[From the Proceedings of the London Mathematical- Society, Vol. n.] 



XXXII. The Construction of Stereograms of Surfaces. 



To make a surface visible, lines must be drawn upon it ; and to exhibit 

 the nature of the surface, these lines ought to be traced on .the surface ac- 

 cording to some principle, as for instance the contour lines and lines of greatest 

 slope on a surface may be drawn. Monge represents surfaces to the eye by 

 their two systems of lines of curvature, which have the advantage of being 

 independent of the direction assumed for the co-ordinate axes. For stereoscopic 

 representation it is necessary to choose curves which are easily followed by the 

 eye, and which are sufficiently different in form to prevent a curve of the 

 one figure from being visually united with any other than the corresponding 

 curve in the other figure. I have found the best way in practice to be as 

 follows : First determine how many curves are to be drawn on the surface, 

 and at what intervals, and find the numerical values of the co-ordinates of 

 points on these curves. A convenient method of drawing the figures according 

 to Cartesian co-ordinates is to draw an equilateral triangle, find its centre of 

 gravity, and take a point about ^ of the side of the triangle distant hori- 

 zontally from the centre of gravity as the origin, and the lines joining this 

 point with the angles as unit axes. For the other figure, the point must be 

 taken on the opposite side of the centre of gravity. I have found that this 

 rule gives a convenient amount of relief to the figures. When the co-ordinates 

 of a point are easily expressed in terms of tetrahedral co-ordinates x, y, z, u\ 

 I draw the two lines (x = 0, y = 0) and (2 = 0, w = 0) in both figures. By means 

 of a sector I. divide the first line in P in the ratio of z to w, and the second 

 in Q in the ratio of x to y ; I then lay a rule along the line PQ (without 

 drawing the line) and divide PQ in the ratio of x + y to z + if>, in order to find 

 the point R in the figure. By drawing the two lines once for all in each 

 figure, and performing the same process of finding ratios in each figure, we get 

 each point without making any marks on the paper, which come to be very 

 troublesome in complicated figures. In this way I have drawn the figures of 

 cyclides, &c. 



