Elliptic Integrals - Map Projections: Cartographic Information Systems - page 697

x =sn u =cn u dn u +cn u dn u,

(C.26)

k 2 sn u cn 2 u,
x = −x (1 − k 2 x 2 ) − k 2 x (1 − x 2 )= −x + k 2 x 3

x = sn u dn 2 u

− −

− k 2 x + k 2 x 3 ,

x = x (1 + k 2 )+2 k 2 x 3

− q . e . d .

End of Proof (Proof of formula ( C.15 )).

Proof (Proof of formula ( C.17 )).

y =cn u =

− sn u dn u

⇒

y =cn u = sn u dn u sn u dn u, − −

y = cn u dn 2 u + k 2 sn 2 u cn u,

− (C.27)

y = −y [1 − k 2 (1 − y 2 )] + k 2 y (1 − y 2 )= −y + k 2 y (1 − y 2 )+ k 2 y (1 − y 2 ) ,
y =

y +2 k 2 y (1 y 2 )= y +2 k 2 y 2 k 2 y 3 ,

− − − −

y = 2 k 2 ) 2 k 2 y 3 − y (1 − − q . e . d .

End of Proof (Proof of formula ( C.17 )).

Proof (Proof of formula ( C.19 )).

z =dn u = k 2 sn u cn u −

⇒

z = k 2 sn u cn u k 2 sn u cn u, − −

(C.28)

z =

k 2 cn 2 u dn u + k 2 sn 2 u dn u,

−

z = [ k 2 z 2 )] z +(1 z 2 ) z = k 2 z +2 z (1 z 2 ) ,

− − (1 − − − −

z =+(2 k 2 ) z 2 z 3

− − q . e . d .

End of Proof (Proof of formula ( C.19 )).