CN106803018A - A kind of personalized orthodontic bow-wire Parameter Expression method - Google Patents

Abstract

A kind of personalized orthodontic bow-wire Parameter Expression method, it is related to orthodontic bow-wire to bend technical field, and the present invention establishes the parametrization Mathematical Modeling of orthodontic bow-wire based on discrete and combination mode with reference to the process for bending orthodontic bow-wire by hand.Technical essential is：Foundation, the expression of offset, the expression of easement curve section, the conversion of every section of arch wire local coordinate system and global coordinate system, the position adjustment of bracket straightway and the bent expression of specific function of patient's tooth position coordinate.The parametrization Mathematical Modeling that personalized orthodontic bow-wire First ray is bent and the special arch function of the second sequence is bent is established, necessary modeling guarantee is provided to realize that robot bends personalized orthodontic bow-wire.

Description

A kind of personalized orthodontic bow-wire Parameter Expression method

Technical field

The present invention relates to a kind of personalized orthodontic bow-wire Parameter Expression method, belong to orthodontic bow-wire and bend technical field.

Background technology

The expression of arch wire shape is to design the important foundation that orthodontic bow-wire bends algorithm, is also to realize that orthodontic bow-wire is digitized Bend and automate the prerequisite for bending.Due to the limitation of bracket groove shape, arch wire is generally-straight with bracket contact-segment Section, because of two-phase neighbour's dental morphology and the personalized difference of position, causes the shape of the arch wire easement curve between the different brackets of connection State is not quite similar.According to patient's maxillary teeth data message, according to orthodontic treatment mechanism by bracket straightway and easement curve section It is linked in sequence, orthodontic bow-wire curve bowed shape can be obtained.Orthodontist by changing the locus of bracket straightway, And then the bowed shape of change orthodontic bow-wire curve, the restoring force produced after being deformed upon by arch wire applies orthodontic force, just The traction on assigned direction is carried out under abnormal power effect to tooth, to reach the purpose of correction.Tooth has edge in correcting process Tooth flank of tooth lateral translation, perpendicular to the translation of dental surface, the displacement and tooth rotation of the rotation of dental surface, and tooth movement Gyration is smaller to wait features of movement, therefore is carried out perpendicular to the movement on surface and around support relative to current location using bracket straight line Two modes of the rotation of end points of groove straightway, the position adjustment to bracket straightway.

Currently in the quantitative research of dental arch shape, the dental arch Mathematical Modeling that research worker proposes has power function mould Type, Beta equation models, catenary equation model and elliptic equation model, can approximate expression orthodontic bow-wire shape.But according to Orthodontic bow-wire Mathematical Modeling and the patient personalized difference of unresolved correction that dental arch shape is set up, it is impossible to which accurate expression is appointed One section of location and shape of arch wire of meaning, to the expression underaction of easement curve, the bent insertion position of specific function and shape are difficult To carry out accurate Parameter Expression.

The content of the invention

Regarding to the issue above, the technical problem to be solved in the present invention is to provide a kind of personalized orthodontic bow-wire Parameter Expression Method, discrete by way of combining, bracket straightway and easement curve section is divided into by dental arch curve, and position shape table reaches standard Really, rationally, clear logic is adapted to the accurate expression of personalization orthodontic bow-wire to parameter setting.

Above-mentioned purpose is mainly realized by following scheme：

A kind of personalized orthodontic bow-wire Parameter Expression method of the invention, it is characterised in that：The specific reality of methods described Now process is：

Step one：The foundation of patient's tooth position coordinate

By taking common 14 teeth of the upper jaw as an example, preform orthodontic bow-wire dental arch curve is carried out into sliding-model control, be divided into 14 bracket straightways and 13 transition linkage sections, if the spatial line segment that two benchmark point coordinates on every tooth are formed is institute In the bracket straightway of tooth position, if connect two neighboring bracket straightway is easement curve section, the FDI teeth adopted international standards Position writing-method, its corresponding bracket straightway is represented with the tooth positional representation of every tooth, is represented with the tooth position of two adjacent teeth Method represents the easement curve section in the middle of it, represents that the tooth position order for importing patient data is 18 using FDI record of tooth position methods ..., 11,21,…,28；If i is expressed as i-th tooth (i=0,1 ..., 13) of the patient according to tooth position order, so 18 correspondences of tooth position I=0, two point coordinates are (x on tooth position 18₀,y₀,z₀) and (x₁,y₁,z₁), 21 corresponding i=7 of tooth position, according to tooth position order Two point coordinates on i-th tooth are (x_2i, y_2i, z_2i) and (x_2i+1, y_2i+1, z_2i+1)；

