Patent Publication Number: US-11648089-B2

Title: Orthodontic appliances and systems with elastic members

Description:
CROSS-REFERENCE 
     This application is a continuation of U.S. application Ser. No. 14/609,970, filed Jan. 30, 2015, which claims the benefit of U.S. Provisional Application No. 61/934,657, filed Jan. 31, 2014, which applications are incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     Orthodontic procedures typically involve repositioning a patient&#39;s teeth to a predetermined arrangement in order to correct malocclusions and/or improve aesthetics. To achieve these objectives, orthodontic appliances such as braces, retainers, shell aligners, and the like can be applied to the patient&#39;s teeth by an orthodontic practitioner. Typically, the appliance is configured to exert force on one or more teeth in order to effect desired tooth movements. The application of force can be periodically adjusted by the practitioner (e.g., by altering the appliance or using different types of appliances) in order to incrementally reposition the teeth to a desired arrangement. 
     In some instances, however, current orthodontic appliances may not be able to effectively generate the forces needed to achieve the desired tooth repositioning, or may not afford sufficient control over the forces applied to the teeth. Additionally, the rigidity of some existing appliances may interfere with the ability of the appliance to be coupled to the patient&#39;s teeth and may increase patient discomfort. 
     SUMMARY 
     Improved orthodontic appliances, as well as related systems and methods, are provided. In many embodiments, an orthodontic appliance configured to be worn on a patient&#39;s teeth includes a discontinuity and an elastic member interacting or configured to interact with the discontinuity. The appliances described herein provide enhanced control over forces exerted onto the teeth, thus enabling improved orthodontic treatment procedures. 
     Accordingly, in one aspect, an orthodontic appliance is provided. The appliance includes a shell having a plurality of cavities shaped to receive teeth and a discontinuity formed in the shell. In many embodiments, an elastic member is directly coupled to the shell at first and second attachment points and positioned to interact with the discontinuity. 
     Other objects and features of the present invention will become apparent by a review of the specification, claims, and appended figures. 
     INCORPORATION BY REFERENCE 
     All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The novel features of the invention are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present invention will be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the invention are utilized, and the accompanying drawings of which: 
         FIG.  1 A  illustrates a tooth repositioning appliance, in accordance with many embodiments; 
         FIG.  1 B  illustrates a tooth repositioning system, in accordance with many embodiments; 
         FIG.  1 C  illustrates a method of orthodontic treatment using a plurality of appliances, in accordance with many embodiments; 
         FIG.  2 A  illustrates an exemplary orthodontic appliance with a coupled elastic member and a discontinuity, in accordance with many embodiments; 
         FIG.  2 B  illustrates the appliance of  FIG.  2 A  when placed over the teeth; 
         FIG.  2 C  illustrates another example of an orthodontic appliance with a coupled elastic member and a discontinuity, in accordance with many embodiments; 
         FIG.  2 D  illustrates the appliance of  FIG.  2 C  when placed over the teeth; 
         FIG.  2 E  illustrates yet another example of an orthodontic appliance with a coupled elastic member and a discontinuity, in accordance with many embodiments; 
         FIG.  2 F  illustrates the appliance of  FIG.  2 E  when placed over the teeth; 
         FIG.  2 G  illustrates an example of an orthodontic appliance having a plurality of elastic members and discontinuities, in accordance with many embodiments; 
         FIG.  2 H  illustrates the appliance of the  FIG.  2 G  when placed over the teeth; 
         FIG.  2 I  illustrates additional exemplary geometries for a discontinuity in an orthodontic appliance, in accordance with many embodiments; 
         FIG.  3 A  illustrates an orthodontic appliance for repositioning teeth, in accordance with many embodiments; 
         FIG.  3 B  illustrates the appliance of  FIG.  3 A  when placed over the teeth. 
         FIG.  4 A  illustrates an orthodontic appliance for repositioning teeth, in accordance with many embodiments; 
         FIG.  4 B  illustrates another orthodontic appliance for repositioning teeth, in accordance with many embodiments; 
         FIG.  5 A  illustrates an orthodontic appliance for repositioning teeth, in accordance with many embodiments; 
         FIG.  5 B  illustrates the appliance of  FIG.  5 A  when placed over the teeth; 
         FIG.  5 C  illustrates the appliance of  FIG.  5 B  after tooth repositioning has occurred; 
         FIG.  6    illustrates an orthodontic appliance including a channel accommodating an attachment on a tooth, in accordance with many embodiments; 
         FIG.  7 A  illustrates another example of an orthodontic appliance for repositioning teeth, in accordance with many embodiments; 
         FIG.  7 B  illustrates the appliance of  FIG.  7 A  when placed over the teeth; 
         FIG.  7 C  illustrates the occlusal surface of the appliance of  FIG.  7 A ; 
         FIG.  7 D  illustrates the appliance of  FIG.  7 B  after tooth repositioning has occurred; 
         FIG.  8 A  illustrates an orthodontic appliance having elastics and associated guide features, in accordance with many embodiments; 
         FIG.  8 B  illustrates the appliance of  FIG.  8 A  when placed over the teeth; 
         FIG.  8 C  illustrates the appliance of  FIG.  8 B  after tooth repositioning has occurred; 
         FIG.  8 D  illustrates an orthodontic appliance having telescopic shell segments, in accordance with many embodiments; 
         FIG.  8 E  illustrates the appliance of  FIG.  8 D  when placed over the teeth; 
         FIG.  8 F  illustrates the appliance of  FIG.  8 E  after tooth repositioning has occurred; 
         FIG.  8 G  is a cross-sectional view of a segment of the appliance of  FIG.  8 C ; 
         FIG.  8 H  is a top view of a telescopic guide feature, in accordance with many embodiments; 
         FIG.  8 I  is a side view of the telescopic guide feature of  FIG.  8 H ; 
         FIG.  9    illustrates an orthodontic appliance for maintaining a current position of the patient&#39;s teeth, in accordance with many embodiments; 
         FIG.  10 A  illustrates an orthodontic appliance with protrusions, in accordance with many embodiments; 
         FIG.  10 B  illustrates the appliance of  FIG.  10 A  when placed over the teeth; 
         FIG.  10 C  illustrates the appliance of  FIG.  10 B  after tooth repositioning has occurred; 
         FIG.  10 D  illustrates an appliance divided into discrete shell segments, in accordance with many embodiments; 
         FIG.  10 E  illustrates the appliance of  FIG.  10 D  when placed over the teeth; 
         FIG.  10 F  illustrates the appliance of  FIG.  10 E  after tooth repositioning has occurred; 
         FIG.  10 G  is a perspective view of the appliance of  FIG.  10 C ; 
         FIG.  11 A  illustrates an orthodontic appliance configured to engage an attachment, in accordance with many embodiments; 
         FIG.  11 B  illustrates the appliance of  FIG.  11 A  when placed over the teeth; 
         FIG.  12 A  illustrates another exemplary orthodontic appliance configured to engage an attachment, in accordance with many embodiments; 
         FIG.  12 B  illustrates the appliance of  FIG.  12 A  when placed over the teeth; 
         FIG.  13 A  illustrates yet another orthodontic appliance configured to engage an attachment, in accordance with many embodiments; 
         FIG.  13 B  illustrates the appliance of  FIG.  13 A  when placed over the teeth; 
         FIG.  14 A  illustrates an orthodontic appliance configured to engage an attachment, in accordance with many embodiments; 
         FIG.  14 B  illustrates the appliance of  FIG.  14 A  when placed over the teeth; 
         FIG.  14 C  illustrates an orthodontic appliance including features for securing an elastic member, in accordance with many embodiments; 
         FIG.  14 D  illustrates another orthodontic appliance including features for securing an elastic member, in accordance with many embodiments; 
         FIGS.  15 A through  15 D  illustrate exemplary flap geometries for orthodontic appliances configured to engage an attachment, in accordance with many embodiments; 
         FIG.  15 E  illustrates an orthodontic appliance including a plurality of flaps for engaging a plurality of attachments on teeth, in accordance with many embodiments; 
         FIG.  15 F  illustrates the appliance of  FIG.  15 E  after tooth repositioning has occurred; 
         FIG.  16 A  is a cross-sectional view of the internal surface profile of an orthodontic appliance including a protrusion, in accordance with many embodiments; 
         FIG.  16 B  is a cross-sectional view of a shell of the appliance of  FIG.  16 A ; 
         FIG.  16 C  illustrates the shell of  FIG.  16 B  when placed over a tooth; 
         FIG.  17 A  is a cross-sectional view of the internal surface profile of another exemplary orthodontic appliance including a protrusion, in accordance with many embodiments; 
         FIG.  17 B  is a cross-sectional view of a shell of the appliance of  FIG.  17 A ; 
         FIG.  17 C  illustrates the shell of  FIG.  17 B  when placed over a tooth; 
         FIGS.  18 A through  18 C  illustrate exemplary orthodontic appliances including protrusions and elastics, in accordance with many embodiments; 
         FIG.  19    illustrates another exemplary orthodontic appliance including protrusions, in accordance with many embodiments; 
         FIG.  20 A  illustrates an orthodontic appliance shell used with an elastic member and attachment, in accordance with many embodiments; 
         FIG.  20 B  illustrates an elastic member with an attachment, in accordance with many embodiments; 
         FIG.  20 C  illustrates the elastic member of  FIG.  20 B  coupled to the appliance of  FIG.  20 A ; 
         FIGS.  21 A through  21 F  illustrate an orthodontic appliance with a plurality of discontinuities, in accordance with many embodiments; 
         FIGS.  22 A through  22 D  illustrate directionality of an elastic member influencing the forces applied to teeth, in accordance with many embodiments; 
         FIGS.  23 A through  23 D  illustrate an orthodontic appliance configured to produce tooth rotation, in accordance with many embodiments; 
         FIGS.  24 A through  24 D  illustrate an orthodontic appliance configured to produce tooth rotation, in accordance with many embodiments; 
         FIGS.  25 A and  25 B  illustrate orthodontic appliances having telescopic guide features, in accordance with many embodiments; 
         FIGS.  26 A through  26 D  illustrate orthodontic appliances with biasing features, in accordance with many embodiments; 
         FIG.  27    is a schematic illustration by way of block diagram of a method for orthodontic treatment, in accordance with many embodiments; 
         FIG.  28    is a schematic illustration by way of block diagram of a method for designing an orthodontic appliance, in accordance with many embodiments; 
         FIG.  29    illustrates a method for digitally planning an orthodontic treatment, in accordance with many embodiments; and 
         FIG.  30    is a simplified block diagram of a data processing system, in accordance with many embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     The orthodontic appliances described herein, along with related systems and methods, can be employed as part of an orthodontic treatment procedure in order to reposition one or more teeth, maintain a current position of one or more teeth, or suitable combinations thereof. Such appliances can include a shell shaped to receive the patient&#39;s teeth, with the geometry of the shell being selected to exert appropriate forces on the teeth in order to achieve the desired positioning of teeth. In many embodiments, the orthodontic appliances described herein utilize one or more elastic members (also referred to herein as “elastics”) acting in conjunction with one or more discontinuities formed in the shell to apply orthodontic forces to the teeth. The geometry and configuration of the one or more discontinuities and/or the one or more elastic members can be selected to control the magnitude and direction of the applied forces. In contrast to existing approaches, in which one or more elastics are fastened to the teeth or to one or more attachments mounted onto the teeth, the appliances disclosed herein employ one or more elastic members directly coupled to the shell and exerting force on the teeth via interaction with the discontinuity. Such appliances may be used to generate larger and/or more precisely controlled forces for orthodontic applications. Furthermore, the geometry and configuration of the one or more discontinuities and/or the one or more elastics can be used to adjust the local compliance of the appliance, thus improving appliance fit and reducing patient discomfort. Additionally, by locally controlling the compliance of the shell, the techniques described herein can be used to ensure that some or all points on the appliance intended to exert forces on the teeth (also known as “active points”) maintain sufficient contact with the teeth throughout the treatment process, thus improving the precision and efficiency of repositioning. The amount of force exerted on the teeth at each active point can vary based on the compliance of the shell, as well as on the configuration of the discontinuities and/or elastics. 
     Thus, in one aspect, an orthodontic appliance can include a shell having a plurality of cavities shaped to receive teeth, and a discontinuity formed in the shell. The appliance also includes an elastic member having a first portion directly coupled to the shell at a first attachment point and a second portion directly coupled to the shell at a second attachment point. The elastic member can be positioned to interact with the discontinuity. For example, the elastic member can interact with regions of the shell on opposing sides of the discontinuity, thereby accommodating changes in configuration and/or size of the discontinuity during mounting of the appliance onto teeth and/or during resulting repositioning of one or more teeth. 
     An orthodontic appliance can be configured to accommodate an attachment coupled to a tooth. A portion of the elastic member between the first and second attachment points can be engaged or engageable with the attachment. 
