the expression for the surface in the local coordinates

We use for the -coordinate of the projection.

Define local coordinates at the projection, and denote . The local coordinate is in the direction.

First, separate the type-a and 9 indices in the local parametrization at the projection. in foo90 indicates the projection point, whereas foo90 itself indicates the -coordinates for a general point.

foo91 is foo90 for a=a'.

foo92 is foo90 for a=9.

Then, is expanded by its expression. is now substituted by to change the origin to the projection. Here we obtain and .

We will equate foo93 with foo95, and foo94 with foo96 below to derive the expression of the surface in the local coordinates. foo95 and foo96 define the surface with the origin at 0. First we consider direction.

Thus, = foo93 as a function of . We make it rule95.

Consider direction.

Using the rule95, we get an expression of foo96 in terms of 's.

Equate foo94⩵foo97 to solve the expression of in terms of .

The following foo99 gives an expression of in terms of .

get the coefficients of and    for .

This is at the projection. We denote it  =foo101.

This is at the projection. We denote it = foo102.

Now the surface is expressed in the local coordinates as .