Modifica di Transverse Hinge Axis (sezione)

==Occlusal Distortions==

In dentistry and gnathology, a precise understanding of mandibular movements is essential for designing accurate dental prostheses. One of the key concepts in this field is <math>_tHA</math>, which represents the axis around which the mandible rotates during the first millimeters of mouth opening. The correct localization of <math>_tHA</math> is crucial for achieving natural mandibular movement and proper dental occlusion. An inaccurate localization of <math>_tHA</math> can lead to significant errors in the final position of the dental cusps. Cusps are the points of the teeth that interlock between the upper and lower teeth during mastication. If <math>_tHA</math> is not correctly localized, especially in the presence of inclined cusps, the error may manifest as a discrepancy between the predicted and actual position of the cusps. This can compromise the effectiveness of dental treatments and lead to malocclusion or discomfort for the patient.

The objective of this simulation is to explore and quantify how the error in localizing <math>_tHA</math> affects the position of the dental cusps, considering two specific scenarios:

*''Mandibular Opening of <math>0</math> <small><math display="inline">mm</math></small>:'' This parameter represents the centric recording model without wax interposed between the arches. With a mandibular opening of ''<math>0</math> <small><math display="inline">mm</math></small>'', the cuspal error is theoretically zero if the cusps were flat. However, if the cusps are inclined, the error may not be zero due to the inclination itself. *''Mandibular Opening of <math>3</math> <small><math display="inline">mm</math></small>:'' This parameter represents the centric recording model with wax interposed between the arches, generating a mandibular opening of ''<math>3</math> <small><math display="inline">mm</math></small>''. In this condition, even a small error in the localization of <math>_tHA</math> can cause a significant cuspal error, exacerbated by the inclination of the dental cusps.

'''Simulation Results'''

To address this problem, we use mathematical models and computer simulations to calculate how an error in the localization of <math>_tHA</math> translates into errors in the final positions of the dental cusps. We use various magnitudes of <math>_tHA</math> localization error (up to ''<math>10</math> <small><math display="inline">mm</math></small>'') and analyze the effect of these discrepancies both with centric recordings without wax and with the mouth closed (''<math>0</math> <small><math display="inline">mm</math></small>'') as well as with a slight mouth opening (''<math>3</math> <small><math display="inline">mm</math></small>''). In addition, we consider the inclination of the dental cusps, simulating errors on inclined cusps of ''<math>0^\circ</math>'' and ''<math>5^\circ</math>'' to make the model more realistic and applicable to actual clinical conditions. This approach allows us to better understand how localization errors affect the final result and provides valuable insights for improving precision in dental practices.

In essence, with an opening of ''<math>0</math> <small><math display="inline">mm</math></small>'' in centric recordings without wax and non-inclined cusps (''<math>0^\circ</math>''), and with a <math>_tHA</math> error of ''<math>10</math> <small><math display="inline">mm</math></small>'' (10 steps from ''<math>0</math> <small><math display="inline">mm</math></small>'' to ''<math>10</math> <small><math display="inline">mm</math></small>''), the cuspal error is ''<math>0</math> <small><math display="inline">mm</math></small>''. With a mandibular opening of '''''<math>3</math> <small><math display="inline">mm</math></small>''''' and flat dental cusps (''<math>0^\circ</math>''), the cuspal error ranges from <math>0</math> ''<small><math display="inline">mm</math></small>'' in a situation of exact <math>_tHA</math> localization to <math>1</math> ''<small><math display="inline">mm</math></small>'' with a localization error of <math>10</math> ''<small><math display="inline">mm</math></small>''. However, even if we were to perform a centric recording without wax with a mandibular opening of ''<math>0</math> <small><math display="inline">mm</math></small>'', with dental cusps inclined by just ''<math>5^\circ</math>'', this condition introduces a vertical cuspal error due to the projection of the <math>_tHA</math> localization error along the inclination of the cusps. In this case, the error ranges from ''<math>0</math> <small><math display="inline">mm</math></small>'' in an exact localization of <math>_tHA</math> to <math>0.87</math> ''<small><math display="inline">mm</math></small>'' with a localization error of <math>10</math> ''<small><math display="inline">mm</math></small>''.




====Python Script: Error Cusp with 0mm Opening and Flat Cusps'====

<br />

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

# Definition of the original data
original_center = np.array([11.56, 61.21])  # Original center of the circle
r = 26.38  # Radius of the original circle
theta = np.linspace(0, 2 * np.pi, 10)  # 10 points along the circumference

# Generation of original points on the circumference
points = np.array([
    original_center + r * np.array([np.cos(angle), np.sin(angle)])
    for angle in theta
])

# Definition of the radius for the position of dental cusps
cusp_radius = 75.0  # Distance of the cusp from the center of rotation

# Generation of the positions of the original dental cusps
cusp_positions = np.array([
    original_center + cusp_radius * np.array([np.cos(angle), np.sin(angle)])
    for angle in theta
])

# Simulation of the errors Δ
errors = np.linspace(0, 10, 10)  # Errors Δ from 0 to 10 mm, 10 steps
max_differences_zero_apertura = []