     An orthodontic appliance can be configured to reduce one or more spaces between teeth. For example, an orthodontic appliance can include one or more elastic members and one or more discontinuities that are configured to elicit a movement of the teeth that reduces the size of an interproximal space between the teeth when the appliance is worn on the teeth. 
     Any suitable configuration and/or number of discontinuities can be employed. For example, the discontinuity can be or include an aperture in the shell, a cut in the shell, or a deformation of the shell. 
     In many embodiments, a portion of the elastic member between the first and second attachment points extends along a surface of the shell such that the portion spans a plurality of the cavities. The discontinuity can include a plurality of openings in the shell disposed between the first and second attachment points. Each of the plurality of openings can be adjacent to or near an interproximal region of the teeth when the appliance is worn on the teeth. 
     In many embodiments, a mesial-distal arch length of the shell is shorter or adapted to be shorter when the appliance is not being worn on the teeth and is longer or adapted to be longer when the appliance is being worn on the teeth. For example, the orthodontic appliance can include one or more discontinuities and one or more elastics such that the arch length of the shell depends on whether or not the appliance is being worn on the teeth. As another example, the discontinuity can divide the shell into discrete segments with one or more elastics coupling the segments, such that the segments are movable relative to each other to enable the arch length of the shell to change depending on whether or not the appliance is being worn on the teeth. 
     The orthodontic appliance may include one or more elastics that span a discontinuity. For example, an orthodontic appliance can include a discontinuity in the form of an elongate opening in the shell, with a portion of the elastic member between the first and second attachment points spanning the elongate opening. 
     In many embodiments, the first and second attachment points are disposed on the shell, such that a portion of the elastic member between the first and second attachment points is adjacent to or near an interproximal region of the teeth when the appliance is worn on the teeth. For example, the first attachment point can be disposed on a lingual surface of the shell and the second attachment point can be disposed on a buccal surface of the shell. In another example, the first and second attachment points can each be disposed on a lingual surface of the shell. As a further example, the first and second attachment points can each be disposed on a buccal surface of the shell. 
     In many embodiments, an appliance includes one or more guide features formed in the shell and configured to guide relative movement between portions of the shell, wherein the relative movement results from a force applied by the elastic member. The one or more guide features can affect at least one of magnitude or direction of the force applied by the elastic member. In some instances, the one or more guide features can include telescopic features formed in the shell. 
     The appliance may include one or more retention features formed in the shell and configured to retain a portion of the elastic member at a specified position relative to the shell. The one or more retention features can include a groove formed in the shell, with the portion of the elastic member retained within the groove. 
     In many embodiments, at least one of the first and second attachment points includes a hook formed in the shell, the hook being configured to fasten the elastic member to the shell. A portion of the elastic member can extend between the first and second attachment points. 
     In many embodiments, the discontinuity forms a flap in a location of the shell configured to accommodate an attachment mounted on a tooth received or receivable within a cavity of the shell. A portion of the elastic member between the first and second attachment points can extend around the flap to engage the attachment, such that the elastic member imparts a force directly on the attachment. As another example, a portion of the elastic member extending between the first and second attachment points can span the flap, such that the elastic member imparts a force on the attachment through the flap. 
     In another aspect, a method of orthodontic treatment includes providing an orthodontic appliance including a shell having a plurality of cavities shaped to receive teeth and a discontinuity formed in the shell. An elastic member can be directly coupled to the shell in a position interacting with the discontinuity, wherein a first portion of the elastic member is directly coupled to the shell at a first attachment point and a second portion of the elastic member is directly coupled to the shell at a second attachment point. The appliance can be placed on a patient&#39;s teeth. Force can be applied to the teeth via the interaction of the elastic member with the discontinuity. 
     In many embodiments, the elastic member and the discontinuity are configured to elicit a movement of the teeth reducing the size of an interproximal space between the teeth. The discontinuity can be an aperture in the shell, a cut in the shell, or a deformation of the shell. In some instances, a portion of the elastic member between the first and second attachment points extends along a surface of the shell such that the portion spans a plurality of cavities. 
     In many embodiments, a mesial-distal arch length of the shell is shorter or adapted to be shorter when the appliance is not being worn on the teeth and is longer or adapted to be longer when the appliance is being worn on the teeth. One or more guide features can be formed in the shell and configured to guide movement of a portion of the shell, wherein the movement results from a force applied to the portion by the elastic member. 
     In another aspect, an orthodontic system includes a plurality of orthodontic appliances each having a shell including a plurality of cavities shaped to receive teeth. The appliances can be adapted to be successively worn by a patient to move one or more teeth from a first arrangement to a second arrangement. At least one of the appliances includes a discontinuity formed in the shell and an elastic member positioned to interact with the discontinuity. The elastic member can have a first portion directly coupled to the shell at a first attachment point and a second portion directly coupled to the shell at a second attachment point. 
     In many embodiments, the discontinuity includes an elongate opening in the shell, with a portion of the elastic member between the first and second attachment points spanning the elongate opening. The first and second attachment points can be disposed on the shell, such that a portion of the elastic member between the first and second attachment points is adjacent to or near an interproximal region of the teeth when the appliance is worn on the teeth. 
     In many embodiments, one or more retention features are formed in the shell and configured to retain a portion of the elastic member at a specified position relative to the shell. In some instances, at least one of the first and second attachment points includes a hook formed in the shell, the hook being configured to fasten the elastic member to the shell. 
     In many embodiments, a portion of the elastic member extends between the first and second attachment points. The discontinuity can form a flap in a location of the shell configured to accommodate an attachment mounted on a tooth received or receivable within a cavity of the shell. 
     Turning now to the drawings, in which like numbers designate like elements in the various figures,  FIG.  1 A  illustrates an exemplary tooth repositioning appliance or aligner  100  that can be worn by a patient in order to achieve an incremental repositioning of individual teeth  102  in the jaw. The appliance can include a shell (e.g., a polymeric shell) having teeth-receiving cavities that receive and resiliently reposition the teeth. In many embodiments, a polymeric appliance can be formed from a sheet of suitable layers of polymeric material. An appliance can fit over all teeth present in an upper or lower jaw, or less than all of the teeth. The appliance can be designed specifically to accommodate the teeth of the patient (e.g., the topography of the tooth-receiving cavities matches the topography of the patient&#39;s teeth), and may be fabricated based on positive or negative models of the patient&#39;s teeth generated by impression, scanning, and the like. Alternatively, the appliance can be a generic appliance configured to receive the teeth, but not necessarily shaped to match the topography of the patient&#39;s teeth. In some cases, only certain teeth received by an appliance will be repositioned by the appliance while other teeth can provide a base or anchor region for holding the appliance in place as it applies force against the tooth or teeth targeted for repositioning. In some cases, many or most, and even all, of the teeth will be repositioned at some point during treatment. Teeth that are moved can also serve as a base or anchor for holding the appliance as it is worn by the patient. Typically, no wires or other means will be provided for holding an appliance in place over the teeth. In some cases, however, it may be desirable or necessary to provide individual attachments  104  or other anchoring elements on teeth  102  with corresponding receptacles or apertures  106  in the appliance  100  so that the appliance can apply a selected force on the tooth. Exemplary appliances, including those utilized in the Invisalign® System, are described in numerous patents and patent applications assigned to Align Technology, Inc. including, for example, in U.S. Pat. Nos. 6,450,807, and 5,975,893, as well as on the company&#39;s website, which is accessible on the World Wide Web (see, e.g., the url “invisalign.com”). 
     In the depiction of  FIG.  1 A , the appliance  100  is designed to fit over a single arch of a patient&#39;s dentition  102 , which may be represented by a positive model of the dentition. The appliance  100  includes a receptacle  106  formed in the shell and configured to accommodate an attachment  104  such as a bracket mounted onto a tooth of the patient (which can correspond to an identical attachment on the tooth of the positive model). When engaged by the appliance  100  (e.g., via the receptacle  106 ), the attachment  104  can transmit repositioning forces exerted by the shell onto the tooth. Additional examples of brackets and other tooth-mounted attachments suitable for use with orthodontic appliances are described in U.S. Pat. Nos. 6,309,215 and 6,830,450. 
       FIG.  1 B  illustrates a tooth repositioning system  110  including a plurality of appliances  112 ,  114 ,  116 . Any of the appliances described herein can be designed and/or provided as part of a set of a plurality of appliances. In such an embodiment, each appliance may be configured so that a tooth-receiving cavity has a geometry corresponding to an intermediate or final tooth arrangement intended to be achieved with the appliance. The patient&#39;s teeth can be progressively repositioned from an initial tooth arrangement to a target tooth arrangement by placing a series of incremental position adjustment appliances over the patient&#39;s teeth. For example, the tooth repositioning system  110  can include a first appliance  112  corresponding to an initial tooth arrangement, one or more intermediate appliances  114  corresponding to one or more intermediate arrangements, and a final appliance  116  corresponding to a target arrangement. A target tooth arrangement can be a planned final tooth arrangement selected for the patient&#39;s teeth at the end of all planned orthodontic treatment. Alternatively, a target arrangement can be one of many intermediate arrangements for the patient&#39;s teeth during the course of orthodontic treatment, which may include where surgery is recommended, where interproximal reduction (IPR) is appropriate, where a progress check is scheduled, where anchor placement is best, where palatal expansion is desirable, etc. As such, it is understood that a target tooth arrangement can be any planned resulting arrangement for the patient&#39;s teeth that follows one or more incremental repositioning stages. Likewise, an initial tooth arrangement can be any initial arrangement for the patient&#39;s teeth that is followed by one or more incremental repositioning stages. 
       FIG.  1 C  illustrates a method  150  of orthodontic treatment using a plurality of appliances, in accordance with many embodiments. The method  150  can be practiced using any of the appliances or appliance sets described herein. In step  160 , a first orthodontic appliance is applied to a patient&#39;s teeth in order to reposition the teeth from a first tooth arrangement to a second tooth arrangement. In step  170 , a second orthodontic appliance is applied to the patient&#39;s teeth in order to reposition the teeth from the second tooth arrangement to a third tooth arrangement. The method  150  can be repeated as necessary using any suitable number and combination of sequential appliances in order to incrementally reposition the patient&#39;s teeth from an initial arrangement to a target arrangement. The appliances can be generated all at the same stage or in sets or batches (e.g., at the beginning of a stage of the treatment), or the appliances can be fabricated one at a time, and the patient can wear each appliance until the pressure of each appliance on the teeth can no longer be felt or until the maximum amount of expressed tooth movement for that given stage has been achieved. A plurality of different appliances (e.g., a set) can be designed and even fabricated prior to the patient wearing any appliance of the plurality. After wearing an appliance for an appropriate period of time, the patient can replace the current appliance with the next appliance in the series until no more appliances remain. The appliances are generally not affixed to the teeth and the patient may place and replace the appliances at any time during the procedure (e.g., patient-removable appliances). The final appliance or several appliances in the series may have a geometry or geometries selected to overcorrect the tooth arrangement. For instance, one or more appliances may have a geometry that would (if fully achieved) move individual teeth beyond the tooth arrangement that has been selected as the “final.” Such over-correction may be desirable in order to offset potential relapse after the repositioning method has been terminated (e.g., permit movement of individual teeth back toward their pre-corrected positions). Over-correction may also be beneficial to speed the rate of correction (e.g., an appliance with a geometry that is positioned beyond a desired intermediate or final position may shift the individual teeth toward the position at a greater rate). In such cases, the use of an appliance can be terminated before the teeth reach the positions defined by the appliance. Furthermore, over-correction may be deliberately applied in order to compensate for any inaccuracies or limitations of the appliance. 
     In many embodiments, an orthodontic appliance includes one or more elastic members. The elastic member can be a band, cord, strip, loop, wire, spring, mesh, membrane, scaffold, layer, or any other suitable elastic connecting element, and can be fabricated from materials such as one or more polymers, one or more metals, or composites. In many embodiments, the elastic member can be fabricated by extrusion, rapid prototyping, spraying, thermoforming, or suitable combinations thereof. The elastic member can be fabricated from a single type of elastic material, or a plurality of different elastic material types. The characteristics of the elastic material (e.g., length, width, thickness, area, shape, cross-section, stiffness, etc.) can be selected based on the desired properties for the elastic member, e.g., magnitude and/or direction of forces to be applied by the elastic member. 
