What defines motion before teeth are added?

01The articulator before the casts are mounted

Imagine an articulator with no casts mounted. Its left and right condylar elements are engaged; its incisal pin rests on its table. Which relative positions of the frames are compatible with those three relations? The question makes sense before anyone chooses the shape of a tooth.

Salikopoulos calls this arrangement the Mechanical Bonwill Triangle. A drawing can show where the elements lie. To decide what a frame can do, we also need to know which relations the mechanism maintains as it moves. Here, “closure” refers to those conditions on admissible configurations; it does not mean bringing teeth into occlusion.

02Where the three constraints come from

The anatomical Bonwill triangle relates the two condyles and the mandibular incisal point. The anterior reference in the paper’s mechanical construction is instead the relation between the incisal pin and its table. In the contact regime considered here, configurations must respect that contact alongside left and right condylar guidance. If the pin loses contact, continuing to impose the same equality requires a new justification.

A primitive source is internal to the instrument, active in defining motion, unchanged by prosthetic reconstruction and independent of the configuration being constructed in the sense that it is not derived from it (§6, Definition 2). Constructed teeth may later introduce contacts. A facebow transfer can locate a cast relative to the frame. Neither supplies the pre-existing mechanical pin–table relation.

03What does the anterior relation exclude?

For both comparisons use the same candidate configuration space X with the same instrument settings, before imposing any of the three equalities. In particular, pin contact must not already be built into X. Let Z_L, Z_R and Z_I be the sets where the left, right and incisal constraint functions respectively vanish (§3.1). Write the posterior set, then the set satisfying all three conditions:

Every state in M belongs to B. To show that adding the anterior relation actually reduces this set, exhibit a state in B outside Z_I. If no such state exists in the declared space, the third condition already follows from the first two there. The intersection describes permitted configurations; it does not select a trajectory through them.

04Sufficiency, minimality and rank

In §6, Definitions 2–4 start from physical sources. A collection is sufficient if it determines the admissible domain; it is minimal if removing any source destroys that sufficiency. When the criterion is recovering the same set M, dropping the anterior relation tests only one deletion. For each of the other two, we also need a configuration satisfying the retained pair that violates the omitted relation.

Section 7, Definitions 8–10, adds different requirements: independent differentials on the admissible set, a bounded and well-defined domain, and loss of closure on reducing the number of constraints. Theorems 7 and 9 identify the minimal structure within the paper’s class, with bilateral posterior guidance and its specified admissible anterior primitive. Counting three contact points does not establish those premises for another mechanism.

05Three equations, one free coordinate

Dropping the left relation admits (1, 0, 0, 0); dropping the right admits (0, 1, 0, 0). Together with the anterior deletion above, these states establish minimality for reproducing this particular set. In fact each deletion frees one whole real coordinate, so the remaining set is also unbounded in this example.

The three coordinate equations have differential rank three. The segment is bounded because the choice of X already limits t. Keep the same equations and take X = ℝ⁴: the rank stays three and the segment becomes the whole unbounded t-axis. Rank alone guarantees neither global bounds nor a unique pose.

06Returning to an instrument

Imagine mounting the casts. If a tooth contact now prevents a configuration allowed by the guides, the new surface restricts motion without becoming the original anterior reference. If the pin lifts off the table, we need to state the contact conditions afresh before carrying the calculation forward.

For the Artex CN, the next question is how a particular anterior guide realizes the declared relation. Establishing that the pin remained in contact throughout a given movement would require observations of that instrument under stated settings. The Re-audit’s distinction between admissibility and realization explains why the equations alone cannot answer it. The next study, CROR: functional compatibility, asks when anterior guidance permits continued posterior participation during motion.

07Source and reading locations

Site study · 7 November 2026. The set comparison and coordinate example are an explanatory development for this chapter. They report no bench measurements.