= f (z) just indicated, families of orthogonal straight lines x=c,, are mapped into orthogonal families of curves in the (u,v)-plane, and likewise the curves u = v = into an orthogonal system in the (x,y) -plane. For example, if f (z) _ we get v= 2xy, and the orthogonal system u i- v= consisting of straight lines parallel to the axes of the (u,v)-plane, is mapped into an orthog onal system of equilateral hyperbolas, This is perhaps the simplest method of securing orthogonal systems of curves, which are of fundamental importance in hydrodynamics, electricity and potential theory where the curves of one system are interpreted as lines of force or stream lines, and those of the orthogonal system as equipotential or level lines.
The uniqueness of the conformal transformation of the region (R) into the circle or its equivalent, the half-plane, can be secured in various ways, for instance, by requiring that three boundary points of (R) shall pass over into three preassigned perimeter points of the circle or half-plane. The analytic function w=f(z) which effects the transformation is thereby uniquely defined. For example, the requirement that a rectangle shall be built upon the half-plane of z above its real axis defines the ana lytic function in which es, e,, are the points on the real z-axis (three of them arbitrarily selected) which correspond to the vertices E,, E, of the rectangle and at which the conformality of the representation breaks down. Properties of the function w=f(z) manifest themselves in the shape of the region (R), and accordingly conformal representation can be used for their study.
In particular, it has been an invaluable aid in the development of the theory of hypergeometric, elliptic modular, and automorphic functions.
Conformal Representation of One Surface upon Another. —It is easily established that any ordinary surface can be built conformally upon any other with, of course, the admission of points where the conformality ceases. For suppose the position of the points on one of these surfaces to be fixed by x = f, (u ,v), xs = , v), x3 = f 3(u , v) ; then the formula for the length of arc on the surface has the form = = 2Fdu dv in which E, F, G are functions of u and v. Let be Edud-Fdv± resolved into its factors E , which the quantity under each radical is necessarily positive, inasmuch as is positive irrespective of the relative values of du and dv. Let µ(u, F iX(u , v) be the integrating factors by which we must multiply these two factors to make them complete differ entials, and denote these differentials by d4(u , v)-± i 1'(u , v) ; then takes the form 4 + d2 . If , lk are plotted as ordinary x2 +µ rectangular co-ordinates in a plane, the corresponding arc in the plane is I Since ds and are proportional, it follows that the surface is built conformally upon the plane. Similarly, the other surface can be built upon the plane. Then, since by Riemann's theorem any two simply connected pieces of the plane can be built conformally the one on the other, it fol lows that the two surfaces can be mapped conformally on each other, as stated above. When isothermic co-ordinates 4) op are used to fix the position of a point on one surface and likewise isothermic co-ordinates cki, tp, on the other, 0+i t' is an analytic function of •i-Filki.
A special case of note is that in which corresponding infinitesi mal arcs ds, on the two surfaces are not merely proportional but equal. The two surfaces are then applicable one to the other; i.e., they can be applied one to the other without stretching, some what as we would roll a leaf of paper into a cylinder. Another important case is that in which a minimal surface is built con formally on a sphere. This is accomplished very simply by draw ing the normal at each point P of the minimal surface, and then taking, as the corresponding point P' on the sphere, the point in which it is cut by a ray drawn through the centre of the sphere, parallel to the normal and like directed. In this and other cases the conformal representation is of value in studying the surface.
Conformal Representation for a Space of Three or More Dimensions.—Quite contrary to what might be naturally ex pected, the conclusions for two dimensions cannot be carried over bodily to a space of three or more dimensions. According to a notable theorem of Liouville the group of all conformal trans formations for n> 3 is identical with the group of reciprocal radii, that is, with the group resulting from combination of the principal group of rigid transformations (translations, rotations, magnifications and reflections) with inversions with respect to hyperspheres E (x, — a,) 2 = (i 2 • • n) . Accordingly there is nothing analogous to the conformal representation of a region of the z-plane upon a w-region by means of an arbitrarily selected analytic function w=f(z).
Klein has pointed out a second noteworthy fact which is also without analogue in two dimensions, viz., for n> 2 the conformal geometry of an Euclidean space of n-dimensions is identical with the projective geometry of a hypersphere in n-1-1 dimensions which is transformed into itself projectively in the most general manner. Various topics have been treated in conformal geometry of n-dimensions such as the possibility of referring non-Euclidean to Euclidean space conformally, but the development must be regarded as fragmentary.