Learning a language starts immediately after birth in the form of repeating basic sounds and gestures that are generated by adults (usually the parents). While teaching is achieved by initial pronunciation through exaggerated gestures and sounds, learning is accompanied by memorizing and comprehension and eventually reproduction of such gestures and sounds. In fact, parents exaggerate speech by breaking up it into lower accepted (to babies) sound and gesture levels that are more accepted by babies. It is the aim of this paper to demonstrate methods in which fundamental articulated phonemes is represented by signatures that reflect dynamic mouth movement contours. The paper starts by explaining the basic (yet limited) Lagrange Interpolation Method to produce fundamental signatures of lip movements by tracking upperlip and corner-lip feature-points. Then the paper proposes a method that produces more compact polynomials by using the Barycentric Lagrange Interpolation Method, which overcomes any limitations of the earlier method.