Modifica di Asse Trasversale Cerniera (sezione)

==Localization Error <math>_tHA</math> from Chord <math>s</math> and Sagitta <math>h</math>==

To achieve an accurate determination of the mandibular rotation axis, the widest possible opening rotation is chosen. The literature describes mouth opening values between 7 and 18 mm, during which an almost pure rotation occurs before further translational movement begins.<ref name="Ferrario1996">Ferrario VF, Sforza C, Miani A Jr, Serrao G, Tartaglia G. "Open-close movements in the human temporomandibular joint: does a pure rotation around the intercondylar hinge axis exist?" J Oral Rehabil. 1996;23:401–408.</ref> While it is debatable whether an opening of 15 mm truly represents pure rotation, this value can be considered the upper limit for optimal hinge axis detection. The radius of movement, particularly of the incisal point, can be calculated using the average Bonwill triangle, which measures approximately 90.7 mm. With this radius and an opening of 15 mm, the opening angle corresponds to about 9.5 degrees. Simulations and calculations were implemented in Python, and the results were geometrically visualized in Geogebra for a spatial representation and better understanding of the phenomenon. Any recorded opening movement from an appropriate measuring device will describe a path that lies on a segment of a circular arc. The task is to determine the radius of this movement or, more precisely, the location of the center of rotation.

In our case (Figure 4), it is possible to identify two arcs of a circle and a center of rotation corresponding to the arc generated by a mandibular opening with forced retrusion. The arc where a presumably precise center of pure rotation, or rather "relatively pure," is found is generated by keeping the mandible forcibly retruded so that it rotates around the temporomandibular ligament, thereby minimizing the translation of the temporomandibular joint (TMJ). This maneuver allows for the distinction of the rototranslational pattern that occurs in the absence of forced retrusion and/or manual guidance (represented by the underlying arc). In Figure 4, the traces differ significantly because the second (lower) one incorporates both rotation and translation simultaneously.

When measuring something, such as the distance between two points on a circle, the measurement is not always perfect; small errors or "noise" can affect the result. We want to explore how these small errors can compromise the estimate of a measurement that should be precise. Figure 5 and the accompanying Python script enhance the representation. This type of analysis is useful because it allows us to assess the reliability of our measurements in real-life situations, where there are always small inaccuracies. It is particularly relevant in fields such as biomechanics or engineering, where measurements need to be highly accurate.

Theoretically, the radius can be recalculated from the circular-like opening movement using the chord <math>s</math> and the sagitta (height) <math>h</math> (Figure 5). From these values, the radius of the circumference inscribed in the pure rotation arc, physically constructed on the plate and labeled as <math>0</math>, can be calculated and verified for discrepancies as <math>0_1</math> using Geogebra.

From the data obtained with Geogebra, it is possible to construct, first, the radius of <math>0_1</math> (forced opening in retrusion, presumably rotational), and, of course, determine the centers of rotation. The mathematical formalism to achieve this is as follows:

<math>r_{calc} = \frac{4h + s^2}{8h}</math>

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|'''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>===

The script simulates the effect of noise on the measurement of the distance between a fixed point and a hinge axis (HA). A distribution of errors caused by noise added to the data is generated, and the 72nd percentile of these errors is analyzed.

The radius (<math>r</math>) is the radius of the circle on which the initial point lies, given by <math>r = 26.30</math> mm. The arc length <math>s</math> is the length of the arc on which the point moves, equal to <math>s = 12.02</math> mm. The angle <math>\alpha</math> is the angle corresponding to the arc, in radians, calculated as:

<math>\alpha = \frac{s}{r}</math>

'''<math>r_{\text{var}}</math>''': This is the distance of the starting point from the hinge axis, equal to <math>r_{\text{var}} = 26.30</math> mm. To calculate the start and end points, we proceed by calculating the point '''<math>p_{\text{start}}</math>''', located at a distance <math>r_{\text{var}}</math> along the <math>x</math>-axis:

<math>p_{\text{start}} = \begin{pmatrix} r_{\text{var}} \ 0 \end{pmatrix}</math>

The terminal point '''<math>p_{\text{end}}</math>''' is defined as the final point after the rotation of an angle <math>\alpha</math>, calculated as:

