A transformation or representation which preserves angles and the shape of infinitesimal 

 figures is called isogonal; if the angles are also not turned over, it is called conformal. Mer- 

 cator's projection represents a conformal mapping of the earth's surface on a plane 



A figure of finite size, however, does not usually retain its shape under a transforma- 

 tion, since the change of scale and the rotation are usually different at different points, be- 

 cause of variation in the value of dw/dz. 



The angle between two curves may fail to be preserved if they intersect either at a 

 singular point, where df/dz does not exist, or at a point at which df/dz = 0. 



It should be noted that as 82 is rotated in direction, by changing its amplitude, Sw 

 rotates in the same direction, and by an equal amount. Hence, as the 2-point traverses a curve 

 in a certain direction and the w-point traverses the corresponding curve on the to-plane, the 

 area on the left-hand side of one curve corresponds to that on the left-hand side of the other, 

 and the area on the right of one to the area on the right of the other. For example, in Figure 23, 

 rotating 8z as shown off the curve and toward the area S causes Sw to rotate toward the area T; 

 this shows that points lying near the curve and in S transform into points in T. Similarly, 

 nearby points in U transform into points in V. This rule is very useful in the study of con- 

 formal mapping. 



The transformation can also be viewed from the inverse standpoint, as a mapping of the 

 wj-plane on the 3-plane by means of the inverse transformation, 



2 ^ F{w) 

 where F is the inverse function obtained by solving Equation [23a] for z. Then 



X ^ X ((fi^il,), y = y(<;6,i/,) 



Two families of curves on the z-plane that are of particular interest are those defined 

 by (f>(x,y) = constant and ip(x,y) = constant. From the conformal property of the transformation, 

 it follows that these two families of curves intersect orthogonally, wherever dw/dz is finite 

 and not zero, as illustrated in Figure 29, page 48. For, this is obviously true of the corres- 

 ponding curves on the w-plane, which are straight lines parallel to the axes. The orthogonality 

 can also be verified directly from Equations [22b, c]. 



Figure 23 - The correspondence of regions adjoining a curve in conformal mapping. 



41 



