Centers of rotation for combined vertical and transverse tooth movements Joseph
J. Hurd,
D.D.S.,
M.S.,*
and
Robert
J. Nikolai,
Ph.D.*
St. Louis, MO.
I
n describing individual orthodontic tooth movements during which the inclination of the long axis changes, the center of rotation may be a useful point of reference. The center of rotation is understood herein to be the edge view of a line which pierces the long axis or its extension and about which the rotational component of the tooth displacement occurs. A number of articles have ap,peared in the literature suggesting the locations of this reference point for various forms of tooth movement. The majority of the research has been theoretical,“-” but experimental and clinical evaluation of the theory has been carried out at least for simple tipping. 4, 5 However, the studies reported to date on centers of rotation have considered only transverse (faciolingual or mesiodistal) tooth movements. not taking into account vertical positional changes which may often occur along with the primary displacement components. For tooth movements produced by transverse crown loading in the absence of a vertical component of force, the accepted location of the center of rotation is the intersection of the two lines coincident with the positions of the long axis before and after the displacement; this point remains stationary during tha movement. Such a tooth movement is depicted on the left in Fig. 1; in the lingual crown tipping of the maxillary incisor, the rotation occurs about a line perpendicular to the long axis and through point C, the center of rotation. Note that, although the vertical elevations of the root apex and incisal edge change relative to the maxilla during the displacement, there is no net vertical whole-body movcment and the distances IC and I’C are equal. On the right in Fig. 1 is an example of a maxillary central incisor which undergoes a movement wherein both lingual tipping and intrusion occur. (Such a combined displacement is often observed during the initial stages of Begg treatment.) It is clear from the illustration that the intersection of the two long-axis *Department
of Orthodontics,
St. Louis
University
Medical
Center.
551
552
Hurd
and Nikolai
Am. J. Orthod. November 1976 I
Fig. and
1. Left, intrusion
lingual crown of the same
tipping tooth.
of
a maxillary
central
incisor;
right,
combined
tipping
positions is not a point fixed in the tooth ; the distances IP and I’P are not of equal magnitude. In this tooth movement apparently no point remains stationary; the center of rotation is itself actually displaced. When the tooth under displacement analysis is part of a fixed multibanded system, only rarely can the movement observed be strictly described as purely transverse ; typically, a vertical displacement component accompanies the transverse movement. Vertical forces are created during treatment in edgewise techniques by ‘(curve of Spee” and second-order bends, in the Begg technique by anchor bends in the arch wire and by intermaxillary elastics, and often by extraoral appliances. Furthermore, the activation of lingual root torque, either by a twist,ed rectangular wire or by a torquing auxiliary, also produces a tendency for vertical movement, as illustrated in simplified form by the cantilever beam of Fig. 2; when the arch wire is held back distally and retained occlusogingivally, a vertical reactive force accompanies the applied lingual root torque and creates a potential for coronal movement of the anterior segment of the arch wire (unless compensated for by some additional applied force). It was consideration of this particular phenomenon which led to the formulation reported in this article. Hypothesis
Examination of only the initial and final positions of a tooth which has experienced a combined vertical and transverse displacement prompts speculation as to the displacement path, Referring to the sketch on the right in Fig. 1 as the example, it is theoretically possible that the center of rotation may have been almost anywhere on the long-axis line. It might be suggested, for instance, that the tooth was tipped lingually about the apex to the final angulation and was
Volume Number
Centers of rotation
70 5
for combined movements
553
Torque Fig. 2. Torsional loading flection of its free end.
of
a cantilever
beam,
resulting
in rotation
and
downward
de-
subsequently bodily translated anteriorly and superiorly to the final position or, alternatively, that the tooth was tipped labially about the incisal edge to the final angulation and was then bodily translated posteriorly and superiorly to the final position. Clearly, neither of these two extremes represents a path likely to be traversed in the usual clinical situation, since considerable mechanical work, over and above the normal capability of the appliance, would be required to accomplish either cited composite displacement. The transverse and vertical force components are typically active concurrently in treatment ; therefore, it reasonably follows that the transverse and vertical movements occur, in large measure, simultaneously. Furthermore, in the purely transverse tooth displacement, the center of rotation remains stationary ; in the combined vertical and bransverse orthodontic movement, minimum energy principles suggest that the center of rotation should be that point on the long-axis line experiencing the least of all pointwise displacements. With this as a hypothetical basis, a method has been deduced and formulas have been derived to locate the center of rotation. Development
and
testing
of
the
theory
Depicted in Fig. 3 are typical initial and final positions of a maxillary central incisor undergoing anterior root torquing in Stage III of Begg treatment. Line IA coincides with the initial position of the long axis of the incisor and line PA with the long-axis position after lingual root movement and slight extrusion. Points I and I’ are the end points of the path of the incisal edge, and points A and A’ are similarly located on the path followed by the root apex; both paths are generally curvilinear. Line LL, defined by the two points which bisect the lines drawn between I and I’ and A and A’, represents the angulation of the incisor midway through the tooth displacement. It is theorized that the center of rotation moves along line LL during the tooth movement. The intersections of LL with IA and I’A’, designated in Fig. 3 as points C and C’, would then correspond to the initial and final positions of the center of rotation. If points C and C’ do represent the initial and final positions of the center of rotation, they must necessarily occupy coincident locations on the long axis of the tooth. That the distances IC and T’C’ are equal is demonstrated initially in
554
Hurd
and Nikolai
Am. J. Orthod. November 1976
4 C’
I
04 Fig. 3. sion.
Root
Fig.