<math>p_{\text{end}} = \begin{pmatrix} r_{\text{var}} \cos(\alpha) \ r_{\text{var}} \sin(\alpha) \end{pmatrix}</math>

The length of the chord '''<math>s_{\text{ref}}</math>''' that connects the initial and final points is given by:

<math>s_{\text{ref}} = \sqrt{(p_{\text{end},x} - p_{\text{start},x})^2 + (p_{\text{end},y} - p_{\text{start},y})^2}</math>

The height of the arc (sagitta) '''<math>h_{\text{ref}}</math>''' is calculated as:

<math>h_{\text{ref}} = \frac{s_{\text{ref}}}{2} \tan\left(\frac{\alpha}{4}\right)</math>

At this point, Gaussian-distributed noise with a standard deviation of 0.01 mm is generated. The noise is added to the start and end points as follows:

<math>pn_{\text{start}} = p_{\text{start}} + noise_{\text{start}}</math>

<math>pn_{\text{end}} = p_{\text{end}} + noise_{\text{end}}</math>

This extends or shortens the chord '''<math>s</math>''' as follows:

<math>\sqrt{(pn_{end,x} - pn_{start,x})^2 + (pn_{end,y} - pn_{start,y})^2}</math>

While the noise on the sagitta '''<math>h_{\text{noise}}</math>''' is calculated as:

<math>h_{\text{noise}} = h_{\text{ref}} + \text{noise}</math>

Once the errors for the chord '''<math>s</math>''' and the sagitta '''<math>h</math>''' are calculated, we can compute the noisy radius '''<math>r_{\text{noise}}</math>''' by reusing the formula:

<math>r_{\text{noise}} = \frac{4 h_{\text{noise}}^2 + s^2}{8 h_{\text{noise}}}</math>

The difference between the radius calculated with noise and the original distance <math>r_{\text{var}}</math> is given by:

<math>\delta r_{\text{noise}} = \left| r_{\text{noise}} - r_{\text{var}} \right|</math>

At the 72nd percentile of the measurement error, this represents the value below which 72% of the observed errors fall.

====Script Python: Errore localizzazione HA da corda <math>s</math> e sagitta <math>h</math>====

<syntaxhighlight lang="python">
import numpy as np
import matplotlib.pyplot as plt

# Constants
r = 90.7  # Radius in mm
s = 12.02  # Length of Arc in mm
alpha = s / r  # Angle in radians
r_var = 26.33  # Distance from the hinge axis in mm

# Calculate start and end points
p_start = np.array([r_var, 0])
p_end = np.array([r_var * np.cos(alpha), r_var * np.sin(alpha)])

# Calculate reference values
s_ref = np.sqrt((p_end[0] - p_start[0])**2 + (p_end[1] - p_start[1])**2)
h_ref = s_ref / 2 * np.tan(alpha / 4)
scale_noise = 0.01  # Reduced noise in mm
samples = 1000  # number of Gaussian samples
noise = scale_noise * np.random.randn(5, samples)

# Add noise to points
pn_start = np.array([p_start[0] + noise[0, :], p_start[1] + noise[1, :]])
pn_end = np.array([p_end[0] + noise[2, :], p_end[1] + noise[3, :]])

# Calculate length of the chord (s) and sagitta (height) (h_noise)
s = np.sqrt((pn_end[0, :] - pn_start[0, :])**2 + (pn_end[1, :] - pn_start[1, :])**2)
h_noise = h_ref + noise[4, :]
r_noise = (4 * h_noise**2 + s**2) / (8 * h_noise)
delta_r_noise = np.abs(r_noise - r_var)

# Calculate the 72nd quantile of the error
error_quantile_reduced_noise = np.quantile(delta_r_noise, 0.72)
print(f'Errore (72° percentile) con rumore ridotto: {error_quantile_reduced_noise:.2f} mm')

# Optional: Plot the distribution of errors
plt.hist(delta_r_noise, bins=30, edgecolor='k', alpha=0.7)
plt.axvline(error_quantile_reduced_noise, color='r', linestyle='dashed', linewidth=1)
plt.title('Distribuzione degli errori di misurazione')
plt.xlabel('Errore (mm)')
plt.ylabel('Frequenza')
plt.show()
</syntaxhighlight>