perpendiculars MP and MQ upon the coordinate axes, then the Lobachevski an coordinates of M are
X
=
O
P
,
Y
=
M
P
{\displaystyle X=OP,\ Y=MP}
Those of Weierstrass are
x
=
sh
M
Q
,
y
=
sh
M
P
,
l
=
ch
O
M
{\displaystyle x=\operatorname {sh} \,MQ,\ y=\operatorname {sh} \,MP,\ l=\operatorname {ch} \,OM}
or
(21)
x
=
sh
ξ
,
y
=
sh
η
,
l
=
ch
r
.
{\displaystyle x=\operatorname {sh} \,\xi ,\ y=\operatorname {sh} \,\eta ,\ l=\operatorname {ch} \,r.}
From the quadrilateral OPMQ of three right angles we obtain
x
=
sh
X
ch
Y
{\displaystyle x=\operatorname {sh} \,X\ \operatorname {ch} \,Y}
in addition we have
(22)
y
=
sh
Y
,
l
=
ch
X
ch
Y
,
{\displaystyle {\begin{array}{l}y=\operatorname {sh} \,Y,\\l=\operatorname {ch} \,X\ \operatorname {ch} \,Y,\end{array}}}
and thus the Weierstrass coordinates are expressed by the Lobachevski an ones.
In the general case we have Fig. 5. N, R, S are the foot points of the three perpendiculars ξ η, ζ of M upon the coordinate plane, then
X
=
O
P
,
Y
=
P
N
,
Z
=
N
M
{\displaystyle X=OP,\ Y=PN,\ Z=NM}
are the Lobachevski an, and
(23)
x
=
sh
ξ
=
sh
X
ch
Y
ch
Z
y
=
sh
η
=
sh
Y
ch
Z
z
=
sh
ζ
=
sh
Z
,
l
=
ch
r
=
ch
X
ch
Y
ch
Z
{\displaystyle {\begin{array}{l}x=\operatorname {sh} \,\xi =\operatorname {sh} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,Z\\y=\operatorname {sh} \,\eta =\operatorname {sh} \,Y\ \operatorname {ch} \,Z\\z=\operatorname {sh} \,\zeta =\operatorname {sh} \,Z,\\l=\operatorname {ch} \,r=\operatorname {ch} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,Z\end{array}}}
are the Weierstrass coordinates of point M .
From the quadrilateral MNRT we have[ 1]
sh
ξ
=
sh
N
T
ch
Z
{\displaystyle \operatorname {sh} \,\xi =\operatorname {sh} \,NT\ \operatorname {ch} \,Z}
while we obtain from OPNT the equation
sh
N
T
=
sh
X
ch
Y
{\displaystyle \operatorname {sh} \,NT=\operatorname {sh} \,X\ \operatorname {ch} \,Y}
From these two relations we obtain the expression for x . From the quadrilateral MNPS we easily find the value for y . The limiting arcs MA, MB and MC are our x, y, and z . We find in addition
l
2
−
x
2
=
ch
2
Y
ch
2
Z
{\displaystyle l^{2}-x^{2}=\operatorname {ch} ^{2}Y\ \operatorname {ch} ^{2}Z}
↑ F. Engel , Nikolaj Iwanowitsch Lobatschefskij. Zwei geometrische Abhandlungen, 1898, p. 347