     An orthodontic appliance can include a shell having teeth receiving cavities as previously described herein and one or more elastic members coupled to the shell. Various configurations for coupling an elastic member to a shell are possible. One or more portions of the elastic member (e.g., portions at or near each end of the elastic member) can be coupled to the shell at a suitable number of attachment points (e.g., one, two, three, four, or more). Alternatively or in addition, one or more portions of the elastic member can be coupled to the shell over a continuous attachment region. Any description herein pertaining to attachment points can also be applied to attachment regions, and vice-versa. Each of the attachment points can be situated on any suitable portion of the shell, such as on a buccal surface, lingual surface, occlusal surface, gingival surface, internal surface (e.g., surface adjacent to or near the teeth), external surface (e.g., surface away from the teeth), or suitable combinations thereof. The position of the attachment points can be selected in order to control the forces (e.g., force magnitude and/or trajectory) applied to the teeth. In many embodiments, the elastic member is directly coupled to the attachment points on the shell without utilizing intervening attachment elements or fasteners. For example, the elastic member can be directly coupled to the shell by adhesives and/or bonding. As another example, the attachment points on the shell can be formed (e.g., integrally formed as a unitary or monolithic piece) with or into one or more hooks, protrusions, apertures, tabs, or other such features suitable for directly fastening the elastic member to the shell. In alternative embodiments, the elastic member may be indirectly coupled to the shell (e.g., via attachment elements or fasteners that are not integrally formed with the shell as a unitary or monolithic piece). In some instances, the elastic member is permanently affixed to the shell. Conversely, the elastic member can be removably coupled or otherwise detachable from the shell. In many embodiments, the elastic member is coupled only to the shell, and not to the teeth of the patient or an attachment mounted on the teeth. 
     The orthodontic appliance described herein can include one or more discontinuities formed in the shell. The one or more discontinuities can include one or more cuts, flaps, apertures (e.g., openings, windows, gaps, notches), and/or deformations (e.g., protrusions, indentations, reliefs) formed in any suitable portion of the shell (e.g., in a buccal, lingual, occlusal, and/or gingival surface). Exemplary geometries for such discontinuities are described in further detail herein. The discontinuities provided herein can be used to control the forces applied to a patient&#39;s teeth by an orthodontic appliance. In many embodiments, one or more discontinuities are used in combination with one or more elastic members in order to produce the desired forces. In alternative embodiments, an orthodontic appliance can include one or more discontinuities without using any elastic members, such that the forces applied to the teeth are modulated through the use of discontinuities alone. 
     In many embodiments, one or more elastic members are positioned to interact with one or more discontinuities in the appliance shell. In some instances, a discontinuity is located between two or more attachment points for an elastic member, such that a portion of the elastic member extending between the attachment points spans the discontinuity (or at least a part of the discontinuity). Alternatively or additionally, a portion of an elastic member between attachment points can extend around the discontinuity (e.g., around the periphery of an aperture or flap of the discontinuity). An elastic member can interact with a discontinuity by exerting forces directly on the discontinuity (e.g., pressing or pulling against a flap, deformation, etc.), as well as by exerting forces on portions of the shell adjacent to the discontinuity (e.g., applying force to portions of the shell surrounding a cut, aperture). Such interactions may comprise, for example, the elastic member applying a force on or in the region of the discontinuity when the appliance is worn (e.g., such that the resulting force is in a direction suitable to change the form of the discontinuity) and/or the elastic member applying a force on the discontinuity when the appliance is not being worn. In many embodiments, the applied force is at least partially generated by deformation (e.g., stretching, compressing, bending, flexing) of the elastic member. In some instances, the deformation of the elastic member can be caused by deformations of the corresponding discontinuity and/or shell, such as deformations occurring when the appliance is placed over teeth, as described in further detail below. 
     The interaction of the elastic member with the discontinuity can result in the application of forces on portions of the appliance shell. Associated resulting forces can be transmitted to the underlying teeth via the shell to elicit tooth movements (e.g., extrusion, intrusion, rotating, torqueing, tipping, and/or translating) towards a specified tooth arrangement. As the teeth move towards the specified arrangement, the deformation of the discontinuity may decrease, until the teeth reach the arrangement and the discontinuity fully reverts to its undeformed state (also known as the “fully expressed” state). In many embodiments, the shell includes a predetermined amount of internal space (e.g., in the teeth-receiving cavities of the shell) to accommodate tooth movements from an arrangement to a subsequent specified arrangement. The size of the internal space can be used to control the extent to which the teeth move. For example, the teeth can be prevented from moving further once they have traversed the available internal space and come into contact with an internal surface of the shell (e.g., the wall of a tooth-receiving cavity). Additionally, the geometry of the discontinuity (e.g., size) can also influence the extent of tooth movement, in that no more tooth movements are produced once the discontinuity has been fully expressed. In some instances, one or more portions of the internal surface can be fabricated from a more rigid material than the rest of the shell to ensure that the teeth are retained at the desired configuration. 
     The magnitude and/or direction of the forces applied to the teeth can be at least partially controlled by, influenced by, or based on the geometry of the discontinuity, as well as its positioning relative to the elastic member. The dimensions (e.g., length, width, depth, surface area, etc.) and/or the shape of the discontinuity can be calculated, for instance, to achieve a specified degree of appliance compliance. For example, portions of the shell adjacent to the discontinuity may be more compliant, while portions of the shell away from the discontinuity may be more rigid. In many embodiments, the discontinuity is configured to be deformable (e.g., changeable in shape, size) and/or displaceable, thereby increasing the local compliance of the appliance. The local compliance of various portions of the shell can be used to control the resulting forces exerted on the underlying teeth. 
     The forces applied to the teeth can also be influenced by characteristics of the elastic member (e.g., length, width, thickness, area, shape, cross-section, number, elastic coefficient and other material properties, etc.). Any suitable combination of characteristics can be used in order to elicit the desired tooth movements, and such characteristics can be homogeneous or variable within the elastic. In many embodiments, the elasticity of the elastic member can vary based on the direction of deformation of the elastic member (anisotropic elasticity). For example, an elastic member can be configured to be more compliant when deformed along one or more specified directions (e.g., longitudinal, lateral, etc.), and less compliant (or noncompliant) when deformed along all other directions, or vice-versa. The directionality of the elasticity can be used to control the resultant forces applied to the teeth. 
     Optionally, the elastic member can be deformed before being coupled to the appliance and/or before the appliance is worn by the patient (e.g., due to the placement of the attachment points and/or discontinuity), such that there is an initial “pre-loading” force or tension in the elastic member. The use of pre-loading can be used to produce a substantially constant force on the teeth throughout the treatment duration. Moreover, the use of pre-loading can ensure that sufficient force is applied to the teeth, e.g., in accordance with a desired treatment plan. Alternatively, the elastic member can be relaxed prior to attachment to the appliance and/or wearing of the appliance, such that there is no pre-loading force before the appliance is placed on the teeth. 
       FIG.  2 A  and  FIG.  2 B  illustrate an orthodontic appliance  200  with a coupled elastic member  202 , in accordance with many embodiments. The elastic member  202  is depicted as an elongate band or strip having two opposing ends. The ends of the elastic member  202  are attached to the exterior of a shell  204  shaped to receive teeth of a single dental arch. In  FIG.  2 A  and  FIG.  2 B , the elastic member  202  spans a discontinuity  206  formed in the shell  204 , with the ends of the elastic member  202  attached to the shell  204  on either side of the discontinuity  206 . The discontinuity  206  includes an elongate cut  208  which optionally terminates at either end in a circular aperture  210 . The circular apertures  210  can be used to prevent undesirable lengthening of the cut  208  when force is applied on the shell  204 . In alternative embodiments, other types of aperture shapes (e.g., oval apertures) can be used instead of circular apertures. In many embodiments, when the appliance  200  is placed on the teeth of a patient&#39;s dental arch  212  (as illustrated in  FIG.  2 B ), at least some portions of the shell  204  are deformed by the forces generated by the deliberately designed mismatch between the patient&#39;s current tooth configuration and the tooth arrangement specified by the geometry of the appliance  200 , resulting in a corresponding deformation of the discontinuity  206 . For example, stretching of the shell  204  can cause the elongate cut  208  to widen into an elongate aperture  214 . The deformation of the discontinuity causes the geometry of the appliance to more easily comply with the current positions of the patient&#39;s teeth, thereby reducing the discomfort experienced by the patient when wearing the appliance. Additionally, the deformation of the discontinuity can enable the appliance to accommodate the patient&#39;s teeth even in situations where the teeth are not in an ideal arrangement relative to the configuration of the appliance (e.g., due to inaccuracies in appliance fabrication, inaccurate measurement data of the initial teeth arrangement, tooth movements lagging behind or not conforming to the treatment plan, etc.). Furthermore, the deformation can allow the appliance to effect larger tooth movements, thus enabling the appliance to be used for a longer time. 
     The deformation of the discontinuity  206  and/or shell  204  generally results in deformation of the elastic member  202 . For example, the elastic member  202  can be stretched by the widening of the discontinuity  206 . The tension in the elastic member  202  generated by such deformation can be reacted to as a continuous force by portions of the shell  204 , such as portions of the shell  204  adjacent the discontinuity  206 , in many embodiments. Associated resulting forces can be transmitted by the shell  204  to the underlying teeth so as to elicit tooth movements repositioning the teeth to a desired predetermined arrangement. For example, since the discontinuity  206  is situated adjacent the tooth  216 , the appliance  200  can exert forces on the tooth  216  and its neighbor  218 , causing them to move towards each other (see, e.g., arrows  220 ). This movement can reduce the interproximal space between the teeth  216 ,  218 , thereby shortening the mesial-distal length of the arch  212  (see, e.g., arrow  222 ). The deformation of the shell  204 , discontinuity  206 , and/or elastic member  202  can decrease as the repositioning of the teeth reduces the mismatch between the tooth arrangement and appliance geometry, thus diminishing the amount of force expressed on the teeth by the appliance  200 . 
       FIG.  2 C  and  FIG.  2 D  illustrate an orthodontic appliance  230  with a coupled elastic member  232  and a discontinuity  234  formed within a shell  236 , in accordance with many embodiments. The discontinuity  234  is similar to the discontinuity  206  of  FIG.  2 A , except that the cut  208  is replaced with a narrow elongate aperture, which can be formed in any suitable manner, such as by removing material from the shell  236 . As used herein, narrow may mean, for example, that the aperture has an extension in one direction of more than twice, e.g., more than four times, its dimension in a second, e.g., perpendicular, direction. When placed on a patient&#39;s arch  238 , as depicted in  FIG.  2 D , the discontinuity  234  and the elastic member  232  are situated adjacent to a tooth  239 . The elongate aperture of the discontinuity  234  can be deformed when worn (e.g., the size of the aperture increases), generating tension in the elastic member  232  and causing it to exert forces on portions of the shell  236  disposed on opposite sides of the discontinuity  234 . Associated resulting forces can be applied to the underlying teeth to close an interproximal space (see, e.g., arrows  240 ) and thereby reduce the overall arch length (see, e.g., arrow  242 ). 
       FIG.  2 E  and  FIG.  2 F  illustrate an orthodontic appliance  250  with a coupled elastic member  252  and a discontinuity  254 , in accordance with many embodiments. The discontinuity  254  can be formed as an elongated cut in the shell  256 , similar to the discontinuity  206  of the appliance  200 . When the appliance  250  is worn (as depicted in  FIG.  2 F ), the discontinuity  254  can be situated adjacent the interproximal space between tooth  258  and tooth  260 . The elastic member  252  can be attached to the shell  256  at attachment points adjacent the teeth  258 ,  260  when the appliance  250  is worn. The principle of operation of the appliance  250  is similar to that of the appliances  200 , and  230 , in that the elastic member  252  interacts with the discontinuity  254  to elicit tooth movements (see, e.g., arrows  262 ) that reduce the interproximal space between the teeth  258 ,  260 . 
       FIG.  2 G  and  FIG.  2 H  illustrate an example of an orthodontic appliance  270  having a plurality of elastic members  272  and discontinuities  274  formed within a shell  276 . Each elastic member  272  is positioned to span one of the plurality of discontinuities  274 , which are depicted as cuts in the shell  276 . The discontinuities  274  are disposed adjacent to the interproximal regions when the appliance  270  is worn over the arch  278  (as illustrated in  FIG.  2 H ). The interactions between the elastic members  272  and discontinuities  274  can produce forces for repositioning the teeth to reduce an interproximal space (see, e.g., arrows  280 ). Although the elastic members  272  and discontinuities  274  are depicted in  FIG.  2 G  and  FIG.  2 H  as situated solely on the buccal surface of the appliance, they can also be situated on other surfaces, such as on the lingual surface or on the occlusal surface, as well as combinations of any these surfaces. For example, an appliance can include some discontinuities and elastics situated on a lingual surface and some discontinuities and elastics situated on a buccal surface. In this configuration, forces are applied to the underlying teeth via both surfaces of the shell, thereby increasing the repositioning efficiency. One, two, three or more discontinuities may additionally or alternatively be disposed in other regions than the regions adjacent to the interproximal regions, and each discontinuity may optionally be spanned by none, one, or more elastic members. 