4.
Analysis
Fig.
3.
torquing of
of
a maxillary
central
displacement
vectors
of
I’
’
L
incisor
accompanied
long-axis
points
by
for
the
simultaneous
tooth
movement
extruof
the Appendix, thus giving credence to the theory. The following formula is then derived which yields the location apical of the incisal edge and along the long axis of the center of rotation: Ic
=
I,c,
=
II’
sin (@ + @/2) 2 sin e/z .
In the above equation II’ is the straight-line distance between the initial and final incisal edge locations, 0 is the change in angular position of the long axis during the tooth displacement, and 4 is the angle between the incisal edge displacement vector and the initial long-axis position. Hypothesized was the center of rotation undergoing the smallest displacement of any point on the tooth or its extension. In the interest of testing whether the point located by the above theory and formula satisfies this condition, the graphical construction shown in Fig. 4 is offered. Illustrated are the displacement vectors of a number of long-axis points of a tooth experiencing combined tipping and extrusion identical to that of Fig. 3. Measurements of the lengths of the vectors reveal that the one vector along line LL, extending from point C to point C’, is the shortest of the set. All displacement vectors of points on the long axis extension are clearly of greater length than those shown. Discussion
Several observations are pertinent and in order here regarding the accuracy and the extent of applicability of the theory and formula presented above.
Volume Number
Fig.
Centers of rotation
50 5
5. Geometry
of the
displacement
of the
long
axis
for
for
combined
the
tooth
movements
movement
of
555
Fig.
3.
It may be argued, relevant to the accuracy of the theory, that a line defining the tooth angulation midway through the tooth movement might better be determined by bisecting the arcs (paths) followed by the incisal edge and apex rather than their displacement vectors. These arcs, however, are largely indeterminant. Furthermore, there is no firm reason to suspect that the incisal edge and apex move on smooth paths of specific curvatures, particularly if several appliance adjustments are made between the initial and final tooth positions. Cursory evauation employing various arcs in determining the center-of-rotatiolr location yielded results varying less than 0.5 mm. from those found by using the displacthment vectors. Determination of the center of rotation via the theory presented herein obviously depends upon the ability to locate the initial and final incisal edge and long-axis positions accurately relative to one another. In instances where the transverse component of the tooth movement, is anteroposterior, the use of prt’and postdisplacement cephalograms is advocated. Greatest accuracy is probably attained by tracing the initial incisal edge and long-axis positions together with the immediately surrounding skeletal anatomy in as much detail as possible, then superimposing the drawing over the anatomy of the final cephalogram and tracing the final incisal edge and long-axis positions. In regard to the extent of applicability, the theory and formulas present4 are valid for mandibular as well as maxillary tooth movements. Although esamples given herein employ lateral views, the technique might also be applied to posteroanterior views ; however, the theory is limited to two-dimensional analyses and thus assumes that the long-axis angulation in the t,hird dimension remains essentially constant. In lieu of the formula given above, an alternate graphic/cephalometric analysis may be employed to closely approximate the center-of-rotation location. He-
A,,,. J. Orthod. NovenlbeY1976
quired is the intersection of the initial and final long-axis positions-point P in Fig. 1, right, and in Fig. 3. Kefcrring specifically to Fig. 3, the caentct’ of rotation appears to he apical of the incisal edge a distance equal to the average of the measured lengths of ZP and Z’P, that is: IC
=
IV’
=
%l(IP
+ I’P).
