x and y are coordinates along the 'asymptotic lines' where the normal curvature is zero, and the following are the three coordinates of the corresponding point on the hyperbolic square. Note that the derivative with respect to y along the x-axis is (0, cos(x), sin(x)), showing that the surface is rotating at uniform spped as we move along the axis.
x
– 1/2! xy2
+ 1/4! xy4
+ 1/3·3! x3y4 – 1/6! xy6
– 1/30·3! x3y6 + 1/8! xy8
– 1/25·3! x5y6 + 1/7! x3y8 – 1/10! xy10
+ 1/63·4! x5y8 – 1/54·7! x3y10 + 1/12! xy12
+ 157/33075·3! x7y8 – 1/42525·1! x5y10 + 1/66·9! x3y12 – 1/14! xy14
– 2/25515·1! x7y10 + 13/5940·7! x5y12 – 1/78·11! x3y14 + 1/16! xy16
– 754/8037225·1! x9y10 + 67/200475·5! x7y12 – 23/117·11! x5y14 + 1/90·13! x3y16 – 1/18! xy18
+ 10957/9823275·5! x9y12 – 3151/12162150·7! x7y14 + 53/225·13! x5y16 – 1/102·15! x3y18 + 1/20! xy20
y
– 1/2! x2y
+ 1/4! x4y
– 1/6! x6y + 1/3·3! x4y3
+ 1/8! x8y – 1/30·3! x6y3
– 1/10! x10y + 1/7! x8y3 – 1/25·3! x6y5
+ 1/12! x12y – 1/54·7! x10y3 + 1/63·4! x8y5
– 1/14! x14y + 1/66·9! x12y3 – 1/42525·1! x10y5 + 157/33075·3! x8y7
+ 1/16! x16y – 1/78·11! x14y3 + 13/5940·7! x12y5 – 2/25515·1! x10y7
– 1/18! x18y + 1/90·13! x16y3 – 23/117·11! x14y5 + 67/200475·5! x12y7 – 754/8037225·1! x10y9
+ 1/20! x20y – 1/102·15! x18y3 + 53/225·13! x16y5 – 3151/12162150·7! x14y7 + 10957/9823275·5! x12y9
xy
– 1/3! x3y – 1/3! xy3
+ 1/5! x5y – 1/2·3! x3y3 + 1/5! xy5
– 1/7! x7y + 1/2·4! x5y3 + 1/2·4! x3y5 – 1/7! xy7
+ 1/9! x9y – 1/36·4! x7y3 + 29/4·6! x5y5 – 1/36·4! x3y7 + 1/9! xy9
– 1/11! x11y + 1/48·6! x9y3 – 43/24·6! x7y5 – 43/24·6! x5y7 + 1/48·6! x3y9 – 1/11! xy11
+ 1/13! x13y – 1/60·8! x11y3 + 19/192·6! x9y5 – 1447/30·8! x7y7 + 19/192·6! x5y9 – 1/60·8! x3y11 + 1/13! xy13
– 1/15! x15y + 1/72·10! x13y3 – 71/28800·6! x11y5 + 25699/24·10! x9y7 + 25699/24·10! x7y9 – 71/28800·6! x5y11 + 1/72·10! x3y13 – 1/15! xy15
+ 1/17! x17y – 1/84·12! x15y3 + 17/96·10! x13y5 – 4727/80·10! x11y7 + 2075513/4032·10! x9y9 – 4727/80·10! x7y11 + 17/96·10! x5y13 – 1/84·12! x3y15 + 1/17! xy17
– 1/19! x19y + 1/96·14! x17y3 – 11/6720·10! x15y5 + 4237/2880·10! x13y7 – 5119691/40320·10! x11y9 – 5119691/40320·10! x9y11 + 4237/2880·10! x7y13 – 11/6720·10! x5y15 + 1/96·14! x3y17 – 1/19! xy19
+ 1/21! x21y – 1/108·16! x19y3 + 113/80640·12! x17y5 – 17747/846720·10! x15y7 + 10170437/1451520·10! x13y9 – 2436660073/302400·12! x11y11 + 10170437/1451520·10! x9y13 – 17747/846720·10! x7y15 + 113/80640·12! x5y17 – 1/108·16! x3y19 + 1/21! xy21