       FIG.  2 I  illustrates additional example geometries for one or more discontinuities in an orthodontic appliance shell, in accordance with many embodiments. As previously mentioned, a discontinuity can have any suitable configuration, for example, such as a cut, flap, aperture, deformation, and the like. For example, a discontinuity can include a cut in the shell, and the cut can include linear portions and/or curved portions (e.g., curvilinear cut  290 ). As another example, the discontinuity can include an aperture formed in a suitable shape, such as a circle, ellipse (e.g., elliptical apertures  292 ,  294 ), triangle, square, rectangle (e.g., rectangular aperture  296 ), or other polygonal shape, and/or suitable combinations thereof. The discontinuities and/or elastics can be positioned in any suitable orientation. For example, the elastic member can extend vertically (along a occlusal-gingival direction), horizontally or longitudinally (along a mesial-distal direction), or any other suitable orientation. Similarly, the discontinuity may extend vertically (e.g., discontinuities  290 ,  294 ,  296 ), horizontally or longitudinally (e.g., discontinuity  292 ), or any other suitable orientation. The orientation of the elastic member and/or discontinuity can be selected based on the desired tooth movements. In some instances, different orientations can be used to produce different types of movements. 
     In many embodiments, a discontinuity can be composed of a plurality of individual elements arranged in a suitable configuration (e.g., plurality of circular apertures  298 ). An appliance can incorporate any suitable number and type of discontinuities, and the discontinuities can interact with any suitable number of elastic members. For example, a single elastic member can be paired with a single discontinuity. Alternatively, a plurality of elastic members can interact with a single discontinuity. Conversely or additionally, a single elastic member can interact with a plurality of discontinuities. The discontinuities described herein, along with their corresponding elastic member(s), can be arranged on the shell in any suitable manner relative to the underlying dentition (e.g., adjacent to one or more teeth, one or more interproximal regions, etc.) and to each other. 
       FIG.  3 A  and  FIG.  3 B  illustrate an orthodontic appliance  300  for repositioning teeth  310 , in accordance with many embodiments. For example, the appliance  300  can be used to reduce interproximal space between the teeth  310 . The orthodontic appliance  300  includes a shell  304  and an elastic member  302  coupled with the shell  304 . The shell  304  has a plurality of discontinuities  308  formed in the shell. The length of the elastic member  302  extends along the surface of the shell  304  spanning a plurality of teeth-receiving cavities  306 . The elastic member  302  spans the discontinuities  308 , depicted herein as cuts, although other geometries can also be used. When placed on the teeth  310  of the patient as depicted in  FIG.  3 B , the discontinuities  308  can deform to form a plurality of openings. The orthodontic appliance  300  can be configured such that each of the openings of discontinuities  308  is positioned over or adjacent to a respective interproximal region of the teeth  310 . The elastic member  302  can exert forces on the shell  304  such that resulting associated forces are applied to the teeth  310 , thereby eliciting tooth movements to reduce the size of the interproximal space(s) between the teeth  310 . 
     In many embodiments, the appliance includes one or more retention features that are formed in the shell (e.g., grooves, ridges, protrusions, indentations, etc.) to retain the elastic member (or suitable portions thereof) at a specified position relative to the shell. The retention features may be beneficial in instances where the elastic member is relatively long and therefore more prone to slippage relative to the shell  304 . For instance, the shell  304  of the appliance  300  can include a groove (not shown) configured to constrain the elastic member  302  to a configuration spanning the teeth-receiving cavities  306  and the discontinuities  308 . Such retention features can be used to prevent the accidental displacement or release of the elastic member from the desired position, thereby ensuring that the appropriate therapeutic force is maintained. 
       FIG.  4 A  illustrates configurations of an orthodontic appliance  400  for repositioning teeth, in accordance with many embodiments. For example, the appliance  400  can be used to reduce interproximal space between teeth. The appliance  400  includes a shell  406 , a plurality of elastic members  402 , each of which spans one of a plurality of discontinuities  404  (depicted as cuts terminating in circular apertures) formed in the shell  406 . The elastic members  402  and discontinuities  404  are situated on the occlusal surface of the shell  406  near the interproximal region between teeth  408  and  410 . The appliance  400  is configured to reduce the size of the interproximal space between teeth  408 ,  410 . In many embodiments, the mesial-distal arch length of the shell  406  is shorter when the appliance is not being worn by a patient (configuration  412 ) compared to when it is being worn (configuration  414 ), e.g., by an amount  415 , due to the increased interproximal space in the patient&#39;s initial tooth arrangement versus the tooth positions of the appliance  400 . The discontinuities  404  can be deformable to contribute to the compliance of the appliance  400  and relieve some of the initial forces generated by the mismatch between the geometry of the patient&#39;s teeth and the geometry of the appliance  400 . Similar to the other embodiments described herein, the elastic members  402  can apply a continuous force between portions of the shell  406  to elicit tooth movements (see, e.g., arrows  416 ) that reduce and may eliminate the interproximal space between teeth  408 ,  410 . 
       FIG.  4 B  illustrates configurations of an orthodontic appliance  450  for reducing an interproximal space between teeth, in accordance with many embodiments. The appliance  450  includes a shell  456  and a plurality of elastic members  452  spanning a single discontinuity  454  (depicted as a single cut) formed in the shell  456 . The discontinuity  454  can be a complete cut in the shell  456  separating it into discrete segments, or it can be a partial cut such that the shell  456  remains a single segment. Similar to the appliance  400 , the discontinuity  454  can be deformed (e.g., widened from a cut into an elongate aperture) when the appliance  450  is placed on the teeth of a patient, such that the mesial-distal arch length of the shell  456  is shorter in the unworn configuration  458  than in the worn configuration  460 , e.g., by an amount  461 . As previously described, the elastic members  452  exert repositioning forces causing closure of the interproximal space (see, e.g., arrows  462 ). 
       FIG.  5 A  through  FIG.  5 C  illustrate an orthodontic appliance  500  for repositioning teeth of a dental arch, in accordance with many embodiments. In the depiction of  FIG.  5 A  through  FIG.  5 C , the appliance  500  is configured to increase a space between teeth of a lower dental arch, although the concepts presented herein can also be applied to space expansions in the upper dental arch. Space expansion (which can involve expansion of interproximal spaces between adjacent teeth, as well as expansion of spaces resulting from tooth removal) can be beneficial for various dental procedures (e.g., implants, treatment of impacted teeth). The techniques disclosed herein, however, can also be used for other applications, such as decreasing a space between teeth, moving a tooth, tipping a tooth, rotating a tooth, and so on. Any description herein referring to space expansion can also be applied to other types of orthodontic repositioning, and vice-versa. The appliance  500  includes a shell  501  and first and second elastic members  502 ,  504 , interacting respectively with first and second discontinuities  506 ,  508  formed in the shell  501 . In alternative embodiments, instead of one elastic member per discontinuity, two or more elastic members may be used for each discontinuity. The elastic members  502 ,  504  and associated respective discontinuities  506 ,  508  can be situated over teeth  510 ,  512  immediately adjacent to a space  514  when the appliance  500  is placed on a patient&#39;s lower arch  516  (depicted in  FIG.  5 B ). The elastic members and the discontinuities can be configured in any manner suitable for producing space-expanding tooth movements. For example, as illustrated in  FIG.  5 A  through  FIG.  5 C , the discontinuities  506 ,  508  can each be configured as an aperture positioned over the tooth surfaces adjacent to the space  514 . Each aperture extends towards the crown of each tooth and is spanned by the elastic member. The respective elastic member can extend around the entire circumference of the tooth and be attached to the shell over the same tooth (see, e.g., elastic member  502 ), or extend partially around the circumference of the tooth and be attached to the shell over an adjacent tooth (see, e.g., elastic member  504 ). In either case, the ends of the elastic member  504  can be respectively attached to the buccal and lingual sides of the shell  501  such that the teeth  510 ,  512  are moved to increase the space between the teeth  510 ,  512  (e.g., in the direction indicated by arrows  518 ). For example, the tooth  512  can be moved so as to reduce and often eliminate an interproximal space  519  between the tooth  512  and the adjacent tooth so as to reposition the teeth as illustrated in  FIG.  5 C . 
     When the appliance  500  is placed over the arch  516 , the elastic members  502 ,  504  interact with the discontinuities  506 ,  508  to apply forces on the teeth  510 ,  512 , thereby moving the teeth  510 ,  512  in desired directions (see, e.g., arrows  518 ) so as to expand the space  514 . In many embodiments, the extent of the movement can be varied based on the size of the discontinuities  506 ,  508 .  FIG.  5 C  illustrates the tooth configuration of the lower arch  516  after repositioning, with an expanded space  514 . The repositioning of the teeth  510 ,  512  reduces the mismatch between the patient&#39;s teeth arrangement and the appliance geometry, thereby causing the deformation of the discontinuities  506 ,  508  to be reduced relative to the previous configuration depicted in  FIG.  5 B . 
       FIG.  6    illustrates an orthodontic appliance  600  including a channel  602  accommodating an attachment  604  mounted on a tooth  606 , in accordance with many embodiments. The channel  602  can be formed within the internal cavity of the shell  608  of the appliance  600 , such that the attachment  604  is received within the channel  602  when the appliance  600  is placed over the patient&#39;s arch  610 . The channel  602  can be configured to guide the movement of the tooth  606  as it is repositioned due to forces applied by the elastic member  612  on and/or near the discontinuity  614 . For example, the geometry of the channel  602  can be used to constrain the movement of the tooth  606  along a predetermined trajectory (e.g., a trajectory substantially parallel to the channel  602 ). Additionally, the channel  602  can be used to produce intrusion or extrusion of the tooth as it moves along the trajectory. Although the channel  602  is depicted herein as extending along a mesial-distal direction, other orientations can also be used, such as an occlusal-gingival direction (e.g., to produce intrusion, extrusion, leveling, etc.). In many embodiments, an appliance may include a plurality of channels receiving a plurality of corresponding attachments, such as a buccal channel and a lingual channel respectively accommodating a buccal attachment and a lingual attachment on the underlying tooth. The use of multiple channel-attachment pairs can be used to increase the efficiency and accuracy of tooth repositioning. Furthermore, the materials of the channels and attachments can be selected to optimize force expression and tooth repositioning. For example, the channel and the attachment can each be fabricated from different materials. In many embodiments, the materials can be selected to minimize the frictional coefficient between the channel and attachment, so that the attachment can be moved freely within the channel. 
       FIG.  7 A  through  FIG.  7 D  illustrate an orthodontic appliance  700  for repositioning teeth of a dental arch, in accordance with many embodiments. The appliance  700  includes a shell  710  and first and second pairs of elastic members  702 ,  704 , which interact respectively with the first and second discontinuities  706 ,  708  formed in the shell  710 . In alternative embodiments, a different number of elastic members can be used for each discontinuity, e.g., a single elastic member, or more than two elastic members. When the appliance  700  is placed on the arch  712  (depicted in  FIG.  7 B ), the elastic member pairs  702 ,  704  and the discontinuities  706 ,  708  are situated on the teeth  714 ,  716  immediately flanking the space  718 . As illustrated in  FIG.  7 C , when the appliance  700  is worn, the discontinuities  706 ,  708  are deformed to form gaps in the shell  710  extending over the occlusal surfaces of the teeth  714 ,  716 , and the elastic members  702 ,  704  extend from the lingual surfaces to the buccal surfaces of the teeth  714 ,  716 . The interaction between the elastic members  702 ,  704  and the discontinuities  706 ,  708  result in tooth movements (see, e.g., arrows  720 ) expanding the size of the space  718 . As previously described, the magnitude of the tooth movements can be influenced by the size of the discontinuities  706 ,  708 .  FIG.  7 D  illustrates the repositioned arch  712 , in which the space  718  has been expanded and the discontinuities  706 ,  708  have reverted to their respective undeformed configuration (the fully expressed state). 
     In many embodiments, the orthodontic appliances presented herein can include a shell that is separated into two or more discrete segments, which may be referred to as “segmented orthodontic appliances.” A shell can be separated into any suitable number of segments, e.g., two, three, four, five, or more. The shell can be separated into two or more horizontal (mesial-distal) segments. Alternatively or in addition, the shell can be separated into two or more vertical (occlusal-gingival) segments. Each shell segment can receive a different subset of the patient&#39;s teeth. Different segments can receive different numbers of teeth. Alternatively, some or all of the segments can receive the same number of teeth. The shell segments can be joined to each other via one or more elastic members so as to form a single orthodontic appliance. The elastic members can permit movement of the shell segments relative to each other, and the direction of permitted movement can be determined based on the desired tooth movements to be achieved (e.g., extrusion, intrusion, translation, etc.). In many embodiments, the segments can move relative to each other along a plurality of different directions. Alternatively, the segments may be constrained to move along a single direction. For example, the shell segments can be movable relative to each other only along a horizontal (mesial-distal) direction , only along a vertical (occlusal-gingival) direction, or any suitable intermediate angle. Constrained movement can be achieved using various techniques, such as guide features that define the permissible direction(s) of motion. In many embodiments, such guide features include a first element (e.g., a channel or groove) located on a first shell segment and a second element (e.g., a protrusion that first into the channel or groove) located on a second shell segment, such that the shell segments are only permitted to move along certain directions (e.g., along the length of the channel) when the two elements are engaged with each other. Moreover, the guide features can include elastic elements (e.g., spring elements) that apply forces on the segments to displace them relative to each other (e.g., towards each other or away from each other). 