# Cusp error for mandibular opening at 0 mm is zero
for delta in errors:
    max_differences_zero_apertura.append(0)  # Cusp error is zero

plt.plot(errors, max_differences_zero_apertura, 'ro-', label='Opening 0 mm (Mouth closed)')
plt.xlabel('Error Δ (mm) in HA localization')
plt.ylabel('Max Difference in Cusp Positions (mm)')
plt.title('Cusp Error with 0 mm Opening and Flat Cusps')
plt.grid(True)
plt.legend()
plt.show()

</syntaxhighlight><blockquote>
===='''Python Script: Cusp Error with 3mm Aperture and Flat Cusps'''====
</blockquote><br />

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

# Definition of the original data
original_center = np.array([11.56, 61.21])  # Original center of the circle
r = 26.38  # Radius of the original circle
theta = np.linspace(0, 2 * np.pi, 10)  # 10 points along the circumference

# Generation of original points on the circumference
points = np.array([
    original_center + r * np.array([np.cos(angle), np.sin(angle)])
    for angle in theta
])

# Definition of the radius for the position of dental cusps
cusp_radius = 75.0  # Distance of the cusp from the center of rotation

# Generation of the positions of the original dental cusps
cusp_positions = np.array([
    original_center + cusp_radius * np.array([np.cos(angle), np.sin(angle)])
    for angle in theta
])

# Simulation of the errors Δ
errors = np.linspace(0, 10, 10)  # Errors Δ from 0 to 10 mm, 10 steps
max_differences_apertura_3mm = []

# Calculation of the cusp error for a mandibular opening of 3 mm
for delta in errors:
    new_center = original_center + np.array([delta, 0])  # Shift along the x-axis
    
    # Calculate the new positions of the dental cusps with error Δ
    new_cusp_positions = np.array([
        new_center + cusp_radius * np.array([np.cos(angle), np.sin(angle)])
        for angle in theta
    ])
    
    # Calculate the spatial differences between the original and calculated positions
    differences = np.linalg.norm(cusp_positions - new_cusp_positions, axis=1)
    max_difference = np.max(differences) * (3 / 30)  # Scaling the error for a 3 mm opening out of 18 mm maximum
    max_differences_apertura_3mm.append(max_difference)
    
    print(f"Error Δ = {delta:.1f} mm: Max Difference in Cusp Positions (3 mm opening) = {max_difference:.2f} mm")

plt.plot(errors, max_differences_apertura_3mm, 'bo-', label='Opening 3 mm')
plt.xlabel('Error Δ (mm) in HA localization')
plt.ylabel('Max Difference in Cusp Positions (mm)')
plt.title('Cusp Error with 3 mm Opening and Flat Cusps')
plt.grid(True)
plt.legend()
plt.show()

</syntaxhighlight><blockquote>
===='''Python Script: Effect of Rotation Center Localization Errors with 0mm Aperture and Inclined Cusps ''<math>5^\circ</math>'''''====
</blockquote><syntaxhighlight lang="python">
import numpy as np
import matplotlib.pyplot as plt

# Definition of the original data
original_center = np.array([11.56, 61.21])  # Original center of the circle
r = 26.38  # Radius of the original circle
theta = np.linspace(0, 2 * np.pi, 10)  # 10 points along the circumference

# Generation of original points on the circumference
points = np.array([
    original_center + r * np.array([np.cos(angle), np.sin(angle)])
    for angle in theta
])

# Definition of the radius for the position of dental cusps
cusp_radius = 75.0  # Distance of the cusp from the center of rotation

# Generation of the positions of the original dental cusps
cusp_positions = np.array([
    original_center + cusp_radius * np.array([np.cos(angle), np.sin(angle)])
    for angle in theta
])

# Simulation of the errors Δ
errors = np.linspace(0, 10, 10)  # Errors Δ from 0 to 10 mm, 10 steps
max_differences_inclined_cusp = []

# Definition of the angle of the dental cusps (in degrees)
cusp_angle_deg = 5  
cusp_angle_rad = np.deg2rad(cusp_angle_deg)  # Conversion to radians

# Calculation of cusp error for mandibular opening at 0 mm with inclined cusps
for delta in errors:
    # Calculation of vertical error due to cusp inclination
    vertical_error_due_to_inclination = delta * np.tan(cusp_angle_rad)
    
    # Save the maximum error in this specific case
    max_differences_inclined_cusp.append(vertical_error_due_to_inclination)
    
    print(f"Error Δ = {delta:.1f} mm: Max Difference in Cusp Positions with inclined cusps = {vertical_error_due_to_inclination:.2f} mm")

plt.plot(errors, max_differences_inclined_cusp, 'ro-', label=f'Inclined Cusps ({cusp_angle_deg}°)')
plt.xlabel('Error Δ (mm) in HA localization')
plt.ylabel('Max Difference in Cusp Positions (mm)')
plt.title(f'Effect of Rotation Center Localization Errors with Inclined Cusps ({cusp_angle_deg}°)')
plt.grid(True)
plt.legend()
plt.show()

</syntaxhighlight>

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