### Maxillo-Facial Frames

There are eight nodes in the Treil maxillo-facial frame [4, 5]. Only one shape function will be evaluated in the `(x,y,z)` coordinates system following a suitable method, such as Taig's isoparametric formulation [6] or Wachspress' rational polynomial interpolation [7, 8]. (Mathematica coding of these two popular and effective procedures are included in the appendix.) This initial shape function in terms of `(x,y,z)` will be indicated by the special symbol `s[0]`, and the associated coordinates will be denoted by `(x0,y0,z0)`. Here the string `"0"` reflects the starting choice for the proposed algorithm. For the eight-node maxillo-facial frame the remaining seven interpolants, `Array[s,7]`, will be calculated to reproduce the set of exact fields denoted by `f` as a list of functions in `(x,y,z)`. In particular, here: `f={1,x,y,z,x*y,y*z,z*x}`. Its algebraic relations are crucial to this presentation, as explained below.

Consider a pre-assigned field `u[x,y,z]` to be an element of the list `f`. With the prescribed nodal values `u[x[i],y[i],z[i]]`, the set of shape functions `s[i]` in terms of `(x,y,z)` should lead to

The aforementioned relation will be exactly satisfied in terms of the coordinate variables `(x,y,z)`. Hence the name "exact interpolated fields" for `f`.

#### Mathematica Example

The following code generates the coefficient matrix `mat` and the right-hand side `vec` for `LinearSolve`:

Execution

A Symmetric Case

The anatomical points are symmetrically located on the left and right side of the face. The canonical maxillo-facial frame is then symmetric about x-axis. Hence the following is a specialization for which a symmetric frame is generated according to the rule.

Now the coefficient matrix and the right-hand side vector, `matr`, `vecr`, respectively, become:

Before attempting to solve for the unknown shape functions, the consistency of the assumption of the interpolated field should be established.

In general, the nonzero determinant signifies that the assumption:

could be acceptable and the matrix `matr` is indeed invertible.

For given numerical data for the nodal coordinates, the shape functions will be obtained from the expression below, after removing the comments `(*` and `*)`.

#### Failure to Interpolate: --An Example of Inconsistent Data

The Symmetric Maxillo-facial Frame

Hence the following evaluation is not attempted since the coefficient matrix is not invertible.