       FIG.  8 A  through  FIG.  8 C  and  FIG.  8 G  illustrate an orthodontic appliance  800  that includes a first shell segment  806  and a second shell segment  808 , which can be viewed as being separated by a discontinuity  804  (e.g., the separation between the two segments  806 ,  808 ). As depicted herein, the segments  806 ,  808  of the appliance  800  are horizontal (mesial-distal) segments. The first and second shell segments  806 ,  808  have guide features  802 . The first and second segments  806 ,  808  are movable relative to each other. A plurality of elastic members  810  spans the discontinuity  804  and is coupled to the first and second segments  806 ,  808 . In many embodiments, the first and second segments  806 ,  808  are configured to overlap, with a portion of the first segment  806  positioned over a portion of the second segment  808 , such that the two segments  806 ,  808  can telescopically slide relative to each other. 
     The guide features  802  formed in the segments  806 ,  808  are configured to guide the movement of the segments  806 ,  808  relative to each other. For example, the guide features can include mating telescopic features (e.g., protrusions  812  sliding within channels  814 ) that constrain the relative motion between the segments  806 ,  808  along a specified direction.  FIG.  8 H  and  FIG.  8 I  illustrate a top view and side view, respectfully, of an exemplary telescopic guide feature  870  including a piston element  872  and spring element  874 , in accordance with many embodiments. The piston  872  can slide telescopically within a channel  876 . The spring  874  can be any suitable elastic piece or element. In many embodiments, the spring  874  is disposed within the channel  876 , with its ends coupled respectively to the interior of the channel  876  and one end of the piston  872 , such that the elasticity of the spring  874  controls the amount of force needed to displace the piston  872  relative to the channel  876  (e.g., inwards and/or outwards). 
     The guide features described herein can be integrally formed with the appliance shell, or provided as separate elements that are attached to the shell. In many embodiments, the guide feature  870  can be installed within the channels  814  of the appliance  800 . Alternatively or in addition, the guide feature  870  can be installed on the shell segments  806 ,  808  of the appliance  800  without requiring the channels  814 . For example, the guide feature  870  may be provided as a separate element and fastened to the appliance  800  using one or more fasteners  878  (e.g., rivets, screws, pins, etc.). Any suitable configuration and/or number of telescopic features (or other guide features) can be used in conjunction with any suitable configuration and/or number of elastic members and discontinuities.  FIG.  8 G  illustrates a cross-section of segment  806  in which the telescopic channels  814  and the elastic members  810  are interspersed with each other. The guide features and the elastic members can be situated on any suitable portion of the appliance, such as the lingual, occlusal, and/or buccal surfaces of the appliance. 
     When the appliance  800  is placed over an arch  816  (as illustrated in  FIG.  8 B ), the segments  806 ,  808  may be displaced relative to each other (e.g., moved apart). The elastic members  810  can exert a force on the segments  806 ,  808  resisting the displacement and pulling the segments  806 ,  808  toward each other. The resulting associated forces applied to the teeth induce repositioning of the teeth of the arch  816  (see, e.g., arrow  818 ) so as to reduce the arch length (e.g., by closing the interproximal space  820 ). The guide features  802  can act in parallel with the elastic members  810  to control the magnitude and/or direction of the forces expressed on the teeth.  FIG.  8 C  illustrates the teeth of the arch  816  after repositioning, with the space  820  closed and the two segments  806 ,  808  returned to the original configuration of  FIG.  8 A . 
       FIG.  8 D  through  FIG.  8 F  illustrate an orthodontic appliance  850  with telescopic shell segments  852 ,  854 , in accordance with many embodiments. The shell segments  852 ,  854  are depicted herein as being vertical (occlusal-gingival) segments. The first shell segment  852  and the second shell segment  854 , which can be viewed as being separated by a discontinuity  856 , are movable relative to each other, such that the first segment  852  overlaps and slides telescopically over the second segment  854 . A plurality of elastic members  858  spans the discontinuity  856  and is coupled to the first and second segments  852 ,  854 . When the appliance  850  is placed over an arch  860  (as illustrated in  FIG.  8 E ), the segments  852 ,  854  may be displaced relative to each other (e.g., moved apart). The elastic members  858  can resist the displacement and pull the segments  852 ,  854  towards each other, causing repositioning of the teeth of the arch  860  (e.g., intrusion of the teeth, as illustrated in  FIG.  8 F ). In many embodiments, the orthodontic appliance  850  can include one or more of the guide features described herein in order to more precisely direct the relative movements of the segments  852 ,  854 . 
       FIG.  25 A  illustrates an orthodontic appliance  2500  having a telescopic guide feature  2502 , in accordance with many embodiments. The appliance  2500  includes a shell  2504  that is separated into discrete segments  2506 ,  2508 , with the guide feature  2502  joining the two segments,  2506 ,  2508 . The two segments  2506 ,  2508  can be configured to move relative to each other without sliding telescopically over each other. In alternative embodiments, the segments  2506 ,  2508  can be configured for telescopic sliding, similar to the embodiments of  FIGS.  8 A through  8 F . The guide feature  2502  can be used to constrain the relative movement of the shell segments  2506 ,  2508  along a specified direction of motion. In many embodiments, the guide feature  2502  includes an elastic member (e.g., a spring element) that provides the force for eliciting tooth movements. For example, the guide feature  2502 , can include a slidable piston element  2510  coupled to an elastic spring element  2512 , similar to the guide features previously described herein with respect to  FIGS.  8 H and  8 I . The guide feature  2502  can be arranged such that when the appliance  2500  is placed on the teeth  2514 , the spring element  2512  is compressed by the piston  2510 , and thus exerts forces (indicated by arrows) to displace the shell segments  2506 ,  2508  away from each other. The resultant forces exerted on the teeth  2514  can be used to move teeth apart, e.g., to increase a space between teeth. 
       FIG.  25 B  illustrates an orthodontic appliance  2550  having a telescopic guide feature  2552 , in accordance with many embodiments. Similar to the appliance  2500 , the appliance  2550  includes a shell  2554  having discrete segments  2556 ,  2558  joined by the guide feature  2552 . The guide feature can include a slidable piston element  2560  coupled to a spring element  2562 . The guide feature  2552  can be arranged such that when the appliance  2550  is placed on the teeth  2564 , the spring element  2562  is stretched by the piston  2560 , and thus exerts forces (indicated by arrows) to displace the shell segments  2556 ,  2558  towards each other. The resultant forces exerted on the teeth  2564  can be used to move teeth together, e.g., to reduce a space between teeth. 
     In many embodiments, the orthodontic appliances described herein can be configured to maintain a current position of a patient&#39;s teeth, rather than repositioning the teeth. Such tooth retaining appliances, also known as retainers, are generally similar to the tooth repositioning appliances described herein, except that the appliance geometry is selected to exert forces on the teeth without causing repositioning of the teeth. In such embodiments, the tooth arrangement specified by the appliance geometry can be substantially similar to the current tooth arrangement of the patient. A retaining appliance may be worn by a patient, for instance, after orthodontic treatment is complete, in order to reduce or prevent movement of the teeth away from the corrected configuration. Any description herein relating to tooth repositioning appliances can also be applied to tooth retaining appliances, and vice-versa. 
       FIG.  9    illustrates an orthodontic appliance  900  configured to maintain a current position of the patient&#39;s teeth, in accordance with many embodiments. The appliance  900  includes a shell  906  and one or more elastic members  902  interacting with a discontinuity  904  formed in the shell  906 . For example, the discontinuity  904  can include one or more cuts in the shell  906 . The discontinuity  904 , e.g., cuts, may extend to a peripheral edge of the shell  906  (e.g., a gingival edge). As illustrated in  FIG.  9   , the elastic members  902  can be attached on the lingual and buccal surfaces of the shell  906  and span the discontinuity  904 . When worn on an arch  908 , the appliance  900  can exert a continuous force on one or more teeth to prevent the teeth from moving out of their current arrangement. The magnitude of such forces can be smaller than the magnitude of forces for eliciting tooth movements. Furthermore, the elastic members  902  can function as clasps to prevent the appliance  900  from moving or becoming dislodged from the teeth. The configuration of the shell, elastic members and/or the discontinuity can be selected to prevent inadvertent tooth repositioning. 
     In many embodiments, in order to improve control over the forces applied to teeth by an orthodontic appliance, the appliance shell can include features such as dimples, ridges, protrusions, etc. that contact teeth at a specified point or region so as to selectively apply force to that point or region. This approach can increase control over the magnitude and/or direction of force application to the teeth, thereby producing more controlled tooth movements and enabling the application of more complex force systems. 
       FIG.  10 A  through  FIG.  10 C  illustrate an orthodontic appliance  1000  with a lingual  1002  and buccal protrusion  1004 , in accordance with many embodiments. The lingual protrusion  1002  and buccal protrusion  1004  are formed as curved surfaces on the lingual and buccal surfaces of the shell  1006 , respectively, and protrude into the internal cavity of the shell  1006 . The shell  1006  can include a pair of discontinuities  1008  formed on the lingual and buccal surfaces, respectively. Each of the discontinuities  1008  can be formed as a cut in the shell  1006  defining a flap surrounding the corresponding protrusion (as illustrated in  FIG.  10 G ) and can be spanned by a pair of elastic members  1010 . When placed on an arch  1012  of a patient (as illustrated in  FIG.  10 B ), the protrusions  1002 ,  1004  are deflected outwards by the underlying tooth  1014 . The elastic members  1010  can resist the deflection by exerting forces that are applied inwards against the tooth  1014  by the protrusions  1002 ,  1004 , thereby causing tooth movement (see, e.g., arrow  1016 ).  FIG.  10 C  illustrates the appliance  1000  and the arch  1012  after repositioning of the tooth  1014  has occurred. 
       FIG.  10 D  through  FIG.  10 F  illustrate an orthodontic appliance  1050  divided into discrete shell segments  1052 ,  1054 , in accordance with many embodiments. The appliance  1050  can be used to increase the size of a space between teeth, for instance, to accommodate installation of a dental prosthesis such as an implant  1056 . The shell segments  1052 ,  1054  are coupled to each other by elastic members  1058 ,  1060  spanning the discontinuities  1062 ,  1064 , respectively. When placed on an arch  1066  (as illustrated in  FIG.  10 E ), the segments  1052 ,  1054  are moved apart from each other due to the arrangement of the underlying teeth, thereby stretching the elastics  1058 ,  1060 . The tension in the elastics  1058 ,  1060  can result in application of repositioning forces to the teeth. For example, the tooth  1068  can be repositioned to increase space for the implant  1056 . In many embodiments, the shell segments, discontinuities, and elastic members can be configured to reposition the tooth  1068  in a plurality of phases. In a first phase, the tooth  1068  can be translated along a mesial direction (see, e.g., arrows  1070 ). In a second phase, the tooth  1068  can be rotated (see, e.g., arrows  1072 ). The phases may occur sequentially, such that the tooth  1068  is first translated then rotated, or vice-versa. Alternatively, in some instances, the first and second phases can overlap or occur simultaneously, such the tooth  1068  is translated and rotated at the same time.  FIG.  10 F  illustrates the arch  1066  after repositioning, in which the tooth  1068  has been moved to expand the space available for the implant  1056 . 
       FIG.  16 A  through  FIG.  16 C  illustrate an orthodontic appliance  1600  including a protrusion  1602  for applying force to a tooth, in accordance with many embodiments. As illustrated in  FIG.  16 A , the internal surface profile of the appliance  1600  has a curved surface that forms the protrusion  1602 , which extends into the interior of the appliance.  FIG.  16 B  illustrates a cross-sectional view of a shell  1604  of the appliance  1600 , in which the protrusion  1602  is implemented as a curved portion  1606  of the shell  1604 . The curved portion  1606  is situated adjacent to or near a discontinuity  1608  in the shell  1604 , depicted herein as a cut formed in the shell  1604 . An elastic member  1610  is coupled to the shell  1604  spanning the discontinuity  1608 , such that one end of the elastic member  1610  is attached to or near the curved portion  1606 .  FIG.  16 C  illustrates a tooth  1612  received within the shell  1604  and displacing the curved portion  1606  outward relative to its initial configuration. The elastic member  1610  can exert force on the curved portion  1606  resisting the displacement (see, e.g., arrow  1614 ). In many embodiments, the exerted force results in associated force being transmitted to the tooth  1612  at a contact point by the curved portion  1606 . Application of force to the contact point can be used, for example, to elicit a tipping movement of the tooth  1612 . 
