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Fig. 3 | Progress in Orthodontics

Fig. 3

From: The inflection point: a torque reference for lingual bracket positioning on the palatal surface curvature of the maxillary central incisor

Fig. 3

x1CP, x2CP, x3CP, and x4CPstand respectively for the x coordinate value of the first, second, third, and fourth control points of the respective cubic poly-Bezier curve (first or second). y1CP, y2CP, y3CP, and y4CP stand respectively for the y coordinate value of the first, second, third, and fourth control points of the respective cubic poly-Bezier curve. The parameter t in all the equations below lies between 0 ≤ t ≤ 1. The first and second equations are the equations of the x and y components of the cubic Bezier curve respectively, while the third and fourth equations are their first derivatives, respectively, and the fifth and sixth equations are the second derivatives. The position of the inflection points of a parametric cubic Bezier curve are among the solutions of the equation: B(tX B(t) = B′x(t) • B″y(t) ‐ B′y(t) • B″x(t), where B′(t) and B″(t) stand for the first and second derivative vectors, respectively, of Bezier curve and X stands for the cross product between the two vectors. It should be noted that Eq. 7 was written under these two considerations: the inflection point of PSC in this study was always located in the first cubic poly-Bezier and P1, the first control point of the first cubic poly-Bezier, has an x and y coordinate equal to zero (any term multiplied by zero is eliminated)

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