Store:Asse Cerniera Verticale parte 1a

Scopo dello studio

La dinamica mandibolare rappresenta uno dei campi più complessi nello studio della biomeccanica articolare, caratterizzandosi per movimenti spaziali descritti da sei gradi di libertà: tre traslazioni e tre rotazioni. Questi movimenti, regolati dai condili mandibolari, interagiscono con la struttura cranio-facciale attraverso traiettorie che possono essere descritte matematicamente e geometricamente. In questo contesto, la conica emerge come modello ideale per analizzare e rappresentare le rototraslazioni dei condili e il loro ruolo nella funzione masticatoria.

L'obiettivo principale di questo studio è duplice:

• Descrivere le traiettorie condilari attraverso i tre assi principali di movimento — latero-mediale (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} ), verticale (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} ) e antero-posteriore (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ) — e analizzare come queste traiettorie generino superfici rigate che descrivono il comportamento dinamico articolare ed inparticolare dell'asse verticale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _vHA} nel piano assiale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_p} .
• Quantificare le distanze ed analizzare le velocità nei vari punti del sistema masticatorio
• Integrare il modello della conica per comprendere come i condili descrivano superfici coniche durante i movimenti di rototraslazione, fornendo una rappresentazione precisa e generalizzabile delle dinamiche masticatorie.

Relazione con la Conica

Le coniche, quali 'ellissi', 'parabole' e 'iperboli', si adattano perfettamente alla descrizione delle traiettorie articolari condilari. Nella cinematica mandibolare:

• 'Ellissi': rappresentano il movimento rotatorio principale dei condili attorno agli assi verticali e orizzontali .
• 'Parabole': descrivono movimenti di apertura e chiusura della mandibola, con particolare attenzione alle variazioni dell'asse antero-posteriore .
• 'Iperbole': emergono nelle traiettorie di lateralità condilare, con particolare enfasi sulla combinazione di rotazione e traslazione del condilo lavorante e mediotrusivo .

Conclusione

Il presente studio propone un’analisi innovativa delle dinamiche mandibolari, considerando la conica come strumento geometrico fondamentale per descrivere e comprendere le traiettorie condilari. Questa prospettiva non solo arricchisce la comprensione teorica della cinematica mandibolare, ma fornisce anche una base per lo sviluppo di tecnologie diagnostiche avanzate e terapie personalizzate.

Misurazioni e Conversione da Pixel a Millimetri

L’analisi dei movimenti condilari richiede misurazioni precise, ottenute tramite **calibrazione dell’immagine**. Calcolo della distanzaCalcolo della Distanza tra i Punti Le coordinate dei punti sono: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_2(525.3, -406)} e Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_2(764.4, -407.1)} . La formula per la distanza euclidea è: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}} . Sostituendo i valori: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(764.4 - 525.3)^2 + (-407.1 - (-406))^2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(239.1)^2 + (-1.1)^2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{57121.81 + 1.21} = \sqrt{57123.02} \approx 239.02 \, \text{pixel}} . Conversione della Scala in mm: Dato che Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 239.02 \, \text{pixel}} equivale a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \, \text{cm} = 10 \, \text{mm}} , calcoliamo la conversione in mm/pixel: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Scala in mm/pixel} = \frac{\text{Lunghezza reale (in mm)}}{\text{Distanza in pixel}} = \frac{10}{239.02} \approx 0.04184 \, \text{mm/pixel}} . Quindi, ogni pixel nella figura corrisponde a circa Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.04184 \, \text{mm/pixel}} . Esempio di Applicazione: Conversione Distanza in mm Se Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 100 \, \text{pixel}} , allora: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_\text{mm} = 100 \cdot 0.04184 \approx 4.184 \, \text{mm}} .

Fattore di scala utilizzato: Distanze condilariCalcolo delle distanze tra i punti Le coordinate dei punti estrapolate da Geogebra dopo calibrazione, per il condilo laterotrusivo, sono: 1L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (58.3, -50.9)} , 2L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (59, -92.3)} , 3L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (46.3, -169.5)} , 4L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (44.1, -207.7)} , 5L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (38.4, -136.2)} , 6L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (36.4, -48.2)} , 7L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (44, -34.9)} , 8L: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (52.9, -48)} . Fattore di scala: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.04184 \, \text{mm/pixel}} . Distanze rispetto a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1L_c} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(59 - 58.3)^2 + (-92.3 - (-50.9))^2} \approx 41.41 \, \text{pixel}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 41.41 \cdot 0.04184 \approx 1.734 \, \text{mm}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2L_c} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(46.3 - 58.3)^2 + (-169.5 - (-50.9))^2} \approx 119.17 \, \text{pixel}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 119.17 \cdot 0.04184 \approx 4.99 \, \text{mm}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3L_c} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(44.1 - 58.3)^2 + (-207.7 - (-50.9))^2} \approx 157.43 \, \text{pixel}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 157.43 \cdot 0.04184 \approx 6.59 \, \text{mm}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4L_c} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(38.4 - 58.3)^2 + (-136.2 - (-50.9))^2} \approx 87.6 \, \text{pixel}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 87.6 \cdot 0.04184 \approx 3.66 \, \text{mm}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5L_c} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(36.4 - 58.3)^2 + (-48.2 - (-50.9))^2} \approx 22.06 \, \text{pixel}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 22.06 \cdot 0.04184 \approx 0.923 \, \text{mm}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6L_c} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(44 - 58.3)^2 + (-34.9 - (-50.9))^2} \approx 21.47 \, \text{pixel}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 21.47 \cdot 0.04184 \approx 0.898 \, \text{mm}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7L_c} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \sqrt{(52.9 - 58.3)^2 + (-48 - (-50.9))^2} \approx 6.13 \, \text{pixel}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = 6.13 \cdot 0.04184 \approx 0.257 \, \text{mm}} .Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8L_c}

• • • 1 cm = 10 mm = 239.02 pixel**
    • Scala in mm/pixel:** Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.04184 \, \text{mm/pixel}}

Punti	Coordinate (x, y)	Distanza (pixel)	Distanza (mm)
1L → 2L	(58.3, -50.9) → (59, -92.3)	41.41 px	1.734 mm
1L → 3L	(58.3, -50.9) → (46.3, -169.5)	119.17 px	4.99 mm
1L → 4L	(58.3, -50.9) → (44.1, -207.7)	157.43 px	6.59 mm

Movimenti Condilari: Traslazioni e Rotazioni

Vettore di Posizione del Condilo Laterotrusivo

Il condilo laterotrusivo (lato del movimento) è descritto dal vettore:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_l(t) = [X_l(t), Y_l(t), Z_l(t), \theta_l(t), \phi_l(t), \psi_l(t)] }

Dove:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_l, Y_l, Z_l} : spostamenti lineari.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_l, \phi_l, \psi_l} : rotazioni sugli assi cartesiani, secondo gli **angoli di Eulero**.

Vettore di Traslazione del Condilo Mediotrusivo

Il condilo mediotrusivo segue una **traslazione antero-mediale**, descritta dal vettore:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_M(t) = \begin{pmatrix} X_M(t) \\ Y_M(t) \\ Z_M(t) \end{pmatrix} }

Conclusioni

L’analisi della cinematica mandibolare a **sei gradi di libertà** permette di ottenere dati affidabili per applicazioni cliniche e protesiche. Nei capitoli successivi approfondiremo questi argomenti non banali.