       FIG.  17 A  through  FIG.  17 C  illustrate an orthodontic appliance  1700  including a protrusion  1702  for applying force to a tooth, in accordance with many embodiments.  FIG.  17 A  illustrates the internal surface profile of the appliance  1700 , including the curved protrusion  1702 , and is similar to the embodiment of  FIG.  16 A .  FIG.  17 B  illustrates a cross-sectional view of a shell  1704  of the appliance  1700  in which the protrusion  1702  is implemented as a knob or button  1706  formed on the interior of the shell  1704 . Similar to the appliance  1600 , the appliance  1700  includes a discontinuity  1708  (e.g., a cut) adjacent to or near the knob  1706 , and an elastic member  1710  spanning the discontinuity  1708  and attached at one end to or near the knob  1706 . When the appliance receives a tooth  1712 , the elastic member  1710  can apply force to the tooth  1712  (see, e.g., arrow  1714 ) at a contact point via the knob  1706 . 
       FIG.  18 A  through  FIG.  18 C  and  FIG.  19    illustrate orthodontic appliances that include protrusions for applying forces to teeth, in accordance with many embodiments. The protrusions can be any suitable feature extending from the shell surface to apply force to a tooth via contact between the protrusion and the tooth, such as the embodiments previously described herein (e.g., curved surface  1606 , knob  1706 ).  FIG.  18 A  illustrates an appliance  1800  including a pair of protrusions  1802  situated over a tooth  1804 . Each of the protrusions  1802  is positioned near a discontinuity, depicted herein as a cut forming a curved flap  1806  in the shell  1808 , such that the protrusion is disposed on the underside of the flap  1806  (extending into the interior of the shell  1808  towards the tooth  1804 ). An elastic member, depicted herein as a band or strip  1810 , is attached to the shell  1808  on opposing sides of the flap  1806  and extends over the flap  1806 .  FIG.  18 B  illustrates an alternative configuration for the appliance  1800 , in which the elastic member is implemented as an elastic membrane or elastic mesh  1812  that connects the edges of the flap  1806  to the adjacent edges of the shell  1804 .  FIG.  18 C  illustrates another exemplary configuration for the appliance  1800 , in which the elastic member includes an elastic membrane or elastic mesh  1814  that is positioned over the entirety of the flap  1806  and a portion  1804  of the shell adjacent to the flap  1806 . In each of the previous examples, the elastic member can generate forces that are applied to the flap  1806  and thereby generate forces that are applied by the protrusion  1802  against the tooth  1804 . The positioning of the protrusions  1802  can be configured to control the tooth movements resulting from the application of these forces. For example, as depicted in  FIG.  19   , an appliance  1900  can include a pair of protrusions  1902  situated on different sides of a tooth  1904  (e.g., on a buccal surface and a lingual surface, respectively). The positioning of the protrusions  1902 , when combined with a suitable set of elastic members and discontinuities (not shown), can be used, for instance, to elicit a rotational tooth movement (see, e.g., arrows  1906 ). The elastics described herein can be coupled to the shell and/or flap using any suitable method. For example, the elastics can be extruded, sprayed, or otherwise directly adhered onto the shell and/or flap. 
       FIG.  20 A  through  FIG.  20 C  illustrate an orthodontic appliance  2000  including an elastic member with an attachment  2002 , in accordance with many embodiments. The elastic member, depicted herein as a mesh or membrane  2004 , is formed with or coupled to an attachment  2002 . The attachment  2002  (e.g., a protrusion, post, stud, button, etc.) is configured to engage and apply force to a tooth  2006 . In many embodiments, the attachment  2002  contacts the tooth  2006  through a discontinuity formed in the shell  2008  of the appliance  2000 , depicted herein as an aperture  2010 . The attachment  2002  can contact the tooth  2006  directly, or indirectly (e.g., via an attachment mounted on the tooth). The elastic member can be coupled to the shell  2008  in a position spanning the discontinuity such that the attachment  2002  extends into the interior of the shell  2008  through the discontinuity. For example, the mesh  2004  is shaped to cover the aperture  2010  and includes an adhesive perimeter  2012  enabling the mesh  2004  to be directly coupled to the shell  2008 . When the mesh  2004  is attached to the shell  2008 , the attachment  2002  protrudes through the aperture  2010  towards the tooth  2006 . 
     In many embodiments, the orthodontic appliance is configured to exert force on a tooth via one or more attachments mounted to the tooth. As previously described herein, an attachment can be coupled to the surface of one or more teeth to transmit forces exerted by the appliance onto the teeth. The geometry of the attachment and its position on the tooth can influence the magnitude and/or direction of the forces applied to the tooth. In many embodiments, the attachment is configured to elicit tooth movements that may be difficult to achieve with the appliance alone (e.g., extrusion). 
     The interactions between the appliance (e.g., an elastic member, a shell, a flap formed in the shell, etc.), attachment (e.g., mounted on a tooth), and teeth can be influenced by friction between these elements. In many embodiments, the frictional coefficient between the appliance and attachment is configured to be smaller than the frictional coefficient between the appliance and the tooth. This arrangement can enable the attachment to move freely relative to the appliance, while increasing the force applied onto the teeth by the appliance. The frictional coefficient can be a function of the material and/or surface properties. In many embodiments, the appliance and the attachment are fabricated using different types of materials, and such materials may be selected based on their material and/or surface properties. Furthermore, the frictional coefficient can be increased or decreased by application of suitable coatings, films, texturing, and the like. 
       FIG.  11 A  through  FIG.  11 B  illustrate an orthodontic appliance  1100  configured to engage an attachment  1102  on a tooth  1106 , in accordance with many embodiments. The appliance  1100  includes a receptacle  1104  that is configured to accommodate the attachment  1102  coupled to the tooth  1106 . For example, the receptacle  1104  can be a protrusion extending outward from the surface of the shell  1108 , with an interior space shaped to receive the attachment  1102 . In many embodiments, the receptacle  1104  is also shaped to accommodate and/or guide the movement of the attachment relative to the shell  1108 , such as movements corresponding to repositioning of the underlying tooth. The receptacle  1104  can include, for example, a sloped lateral wall  1112  along which the attachment  1102  can slide as the tooth  1106  moves upwards or downwards (along a gingival-occlusal axis). 
     The appliance  1100  further includes a discontinuity formed in the shell  1108 , e.g., so as to form a flap  1114 , which can be positioned over and/or against the open upper surface of the receptacle  1104 . An elastic member  1116  is attached to the shell  1108  at attachment points, e.g., on either side of the receptacle  1104 , and can extend over the top of the receptacle  1104  to hold the flap  1114  in place. When the appliance  1100  is placed over the teeth (as illustrated in  FIG.  11 B ), the attachment  1102  is positioned within the receptacle  1104  and can protrude at least partially from the open upper surface, causing the flap  1114  to be displaced from its initial configuration. The elastic member  1116  can push against the flap  1114 , thus imparting a downwards force on the attachment  1102  (see, e.g., arrow  1118 ) that is transmitted to the underlying tooth  1106 , eliciting an intrusive tooth movement. 
       FIG.  12 A  through  FIG.  12 B  illustrate an orthodontic appliance  1200  configured to engage an attachment  1202 , in accordance with many embodiments. The appliance  1200  includes a shell  1206  and a receptacle  1204  formed in the shell  1206  and shaped to receive the attachment  1202 . The receptacle  1204  can include an open lateral surface from which the attachment  1202  can protrude. In many embodiments, the appliance  1200  includes a discontinuity that forms a flap  1208 , which can be positioned over the open lateral surface of the receptacle  1204 . An elastic member  1210  is coupled to the shell  1206  at attachment points situated on opposite sides of the receptacle  1204  and extends over the lateral surface of the receptacle  1204  to hold the flap  1208  in place. Similar to the appliance  1100 , when the appliance  1200  is placed over the teeth, the attachment  1202  protrudes from the lateral surface of the receptacle  1204 , displacing the flap  1208 . The elastic member  1210  exerts a force against the flap  1208  to urge the flap  1208  to its closed configuration (see, e.g., arrow  1212 ), thereby imparting a force onto the attachment  1202  to elicit movement of the underlying tooth  1214 . 
       FIG.  13 A  and  FIG.  13 B  illustrate an orthodontic appliance  1300  configured to engage an attachment  1302 , in accordance with many embodiments. Similar to the appliance  1100 , the appliance  1300  includes a receptacle  1304  with an open upper surface for accommodating the attachment  1302 . A discontinuity of the appliance  1300  can form a flap  1306  positioned over the upper surface of the receptacle  1304 . The flap  1306  is vertically offset from the receptacle  1304  such that only the distal edge  1308  of the flap  1306  contacts the receptacle  1304  when the appliance  1300  is not placed over teeth. The elastic member  1310  is similar to the elastic member  1116  in that it extends over the top of the receptacle  1304  to hold the flap  1306  in place. When the appliance  1300  is placed over the teeth (as illustrated in  FIG.  13 B ), the attachment  1302  is received in receptacle  1304  and protrudes from the upper surface of the receptacle  1304  to displace the flap  1306 . The elastic member  1310  can impart a downwards force on the flap  1306  (see, e.g., arrow  1312 ), thereby imparting a downwards force on the attachment  1302  to reposition the tooth  1314 . 
       FIG.  14 A  and  FIG.  14 B  illustrate an orthodontic appliance  1400  configured to engage an attachment  1402 , in accordance with many embodiments. Similar to the appliance  1200 , the appliance  1400  includes a receptacle  1404  with an open lateral surface for accommodating the attachment  1402 . The appliance  1400  includes a discontinuity that forms a flap  1406  positioned over the open lateral surface of the receptacle  1404 . An elastic member  1408  of the appliance  1400  is configured as an elastic membrane or elastic mesh joining the edges of the flap  1406  to the corresponding edges of the lateral surface of the receptacle  1404 . When the appliance  1400  is worn by the patient (as illustrated in  FIG.  14 B ), the attachment  1404  protrudes through the lateral surface of the receptacle  1404 , displacing the flap  1406 . The resulting stretching of the elastic member  1408  generated by the displacement of the flap  1406  generates a tooth repositioning force that is applied to the tooth  1410  via the attachment  1404 . 
       FIG.  14 C  illustrates an orthodontic appliance  1420  including features for securing an elastic member  1422 , in accordance with many embodiments. The appliance  1420  includes fastening features for coupling the elastic member  1422  to the shell  1424 , exemplarily depicted herein as a pair of posts  1426 . The elastic member  1422  can engage and be secured to the posts  1426  by loops  1428 . The posts  1426  can be formed with the shell  1424 , such that the elastic member  1422  is directly coupled to the shell  1424  by the posts  1426 . The loops  1428  can be situated at any suitable portion of the elastic member  1422 , such as at the ends. Furthermore, the appliance  1420  includes retention features for the elastic member  1422 , depicted herein as a pair of tabs or protrusions  1430  situated on the flap  1432 . As previously described, the retention features can secure the elastic member  1422  at a specified position relative to the shell  1424 . For example, the protrusions  1430  can engage the elastic member  1422  to ensure that at least a portion of its length passes over the flap  1432 , so that the appropriate force (see, e.g., arrow  1434 ) is exerted on the underlying tooth  1436  via the attachment  1438 .  FIG.  14 D  illustrates an orthodontic appliance  1440  including features for securing an elastic member  1442 , in accordance with many embodiments. The elastic member  1442 , depicted herein as an elastic loop, is coupled to the shell  1444  by hooks  1446  formed in the shell  1444 . Similar to the appliance  1420 , the appliance  1440  includes a pair of protrusions  1448  configured to retain the elastic member  1442  in a position spanning the flap  1450  to ensure that the desired force (see, e.g., arrow  1452 ) is applied. 
