define the modified signed distance as a series of v.

Define w = foo45 as a function of v below.

Then consider the inversion v=foo46 as a function of w  below.

The relations between the two sets of the coefficients are given below.

The relation is actually obtained by solving the following coefficients==0.

Checking if the relation is correct by seeing the identity.

Consider the following

Since func48[v] = func48[w] + , and we can ignore the difference between func48[v] and func48[w]. So, if we redefine w = foo45+func48[v], and v=foo46-func48[w], the inversion relation still holds.  We call "v" as the signed distance, and "w" as a modified signed distance characterized by the coefficients cr[r] and br[r].

Jacobian of the transformation from v to w is given below. Here D[func48[w],w] is denoted as

We need the log of the Jacobian for later use. logjvw=log .

Similarly, we write func48[w] as and vinuw is v expressed by u and w.