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Difference between revisions of "Transverse Hinge Axis"
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By substituting <math>s</math> and <math>h</math> with the values obtained from Geogebra, we obtained a maximal error in the localization of Center <math>0_1</math> of approximately <math>\cong4.95</math> <small><math display="inline">mm</math></small> at a percentile of <math>72%</math> (see Figure 6). Measuring the distance with Geogebra between centers <math>0_1</math> and <math>0</math>, it results in <math>\Delta = 1.73</math> <small><math display="inline">mm</math></small>, which corresponds to an error of 30% compared to the maximal value of <math>\cong4.95</math>. This means that visually and manually locating the center of rotation can result in an error of about <math>\cong1.73</math> <small><math display="inline">mm</math></small> compared to a mathematical calculation on the mandibular opening and closing trace. | By substituting <math>s</math> and <math>h</math> with the values obtained from Geogebra, we obtained a maximal error in the localization of Center <math>0_1</math> of approximately <math>\cong4.95</math> <small><math display="inline">mm</math></small> at a percentile of <math>72%</math> (see Figure 6). Measuring the distance with Geogebra between centers <math>0_1</math> and <math>0</math>, it results in <math>\Delta = 1.73</math> <small><math display="inline">mm</math></small>, which corresponds to an error of 30% compared to the maximal value of <math>\cong4.95</math>. This means that visually and manually locating the center of rotation can result in an error of about <math>\cong1.73</math> <small><math display="inline">mm</math></small> compared to a mathematical calculation on the mandibular opening and closing trace. | ||
<br /> <Center> <gallery widths="350" heights="282" perrow="2" mode="slideshow"> File:Hinge axis 1.jpg|''' | <br /> | ||
<Center> | |||
<gallery widths="350" heights="282" perrow="2" mode="slideshow"> | |||
File:Hinge axis 1.jpg|'''Figura 4:''' Tracciati assiografica di apertura e chiusura della bocca in retrusione forzata ( arco di cerchio a sinistra della finestra) e guidata ( sottostante). In seguito verranno descritti più dettagliatamente i tracciati. | |||
File:Figura 6a.jpg|'''Figura 5:''' Determinazione dei parametri <math>s</math> ed <math>h</math> necessari per generare un centro di rotazione. | |||
File:Figura 6b.jpg|'''Figura 6:''' Discrepanza tra asse cerniera generato visivamente dall'operatore e matematicamente in Geogebra. | |||
</gallery> | |||
</Center> | |||
===Mathematical Formalism: Localization Error of HA from Chord <math>s</math> and Sagitta <math>h</math>=== | ===Mathematical Formalism: Localization Error of HA from Chord <math>s</math> and Sagitta <math>h</math>=== | ||
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Figure 7 illustrates the dependency of the hinge axis determination on the distance of the measurement position from the center of rotation. It also highlights that a measurement close to the TMJ (with a small radius) does not offer greater accuracy with the circle fitting method compared to a measurement taken further from the TMJ or closer to the jaw (e.g., at the incisal point). On the contrary, radii smaller than <math>20</math> <small><math display="inline">mm</math></small> significantly increase the imprecision in determining the hinge axis. This result is positive in a clinical context, as it indicates that, even in the presence of noise (as can happen in real measurements), it is possible to accurately estimate the <math>_tHA</math> and the associated radius, which are crucial for patient diagnosis and rehabilitative treatment. | Figure 7 illustrates the dependency of the hinge axis determination on the distance of the measurement position from the center of rotation. It also highlights that a measurement close to the TMJ (with a small radius) does not offer greater accuracy with the circle fitting method compared to a measurement taken further from the TMJ or closer to the jaw (e.g., at the incisal point). On the contrary, radii smaller than <math>20</math> <small><math display="inline">mm</math></small> significantly increase the imprecision in determining the hinge axis. This result is positive in a clinical context, as it indicates that, even in the presence of noise (as can happen in real measurements), it is possible to accurately estimate the <math>_tHA</math> and the associated radius, which are crucial for patient diagnosis and rehabilitative treatment. | ||
<Center> <gallery widths="350" heights="282" perrow="2" mode="slideshow"> File:Official HA 1.jpg| | <Center> | ||
===Mathematical Formalism: Fitting Error=== To illustrate the concept of error due to noise in determining the points of the curve, the preliminary mathematical steps are as follows (refer to Figure 7, represented in Geogebra). | <gallery widths="350" heights="282" perrow="2" mode="slideshow"> | ||
File:Official HA 1.jpg| Figura 7: | |||
</gallery> | |||
</Center> | |||
===Mathematical Formalism: Fitting Error=== | |||
To illustrate the concept of error due to noise in determining the points of the curve, the preliminary mathematical steps are as follows (refer to Figure 7, represented in Geogebra). | |||
The script starts by defining the original circle center with the coordinates: | The script starts by defining the original circle center with the coordinates: | ||