       FIG.  15 A  through  FIG.  15 D  illustrate example flap geometries for orthodontic appliances, in accordance with many embodiments. Similar to the embodiments discussed with respect to  FIG.  11 A  through  FIG.  14 B , the flaps described herein can be formed via a discontinuity in a shell and positioned over an attachment. The flaps can include one or more features for engaging the attachment. For example, in  FIG.  15 A , a flap  1500  includes a protrusion  1502  extending towards the interior of a shell  1504  to contact an attachment  1506  mounted on a tooth  1508 . The elastic member  1510  is held against the flap  1500  by retention features  1514  so that a repositioning force (see, e.g., arrow  1512 ) is applied to the attachment  1506  via the flap  1500 . As another example, in  FIG.  15 B , a flap  1520  includes a relief  1522  shaped to accommodate a corresponding protrusion  1524  on an attachment  1526  of a tooth  1528 . The ends of the elastic member  1530  can be positioned higher than the protrusion  1524 , such that the middle portion of the elastic member  1530  engages the underside of the relief  1522  to apply a force (see, e.g., arrow  1532 ) to the attachment  1526  via the flap  1520 . In a further example, in  FIG.  15 C , a flap  1540  includes an aperture  1542  into which a protrusion  1544  on an attachment  1546  of a tooth  1548  can extend. Similar to the elastic member  1530 , the ends of the elastic member  1550  can be positioned such that the middle portion of the elastic member  1550  engages the protrusion  1544  of the attachment  1546 , producing a corresponding force (see, e.g., arrow  1552 ) directly against the attachment  1546 . Similar to other embodiments of flaps described herein, an appliance can include any suitable number and configuration of flaps. For example, as depicted in  FIG.  15 D , a single appliance can include a plurality of different flap geometries interacting with various types of attachments. 
       FIG.  15 E  and  FIG.  15 F  illustrate an orthodontic appliance  1570  including a plurality of flaps  1572  for engaging a plurality of attachments  1574  mounted on the teeth  1576 . Each flap  1572  can include a relief  1578  shaped to accommodate a protrusion  1580  on the corresponding attachment  1574 . In some instances, the protrusion  1580  is sized to fit tightly within the relief  1578  with little or no room for movement. Alternatively, the relief  1578  can be larger than the protrusion  1580 , such that the relief  1578  includes sufficient space to accommodate movement of the protrusion  1580  within the relief  1578  (e.g., due to movement of the underlying tooth  1584 ). Each of the plurality of elastic members  1582  is angled upwards to pull against the relief  1578 , thereby applying force on the attachment  1574  via the flap  1572 . In many embodiments, the portion of the elastic member  1582  engaging the relief  1578  can be secured to the flap  1572  by adhesives, bonding, retention features, and the like. The configuration of the flaps, attachments, and elastics can be customized for each tooth, such that the applied force and/or resultant tooth movements vary per tooth. For example, the appliance  1570  can be configured to elicit an extrusive movement of the tooth  1584  relative to the other teeth. Similar to a conventional wire-bracket system, after the teeth  1576  have been repositioned (as illustrated in  FIG.  15 F ), the attachments  1582  can be positioned collinearly (or approximately collinearly) with each other along a mesial-distal direction. 
     Although embodiments depicted in  FIGS.  11  through  15    are shown as eliciting intrusive tooth movements, it shall be understood that the configurations presented herein can be modified as necessary in order to produce other types of tooth movements along different directions (e.g., occlusal-gingival, mesial-distal, buccal-lingual). Such modifications can involve changing an orientation, location, size, and/or shape of the various features provided herein. For example, referring again to  FIGS.  11 A and  11 B , the orientation of the attachment  1102 , receptacle  1104 , and flap  1114  can be rotated by any amount (e.g., by 180°) to produce tooth movement in other directions (e.g., tooth extrusion instead of intrusion). 
       FIGS.  21 A through  21 F  illustrate an orthodontic appliance  2100  with a plurality of discontinuities, in accordance with many embodiments.  FIGS.  21 A and  21 B  depict a side view,  FIGS.  21 C and  21 D  depict a top view, and  FIGS.  21 E and  21 F  depict a perspective view. The appliance  2100  includes a shell  2102  with a first plurality of elongate cuts  2104   a  and a second plurality of elongate cuts  2104   b.  The cuts  2104   a,    2104   b  can be substantially parallel to each other. In many embodiments, the cuts  2104   a  are located primarily on the occlusal surfaces of the appliance  2100  and the cuts  2104   b  are located primarily on the lingual or buccal surfaces of the appliance  2100 . Optionally, some portions of the cuts  2104   a  and/or  2104   b  can also extend to other surfaces of the appliance  2100 , e.g., some portions of each cut  2104   a  can extend to the buccal and/or lingual surfaces and portions of each cut  2104   b  can extend to the occlusal surface. The positioning of the cuts  2104   a,    2104   b  relative to the teeth  2106  received by the shell  2102  can be varied as desired. In the depicted embodiments, the cuts  2104   a  are located adjacent to occlusal regions of the teeth  2106  while the cuts  2104   b  are located adjacent to interproximal regions of the teeth  2106 . The cuts  2104   a  can be interspersed with the cuts  2104   b  along the mesial-distal axis of the appliance  2100 , so as to form an expandable “accordion” configuration that allows for mesial-distal elongation of the shell  2102  when placed on the teeth  2106  (depicted in  FIGS.  21 B,  21 D, and  21 F ). The deformation of the cuts  2104   a,    2104   b  when worn over the teeth  2106  can produce forces (e.g., opposing pairs of mesial-distal forces indicated by arrows) to elicit tooth movements that reduce spaces between teeth. Although  FIGS.  21 A through  21 F  depict an appliance  2100  without any elastic members, it shall be understood that alternative embodiments can include one or more elastic members that interact with the cuts  2104   a  and/or  2104   b  (e.g., spanning the cuts  2104   a  and/or  2104   b ) as previously described herein in order to modulate the forces applied to the teeth  2106 . 
     In many embodiments, the directionality of an elastic member influences the directionality of the resultant forces applied to teeth. For example, in embodiments where the elastic member is elongate (e.g., a band or strip) the forces exerted by the elastic member onto the appliance and/or underlying teeth may be aligned with (e.g., substantially parallel to) the length of the elastic member. Moreover, the directionality of the elastic member relative to a discontinuity can influence the forces applied to teeth via the interaction of the elastic member and discontinuity. The directionality of an elastic member can be varied as desired in order to influence the direction of tooth movement, as well as control the portion(s) of teeth the force is exerted upon. For instance, in some instances it may be desirable to apply forces closer to the crown tip of a tooth (e.g., to produce tipping), while in other instances it may be desirable to apply forces closer to the root center of a tooth (e.g., to avoid tipping). 
       FIGS.  22 A through  22 D  illustrate directionality of an elastic member influencing the forces applied to teeth, in accordance with many embodiments. An appliance  2200  includes a shell  2202  and at least one discontinuity  2204   a - b  formed in the shell  2202 , depicted herein as elongate cuts spanning at least the occlusal and buccal surfaces of the appliance  2200 . An elastic member  2206 , depicted herein as an elongate band, is coupled to the buccal surface of the shell  2202  in a position spanning the discontinuity  2204   a.  In the embodiment of  FIGS.  22 A and  22 B , the elastic member  2206  includes a mesial end  2208  that that is closer to the gingival edge of the shell  2202  and a distal end  2210  that is closer to the occlusal surface of the shell  2202 , such that the length of elastic member  2206  is at an angle relative to the mesial-distal axis of the shell  2202  (e.g., is not parallel to the mesial-distal axis). The elastic member  2206  can be arranged such that the length of the elastic member  2206  is non-orthogonal to the length of the discontinuity  2204 . Accordingly, when the appliance  2200  is placed on a patient&#39;s teeth  2212 , the forces exerted on the teeth  2212  (indicated by arrows) are applied closer to the root center of tooth  2214  and closer to the crown tip of tooth  2216 . 
       FIGS.  22 C and  22 D  illustrate an orthodontic appliance  2250  having a shell  2252 , discontinuities  2254   a - b  formed in the shell, and an elastic member  2256  spanning the discontinuity  2254   a.  The components of the appliance  2250  are substantially similar to those of the appliance  2200 , except that the mesial end  2258  of the elastic member  2256  is closer to the occlusal surface of the shell  2252  while the distal end  2260  is closer to the gingival edge. Accordingly, when the appliance  2250  is placed over the teeth  2262 , the resultant forces (indicated by arrows) are applied closer the crown tip of tooth  2264  and closer to the root center of tooth  2266 . 
     In many embodiments, two or more elastic members can be used in conjunction with each other to apply a plurality of forces having different magnitudes and/or directions. For example, a pair of elastic members coupled to opposite sides of a shell can be used to produce a force couple to elicit tooth rotation. The use of multiple elastic members can allow for the generation of more complex force systems to improve control over tooth movement and/or produce more complicated tooth movements. 
       FIGS.  23 A through  23 D  illustrate an orthodontic appliance  2300  configured to produce tooth rotation, in accordance with many embodiments. The appliance  2300  includes a shell  2302  that is separated into a plurality of discrete segments  2304   a - c . The segments  2304   a - c  can be joined to each other by a first pair of elastic members  2306   a - b  and a second pair of elastic members  2308   a - b , thereby forming a single appliance in which the segments  2304   a - c  can move relative to each other. In many embodiments, the segment  2304   a  is coupled to the segment  2304   b  by elastic members  2306   a,    2308   a  and the segment  2304   b  is coupled to the segment  2304   c  by elastic members  2306   b,    2308   b.  The properties (e.g., stiffness, thickness, material type, etc.) of the elastic members  2306   a - b  can differ from the properties of the elastic members  2308   a - b . For example, the stiffnesses (elastic moduli) of the elastic members  2306   a - b  can be less than the stiffnesses (elastic moduli) of the elastic members  2308   a - b . Accordingly, when the appliance  2300  is worn on the patient&#39;s teeth  2310  (as depicted in  FIGS.  23 B,  23 D ), the forces applied to the teeth by the elastic members  2308   a - b  can be greater in magnitude than the forces applied by the elastic members  2306   a - b . In many embodiments, the difference in force magnitudes applied by the respective pairs of elastic members results in application of a force couple on the tooth  2312 , thereby eliciting rotation of the tooth  2312 . 
       FIGS.  24 A through  24 D  illustrate an orthodontic appliance  2400  configured to produce tooth rotation, in accordance with many embodiments. Similar to the appliance  2300 , the appliance  2400  includes a shell  2402  that is separated into discrete segments  2404   a - c . The segments  2404   a - c  are joined by a first elastic member  2406 , depicted herein as a mesh or sheet, in order to form a single appliance  2400  and permit relative movement of the segments  2404   a - c . Additionally, the appliance  2400  includes a second elastic member  2408  coupled to a first side of the appliance  2400  (e.g., a buccal surface) and a third elastic member  2410  coupled to a second, opposing side of the appliance  2400  (e.g., a lingual surface). The second and third elastic members  2406 ,  2408  can be arranged such that when the appliance  2400  is placed on the teeth  2412 , the second and third elastic members  2406 ,  2408  apply a force couple onto the tooth  2414 , thereby eliciting rotation of the tooth  2414 . 
     In order to improve control over the deformation of an orthodontic appliance (e.g., when worn by a patient), biasing features such as perforations, grooves, parallel lines, engraved shapes, and the like can be formed in the shell in order to define specific locations where desired deformations (e.g., bending, flexing, stretching, compression) should occur. The biasing features may penetrate only partially through the thickness of the shell (e.g., a groove) or may penetrate through the entire thickness (e.g., a cut or aperture). Such features can increase the local compliance of the shell to reduce its resistance to deformation at the specified locations and cause it to preferentially deform at those locations when appropriate forces are applied. In many embodiments, one or more biasing features are used in combination with one or more discontinuities (e.g., flaps, cuts, apertures, etc.) in order to modulate the deformation of the discontinuity when the appliance is placed on patient&#39;s teeth. For example, a perforated or engraved line can be used to define a hinge for a flap formed in an appliance. As another example, a plurality of parallel perforated or engraved lines can be used to define a compliant region in the shell that accommodates deformations of the shell (e.g., as force is applied by an elastic member). 
       FIGS.  26 A through  26 D  illustrate orthodontic appliances with biasing features, in accordance with many embodiments. Although the biasing features are depicted herein as perforated lines, it shall be understood that various alternative embodiments provided herein of biasing features can also be used.  FIG.  26 A  illustrates an orthodontic appliance  2600  having a receptacle  2602  and a flap  2604 . A biasing feature  2606  is formed at the hinge of the flap  2604  in order to ensure that the flap  2604  will preferentially bend at that location. Similarly,  FIG.  26 B  illustrates an orthodontic appliance  2610  having a pair of flaps  2612 , with a respective biasing feature  2614  defining the hinge of each flap  2612 .  FIG.  26 C  illustrates an appliance  2620  in which a biasing feature  2622  contacts and extends from one end of a discontinuity  2624  (depicted herein as a cut). The biasing feature  2622  can be aligned with the length of the discontinuity  2624  so as to facilitate deformation of the discontinuity  2624  (e.g., widening) when the appliance  2620  is placed on teeth.  FIG.  26 D  illustrates an orthodontic appliance  2630  in which a plurality of biasing features  2632  are used to define a region of increased compliance near a discontinuity  2634  (depicted herein as an aperture). Accordingly, when the elastic member  2636  applies force to the appliance near the discontinuity  2634 , the appliance  2630  can preferentially bend at the region of increased compliance in order to apply forces to the tooth, rather than at other locations where force application is not desired. 
