  A finite unit norm tight frame is a collection of $r$ vectors in
$\mathbb{R}^n$ that generalizes the notion of orthonormal bases. The affine
finite unit norm tight frame variety is the Zariski closure of the set of
finite unit norm tight frames. Determining the fiber of a projection of this
variety onto a set of coordinates is called the algebraic finite unit norm
tight frame completion problem. Our techniques involve the algebraic matroid of
an algebraic variety, which encodes the dimensions of fibers of coordinate
projections. This work characterizes the bases of the algebraic matroid
underlying the variety of finite unit norm tight frames in $\mathbb{R}^3$.
Partial results towards similar characterizations for finite unit norm tight
frames in $\mathbb{R}^n$ with $n \ge 4$ are also given. We provide a method to
bound the degree of the projections based off of combinatorial~data.
