Asse Cerniera Verticale

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Cinematica Mandibolare: Rotazioni e Traslazioni Condilari

Introduzione

Nel capitolo precedente, Transverse Hinge Axis, abbiamo introdotto la cinematica mandibolare analizzandone i movimenti sul piano sagittale. Durante i movimenti di **protrusione** e **retrusione**, la mandibola non si muove esclusivamente lungo l'asse 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} , ma ruota anche attorno all'asse 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} . Questo genera una traiettoria curvilinea dell’incisivo mandibolare, risultato di un complesso moto spaziale che combina **rotazione e traslazione condilare**.

Uno degli aspetti chiave di questa dinamica è lo **spazio libero interincisivo**, una regione angolare che permette movimenti masticatori fluidi e senza interferenze. Tuttavia, gli strumenti di analisi come il **Sirognatograph** e i sistemi elettromagnetici convenzionali ad effetto Hall tendono a focalizzarsi sulle traslazioni condilari, trascurando la componente rotazionale. Sebbene ciò possa essere sufficiente in alcuni contesti, non è adeguato a rappresentare fedelmente i movimenti mandibolari a sei gradi di libertà.

Cinematica Mandibolare a Sei Gradi di Libertà

Il movimento mandibolare si sviluppa in uno **spazio tridimensionale** e può essere descritto attraverso **sei gradi di libertà**, suddivisi in **tre traslazioni** e **tre rotazioni**.

Ogni condilo si muove rispetto ai seguenti **assi principali**:

- - Asse 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} (latero-mediale):** definisce la rotazione attorno all’asse cerniera trasversale (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 _tHA} , transverse Hinge Axis).
  - Asse 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} (verticale):** definisce la rotazione attorno all’asse cerniera 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} ).
  - Asse 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} (antero-posteriore):** definisce la rotazione attorno all’asse cerniera orizzontale (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 _oHA} ).

A ciascun asse corrisponde un **piano di riferimento anatomico**:

Piano sagittale: mostra il tracciato condilare prodotto dalla **rototraslazione** sull’asse trasversale (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 _tHA} ).
Piano coronale: associato all’asse orizzontale (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 _oHA} ).
Piano assiale: legato alla rotazione sull’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} ).

Nota: un piano non è generato direttamente da un asse, bensì un asse può essere contenuto in un piano o definirne una direzione. Più precisamente, il movimento di un asse genera una superficie rigata, che rappresenta l’insieme delle traiettorie spaziali risultanti.

Asse Cerniera Verticale e Strumenti di Registrazione

L’**asse cerniera 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} ) è particolarmente rilevante per i sistemi di registrazione cinematici, come:

- - Pantografi** (analogici ed elettronici)
  - Elettrongnatografi**
  - Assiografi**

Strumenti di Registrazione e Precisione

Il pantografo analogico è stato a lungo considerato un dispositivo preciso per la riproduzione dei tracciati condilari e il loro trasferimento su un articolatore regolabile.^[1]^[2]^[3]
Il pantografo elettronico, introdotto successivamente, ha dimostrato una precisione comparabile nella registrazione dei determinanti condilari.^[4]
Un parametro controverso nel movimento condilare è la **traslazione laterale immediata mandibolare** (Movimento di Bennett), il cui significato clinico è stato oggetto di dibattito.^[5] Studi recenti indicano che non esistono prove sufficienti a confermare la sua rilevanza clinica.^[6]

Nota sulla Precisione e Sugli Obiettivi dello Studio
Questo studio mira a fornire una comprensione concettuale dei principi cinematici coinvolti nella dinamica masticatoria, con un focus sulla biomeccanica mandibolare. Sebbene i calcoli siano stati eseguiti con rigore, potrebbero verificarsi discrepanze dovute a:

Approssimazioni nei dati numerici: Differenze nei valori cartesiani legate a variabili operative.

Limiti di rappresentazione: Uso di numeri approssimati per motivi pratici.

Finalità cliniche: Lo scopo è descrivere concetti piuttosto che ottenere precisione assoluta.

Figura 1: Cinematica mandibolare sul piano assiale rappresentata dai markers prelevati dallo strumento ogni 20 mSec. Questi punti rappresentano i condili laterotrusivi dal punto 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 L_c } e mediotrusivi 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 M_c } . Il Laterotrusive point (a sinistra) e il Mediotrusive point (a destra) tracciano la posizione dei condili della mandibola durante un movimento masticatorio laterale, che include movimenti complessi di traslazione e rotazione. I punti numerati (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 8L_c } ) seguono il movimento del condilo laterotrusivo nel tempo, mentre i punti 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 1M_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 8M_c } seguono il condilo mediotrusivo. Nell'area del Molar point e dell' Incisal point sono rappresentati i percorsi occlusali durante la masticazione.

Passi Successivi

In questo capitolo, analizzeremo la cinematica 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} ) e il fenomeno masticatorio, rappresentandolo con tracciati estratti da lavori di riferimento come quello di Lund e Gibbs.^[7](Figura 1)

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.

Bibliography & references

↑ Curtis, D.A. & Sorensen, J.A. Errors incurred in programming a fully adjustable articulator with a pantograph. J Prosthet Dent. 1986; 55:427-429.
↑ Cite error: Invalid <ref> tag; no text was provided for refs named :0
↑ Cite error: Invalid <ref> tag; no text was provided for refs named :1
↑ Payne, J. Condylar determinants in a patient population: electronic pantograph assessment. J Oral Rehabil. 1997; 24:157-163.
↑ Bennett, N.G. A contribution to the study of the movements of the mandible. Proc R Soc Med. 1908; 1:79-98.
↑ Taylor, T.D., Bidra, A.S., Nazarova, E. Clinical significance of immediate mandibular lateral translation: A systematic review. J Prosthet Dent. 2016; 115:412-418.
↑ Cite error: Invalid <ref> tag; no text was provided for refs named :2

[1] Curtis, D.A. & Sorensen, J.A. Errors incurred in programming a fully adjustable articulator with a pantograph. J Prosthet Dent. 1986; 55:427-429.

[:0-2] Cite error: Invalid <ref> tag; no text was provided for refs named :0

[:1-3] Cite error: Invalid <ref> tag; no text was provided for refs named :1

[4] Payne, J. Condylar determinants in a patient population: electronic pantograph assessment. J Oral Rehabil. 1997; 24:157-163.

[5] Bennett, N.G. A contribution to the study of the movements of the mandible. Proc R Soc Med. 1908; 1:79-98.

[6] Taylor, T.D., Bidra, A.S., Nazarova, E. Clinical significance of immediate mandibular lateral translation: A systematic review. J Prosthet Dent. 2016; 115:412-418.

[:2-7] Cite error: Invalid <ref> tag; no text was provided for refs named :2

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