     The various embodiments of the orthodontic appliances presented herein can be fabricated in a wide variety of ways. The configuration of an orthodontic appliance can be determined according to a treatment plan for a patient, e.g., a treatment plan involving successive administration of a plurality of appliances for incrementally repositioning teeth. Computer-based treatment planning and/or appliance manufacturing methods can be used in order to facilitate the design and fabrication of appliances. 
       FIG.  27    is a schematic illustration by way of block diagram of a method  2700  for orthodontic treatment, in accordance with many embodiments. The method  2700  can be applied to reposition one or more of a patient&#39;s teeth, maintain one or more of a patient&#39;s teeth in a current configuration, or suitable combinations thereof. The method  2700  can be practiced using any suitable orthodontic appliance, such as suitable orthodontic appliances described herein. 
     In step  2710 , an orthodontic appliance with a discontinuity formed in the shell is provided. The discontinuity can include any embodiments of the various types of discontinuities described herein. In many embodiments, the discontinuity can be formed in the shell (e.g., by cutting, removal of material, deforming a portion of the shell, etc.) after the shell has been fabricated. Alternatively, the discontinuity can be formed simultaneously with the fabrication of the shell. 
     In step  2720 , an elastic member is directly coupled to the shell in a position interacting with the discontinuity. Any embodiment of the elastic members described herein can be combined with any suitable discontinuity. As previously mentioned, the elastic member can be directly coupled to the shell without the use of intervening attachment elements (e.g., fasteners provided separately and coupled to the shell, such as hooks, screws, nails, pins, etc.). The coupling of the elastic member can be performed by an orthodontic practitioner prior to applying the appliance to the teeth. Alternatively, the coupling can be performed by a manufacturer of the appliance, such that the appliance is provided to the practitioner with the coupled elastic member. In many embodiments, the step  2720  is optional, such as where the orthodontic appliance is already provided with the coupled elastic member. 
     In step  2730 , the appliance is placed on the teeth of an arch of the patient. In many embodiments, the appliance is designed to receive teeth from a single dental arch. One or more of the teeth can be coupled to a previously mounted attachment (e.g.,  FIG.  11 A  through  FIG.  15 D ). Alternatively, the appliance can be placed on teeth without any attachments. As previously described herein, placement of the appliance can involve deformation of one or more of the shell, the discontinuity, and the elastic member in order to accommodate the teeth. In some instances, the discontinuity and/or a portion of shell near the discontinuity is displaced when the appliance is worn. For example, the discontinuity can form a flap (e.g.,  FIGS.  11 - 19   ) that is pushed outwards when the appliance is placed on the teeth. As another example, where the appliance includes separate shell segments (e.g.,  FIG.  8    and  FIG.  10   ), the segments can be moved relative to each other when the appliance is placed on teeth. In some instances, the step  2730  can be performed prior to the step  2720 , such that the appliance is placed on the teeth before the elastic member is coupled to the shell. 
     In step  2740 , force is applied to the teeth via the interaction of the elastic member with the discontinuity. As described elsewhere herein, the elastic member can exert a continual force on the shell, and this force can be transmitted via the shell to the underlying teeth. In many embodiments, the force is applied to the teeth via an attachment mounted on one or more of the teeth (e.g.,  FIGS.  11 - 15   ). The applied force can result in repositioning of one or more teeth, as previously described herein. Alternatively, the force can be applied to maintain a current arrangement of the teeth, such that no tooth movements occur. 
       FIG.  28    is a schematic illustration by way of block diagram of a method  2800  for designing an orthodontic appliance, in accordance with many embodiments of the present invention. The steps of the method  2800  can be performed by a suitable system, such as the data processing system described elsewhere herein. 
     In step  2810 , a first position of a tooth of a patient is determined. The first position can be, for example an initial position of the tooth (e.g., the current position of the tooth within the patient&#39;s arch). The position can be determined based on measurement data of the current tooth arrangement of the patient, such as measurement data obtained by scanning of the patient&#39;s teeth or a model of the patient&#39;s teeth. The measurement data can be used to generate a digital representation (e.g., a digital model) of the dentition, from which the first position of the tooth can be determined. 
     In step  2820 , a second position of the tooth is determined. In many embodiments, the second position represents an intermediate or final position of the tooth after orthodontic treatment (e.g., repositioning) has occurred. The second position can, for instance, be selected based on an intermediate or final tooth arrangement specified by an orthodontic practitioner for correcting one or more malocclusions. 
     In step  2830 , a movement path of the tooth from the first position to the second position is calculated. In many embodiments, the movement path is calculated using one or more suitable computer programs, which can take digital representations of the first and second positions as input, and provide a digital representation of the movement path as output. The movement path may also be calculated based on the positions and/or movement paths of other teeth in the patient&#39;s dentition, and such information can also be provided as digital representations. For example, the movement path can be optimized based on minimizing the total distance moved, preventing collisions with other teeth, avoiding tooth movements that are more difficult to achieve, or any other suitable criteria. In some instances, the movement path can be provided as a series of incremental tooth movements that, when performed in sequence, result in repositioning of the tooth from the first position to the second position. 
     In step  2840 , geometry of an appliance shell having a discontinuity is determined based on the movement path, such that an elastic member can be directly coupled to the shell in a position interacting with the discontinuity in order to elicit tooth movement along the movement path. The geometry can be determined by one or more suitable computer programs, such as a computer program configured to accept a digital representation of the movement path as input and provide a digital representation of the shell, discontinuity, and/or elastic member geometry as output (e.g., as digital models). In some instances, the output can be provided to a manufacturing system in order to fabricate a physical model of the shell with the discontinuity, such as a suitable computer-aided manufacturing system. 
     The geometry of the shell, discontinuity, and elastic member can be configured in any manner suitable for generating the tooth movement, such as any of the embodiments described herein. In many embodiments, one or more portions of the shell (e.g., tooth receiving cavities of the shell) can be adapted to include a suitable amount of additional space to accommodate the tooth movement, as previously described herein. In some instances, the step  2840  can further include calculating the geometry of an attachment to be coupled to the tooth, such that the elastic member interacts with the attachment to effect movement of the underlying tooth. 
       FIG.  29    illustrates a method  2900  for digitally planning an orthodontic treatment and/or design or fabrication of an appliance, in accordance with many embodiments. The method  2900  can be applied to any of the treatment procedures described herein and can be performed by any suitable data processing system. 
     In step  2910 , a digital representation of a patient&#39;s teeth is received. The digital representation can include surface topography data for the patient&#39;s intraoral cavity (including teeth, gingival tissues, etc.). The surface topography data can be generated by directly scanning the intraoral cavity, a physical model (positive or negative) of the intraoral cavity, or an impression of the intraoral cavity, using a suitable scanning device (e.g., a handheld scanner, desktop scanner, etc.). 
     In step  2920 , one or more treatment stages are generated based on the digital representation of the teeth. The treatment stages can be incremental repositioning stages of an orthodontic treatment procedure designed to move one or more of the patient&#39;s teeth from an initial tooth arrangement to a target arrangement. For example, the treatment stages can be generated by determining the initial tooth arrangement indicated by the digital representation, determining a target tooth arrangement, and determining movement paths of one or more teeth in the initial arrangement necessary to achieve the target tooth arrangement. The movement path can be optimized based on minimizing the total distance moved, preventing collisions between teeth, avoiding tooth movements that are more difficult to achieve, or any other suitable criteria. 
     In step  2930 , at least one orthodontic appliance is fabricated based on the generated treatment stages. For example, a set of appliances can be fabricated, each shaped to accommodate a tooth arrangement specified by one of the treatment stages, such that the appliances can be sequentially worn by the patient to incrementally reposition the teeth from the initial arrangement to the target arrangement. The appliance set may include one or more orthodontic appliances having a at least one discontinuity and/or at least elastic member described herein. The configuration of the discontinuities and/or elastic members of such appliances (e.g., number, geometry, configuration, material characteristics) can be selected to elicit the tooth movements specified by the corresponding treatment stage. At least some of these properties can be determined via suitable computer software or other digital-based approaches. The fabrication of the appliance may involve creating a digital model of the appliance to be used as input to a computer-controlled fabrication system. 
     In some instances, staging of various arrangements or treatment stages may not be necessary for design and/or fabrication of an appliance. As illustrated by the dashed line in  FIG.  29   , design and/or fabrication of an orthodontic appliance, and perhaps a particular orthodontic treatment, may include use of a representation of the patient&#39;s teeth (e.g., receive a digital representation of the patient&#39;s teeth  2910 ), followed by design and/or fabrication of an orthodontic appliance based on a representation of the patient&#39;s teeth in the arrangement represented by the received representation. For example, a shell may be generated based on the representation of the patient&#39;s teeth (e.g., as in step  2910 ), followed by forming of discontinuities and/or application of elastic members to generate an appliance described in various embodiments herein. 
       FIG.  30    is a simplified block diagram of a data processing system  3000  that may be used in executing methods and processes described herein. The data processing system  3000  typically includes at least one processor  3002  that communicates with one or more peripheral devices via bus subsystem  3004 . These peripheral devices typically include a storage subsystem  3006  (memory subsystem  3008  and file storage subsystem  3014 ), a set of user interface input and output devices  3018 , and an interface to outside networks  3016 . This interface is shown schematically as “Network Interface” block  3016 , and is coupled to corresponding interface devices in other data processing systems via communication network interface  3024 . Data processing system  3000  can include, for example, one or more computers, such as a personal computer, workstation, mainframe, laptop, and the like. 
     The user interface input devices  3018  are not limited to any particular device, and can typically include, for example, a keyboard, pointing device, mouse, scanner, interactive displays, touchpad, joysticks, etc. Similarly, various user interface output devices can be employed in a system of the invention, and can include, for example, one or more of a printer, display (e.g., visual, non-visual) system/subsystem, controller, projection device, audio output, and the like. 
     Storage subsystem  3006  maintains the basic required programming, including computer readable media having instructions (e.g., operating instructions, etc.), and data constructs. The program modules discussed herein are typically stored in storage subsystem  3006 . Storage subsystem  3006  typically includes memory subsystem  3008  and file storage subsystem  3014 . Memory subsystem  3008  typically includes a number of memories (e.g., RAM  3010 , ROM  3012 , etc.) including computer readable memory for storage of fixed instructions, instructions and data during program execution, basic input/output system, etc. File storage subsystem  3014  provides persistent (non-volatile) storage for program and data files, and can include one or more removable or fixed drives or media, hard disk, floppy disk, CD-ROM, DVD, optical drives, and the like. One or more of the storage systems, drives, etc may be located at a remote location, such coupled via a server on a network or via the internet/World Wide Web. In this context, the term “bus subsystem” is used generically so as to include any mechanism for letting the various components and subsystems communicate with each other as intended and can include a variety of suitable components/systems that would be known or recognized as suitable for use therein. It will be recognized that various components of the system can be, but need not necessarily be at the same physical location, but could be connected via various local-area or wide-area network media, transmission systems, etc. 
     Scanner  3020  includes any means for obtaining a digital representation (e.g., images, surface topography data, etc.) of a patient&#39;s teeth (e.g., by scanning physical models of the teeth such as casts  3021 , by scanning impressions taken of the teeth, or by directly scanning the intraoral cavity), which can be obtained either from the patient or from treating professional, such as an orthodontist, and includes means of providing the digital representation to data processing system  3000  for further processing. Scanner  3020  may be located at a location remote with respect to other components of the system and can communicate image data and/or information to data processing system  3000 , for example, via a network interface  3024 . Fabrication system  3022  fabricates appliances  3023  based on a treatment plan, including data set information received from data processing system  3000 . Fabrication machine  3022  can, for example, be located at a remote location and receive data set information from data processing system  3000  via network interface  3024 . 
     The data processing aspects of the methods described herein (e.g., the method  2700 ) can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or suitable combinations thereof. Data processing apparatus can be implemented in a computer program product tangibly embodied in a machine-readable storage device for execution by a programmable processor. Data processing steps can be performed by a programmable processor executing program instructions to perform functions by operating on input data and generating output. The data processing aspects can be implemented in one or more computer programs that are executable on a programmable system, the system including one or more programmable processors operably coupled to a data storage system. Generally, a processor will receive instructions and data from a read-only memory and/or a random access memory. Storage devices suitable for tangibly embodying computer program instructions and data include all forms of nonvolatile memory, such as: semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM disks. 
     As used herein A and/or B encompasses one or more of A or B, and combinations thereof such as A and B. 
     While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. Numerous different combinations of embodiments described herein are possible, and such combinations are considered part of the present disclosure. In addition, all features discussed in connection with any one embodiment herein can be readily adapted for use in other embodiments herein. It is intended that the following claims define the scope of the invention and that methods and structures within the scope of these claims and their equivalents be covered thereby.