For the sake of clarity, the displacement examples in this article all reflect substantial angular positional, changes. However, the theory and formulas apply equally well to all types of movement wherein the long axis experiences a twodimensional displacement, including those having no vertical component and for which points C, C’, and P will all coincide. The formula presented in the preceding section will locate centers of rotation at, and apical to, the incisal edge and is valid for the bodily movement case for which the angle B is zero and points C and C’ are located at, infinity. For those less common displacements for which the center of rotation is occlusal of the incisal edge, an alternate but similar formula, given in the Appendix, must be employed. In these situations the data obtained cephalometrically must permit the complete definition of the root, apex displacement vector. Summa8ty
and
conclusions
For purely transverse orthodontic tooth movements, the center of rotation is defined as that point on the Iong axis or its extension which remains stationary during the movement and around which the rotational component of the tooth displacement takes place. For tooth movements having both vertical and transverse components, no point on the long-axis line remains fixed in space. The twodimensional theory proposed herein suggests the more general definition of the center of rotation as that point on the long-axis line which is displaced the shortest distance during the tooth movement. The center of rotation can be located for the combined transverse and vertical tooth displacement. It is found to move along a path coincident with a segment of a line in a position depicting the tooth angulation midway through the movement. Formulas, which can be used in conjunction with a composite pre- and postdisplacement cephalometric tracing, are presented herein to define the center-ofrotation location for such tooth movements. REFERENCES
1. Burstone,
C. J.: The biomechanics of tooth movement, In Kraus, B. S., and Reidel, R. A. (editors) : Vistas in Orthodontics, Philadelphia, 1962, Lea & Febiger, Publishers, pp. 199204. 2. Gianelly, A. A., and Goldman, H. M.: Biologic Basis of Orthodontics, Philadelphia, 1971, Lea & Febiger, Publishers, pp. 144-154. 3. Nikolai, R. J.: Periodontal ligament reaction and displacements of a maxillary central incisor subjected to transverse crown loading, J. Biomech. 7: 93-99, 1974. 4. Christiansen, R. L., and Burstone, C. J.: Centers of rotation within the periodontal space, Au. J. ORTHOD. 55: 353-369, 1969.
Volzlme 70 Number 5
Centers of rotation
5. Bouldien, G. H.: Locating the fulcrum University of Tennessee, 1971. 3556
for
combined movements
of teeth tipped with the Begg appliance,
557
M.S. thesis,
Caroline St. (63104)
The formulas defining the location of the center of rotation are generated in this appendix. The derivations to follow need not be totally understood by every reader, but their inclusion is deemed warranted by the authors, primarily in the interest of credibility. Fig. 5 depicts the long-axis segment between the root apex and the incisal edge in typical positions preceding (IA) and following (PA’) tooth displacement. The line segment LL bisects the apex and in&al edge displacement vectors AA’ and II'. Points C and C’ are the intersections of LL with IA and PA’, respectively. Initially to be demonstrated is the fact that C and C’ are coincident points on the long axis. A two-dimensional, generally nonorthogonal, coordinate system is established such that the x axis pierces points I and I’ and the y axis contains points A and A’. (The development proceeds most directly via projective geometry theory.) Equations of lines in this system which coincide with the initial long-axis position and the line LL may be written as follows : x +L=l,
2,
-=l. 2Y YA f Y.4,
2x -+ 2, + XI,
Yn
(Al)
Simultaneous solution of these expressions for 5 and y and evaluation point of intersection C of the two lines yields X6 =
XI (Xl + xrl
(VA - Y.4,)
2(~‘4Xr - XIY,.)
yc = ’
Y‘dYA f Y‘v) CG - XI/ 2(YAG - WA.)
at the .
(A21
The equation of the line which coincides with the final long-axis position is 2 1;,-
+Y=l Y.4.
(A31
and simultaneous solution of the second of Equations using Equations (A2), gives at point C’ X0’ =-
XI. XI
xc,
yc,
zz
E Yn
(Al)
ye
with Equation
(A3),
*
Geometric arguments involving similar triangles lead to the fact that the ratio of lengths IC to IA is equal to the ratio of coordinate distances y. to yA. Likewise, the ratios y/6 to yA’ and I’C’ to I’A’ are equal, From these relationships and the second each of Equations (A2) and (A4), it follows that IC
T;i=
(Xl,
- XI) 2(YAC
(Y, + Yx) - XlY.4.)
Z’C =I’B’=n
Z’C’
and therefore shown is the concincidence of points C and C’ on the long axis. Proceeding toward the formula presented in the text of this paper, in Fig. 5
558
Hurd
Am. J. Orthod. November 1976
and Nikolai
6 is the angular displacement of the long axis and equals the sum of the angles tI1 and 19~.For the triangles ICB and BC’I’ the law of sines yields sin
=r’
sin t$
sin
=BI”
sin 8,
(As’)
Since IC = I’C’, IB = BP, where B is the intersection of the bisector LL and the x axis, and angles IBC and I’BC’ are supplementary (sum to 180 degrees), it follows that the angles & and 8, are equal. Finally, solving the first of Equations (A5) for IC and employing a trigonometric relationship among the angles of the triangle ICB, the desired formula is obtained : IC
=
I’C’
II’
=
sin (@ + e/2) 2 sin e/2 *
In a very similar, parallel fashion, the following equation may be derived to locate the center of rotation with respect to the root apex : AC
=
A,c,
=
88’
sin (9 + 2 sin 8/2
e/2)
C-47)
where JI is the angle between the root apex displacement vector and the initial long-axis position. For the sake of completeness both formulas are given, since Equation (A6) is not valid for cases where points C and C’ are occlusal of the incisal edge. Likewise, Equation (A7) cannot be employed to define center-of-rotation locations apical of the root apex. The authors thank John C. Cantwell, Professor assistance in the preparation of the appendix.
of Mathematics,
St. Louis University,
for