Abstract
The center of resistance is a concept in theoretical orthodontics used to describe tooth movement under loads. It is commonly used to qualitatively predict tooth movement without recourse to complex equations or simulations. We start with a survey of the historical origin of the technical term. After this, the periodontal ligament is idealized as a linear elastic suspension. The mathematical formalism of vector and tensor calculus will clarify our reasoning. We show that a point such as the center of resistance basically only exists in two dimensions or in very special symmetric spatial configurations. In three dimensions, a simple counterexample of a suspension without a center of resistance is given. A second more tooth-like example illustrates the magnitude of the effects in question in dentistry. In conclusion, the center of resistance should be replaced by a newer and wider mathematical concept, the “center of elasticity,” together with a limiting parameter, the “radius of resistance.”
Highlights
IntroductionCenter of resistance (CR) is a concept borrowed from the field of mechanics of rigid bodies. It was first introduced into dentistry by G. D. Fish in 1917. Analyzing the tooth movement of a single-root tooth in the mesial plane, he argued, Let us suppose that a horizontal force is to be applied toward the cheek. There is a point C somewhere between the apex and the gingival margin, such that if the force be applied on a line passing through that point there will be no tipping and no rotation of the tooth about its long axis. This point may be called the center of resistance. It is undoubtedly somewhere between the gingival margin and the point half way to the apex. Of course this point may vary with variation of the intensity of force or with change in direction of application of the force.[1]
Reaction forces can be derived from an elastic potential, and the flexibility matrix is symmetric. This completes the proof for the existence of a Center of resistance (CR) using a linear, two-dimensional tooth model
Understanding the flexibility matrix as the outcome of a measurement or a numerical simulation, this assumption of symmetry does not need to be true and a CR would not exist in such a case
Summary
Center of resistance (CR) is a concept borrowed from the field of mechanics of rigid bodies. It was first introduced into dentistry by G. D. Fish in 1917. Analyzing the tooth movement of a single-root tooth in the mesial plane, he argued, Let us suppose that a horizontal force is to be applied toward the cheek. There is a point C somewhere between the apex and the gingival margin, such that if the force be applied on a line passing through that point there will be no tipping and no rotation of the tooth about its long axis. This point may be called the center of resistance. It is undoubtedly somewhere between the gingival margin and the point half way to the apex. Of course this point may vary with variation of the intensity of force or with change in direction of application of the force.[1]
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