Offset refers to the distance for being set in the extension of bracket straightway both sides, and bracket straightway Mathematical Modeling is based on orthodontic arch Silk datum mark is set up, and orthodontic bow-wire uses expression way familiar to doctor in the offset that bracket both sides are added：In i-th bracket Straightway adds nearly midpoint offset a_iWith remote midpoint offset b_i, its Mathematical Modeling is by formula (1) expression

In formula：Nearly midpoint offset a_iWith remote midpoint offset b_iNumerical values recited represent the extension of bracket straightway two Apart from size, (X_2i, Y_2i, Z_2i) represent the starting point and ending point of straightway after addition offset on i-th tooth；

Step 3：The expression of easement curve section

The shape of easement curve section can be adjusted by the size of Set scale parameter, in the easement curve of orthodontic bow-wire curve The selection of section part has 43 rank Bezeir curves at control point, and its expression formula can be reduced to：

In formula：P₀、P₁、P₂、P₃Represent 34 control points of rank Bezeir curves, t ∈ [0,1]；

Two adjacent bracket straightways determine an easement curve section, control point P₀、P₃It is the starting point of easement curve section And terminating point, while being also the end points of bracket straightway after addition offset.Two adjacent bracket straightways are thrown in X/Y plane X, y value of movie queen's intersection point are brought into every section of space line of line segment, obtain their coordinates in Z-direction, the x, y, z tried to achieve Value just constitutes an intermediate point (x, y, z), and the intermediate point is connected with corresponding bracket straightway end points again, and the line segment for being connected is pressed After being distributed according to certain proportion, the coordinate points of its distributing point are two control point P₀、P₁；

It is (X to try to achieve intersection point of the straight line where adjacent brackets straightway under X/Y plane projection₀, Y₀)；

Shown in two space line general equations such as formula (3) of bracket straightway：

By intersecting point coordinate (X₀, Y₀) substitute into formula (3) in the middle of two point coordinates be (X₀, Y₀, Z₁) and (X₀, Y₀, Z₂).So Four control points of easement curve section are：

Wherein：e_i、f_iIt is scale parameter, by setting e_i、f_iNumerical value change Bezeir curve control points P₁、P₂；

Step 4：The conversion of every section of arch wire local coordinate system and global coordinate system

First, the patient's upper jaw data message for being provided according to orthodontist, sets up dental arch curve global coordinate O-XYZ, Then with global coordinate origin of coordinates O as origin, it is O1-UVW to set up local mark, reference axis successively with global coordinate coordinate Overlapping of axles, then determines dental arch curve target insertion point position, here by any linkage section left and right sides end points P₂、P₃Coordinate Try to achieve, finally, local coordinate system O1-UVW is converted into target location through coordinate transform, so as to realize the second sequence specific function The digital expression of bowl line.P₁(x₁, y₁, z₁)、P₂(x₂, y₂, z₂)、P₃(x₃, y₃, z₃)、P₄(x₄, y₄, z₄) it is given bracket Point coordinates.

Local coordinate system O1-UVW changes to target location under global coordinate O-XYZ, need to be revolved by three coordinate transforms Turn-translation-rotation.Coordinate system O1-UVW is first by around own axes O1W axle rotation alphas angle (transformation matrix R₁) to coordinate system O1'-U'V'W', coordinate system O1'-U'V'W' are by translation transformation (transformation matrix R₂) to coordinate system O1 "-U " V " W ", coordinate system O1 "-U " V " W " are around own axes O1 " V " rotation β angular transformation (transformation matrix R₃) to target location O1 " '-U " ' V " ' W " '.Assuming that Specific function arch dental arch curve local coordinate system drag equation matrix is C, and it is converted into target location under global coordinate Matrix equation be D, then：

D=R₂×(C×R₁)×R₃ (5)

In formula：

Step 5：The mathematical expression of the position adjustment of bracket straightway

According to the distance of adjustment translation, the anglec of rotation and rotation mode realize bracket straightway in the sequence of orthodontic bow-wire first Position adjustment in the bent plane of row, the coordinate of the bracket straightway after gained position adjustment under global coordinate O-XYZ is by formula (5) Represent, dragging groove straightway local coordinate system O1 "-U " V " W " in, the translation of bracket straightway and the Mathematical Modeling rotated around Q2 It is A, the translation of bracket straightway and the Mathematical Modeling rotated around Q1 are A ', as shown in formula (6),

In formula：Q represents the bracket straightway coordinate points equation under local coordinate system, and h represents that bracket straightway is moved in parallel Distance, ω represents the angle of rotation, and l represents that the mould of Q1Q2 is long；

Step 6：The bent expression of specific function

Specific function song Parameter Expression method is coordinate system where the patient teeth's data message provided with orthodontist O-XYZ is global coordinate, with the local coordinate system O1-UVW that easement curve two end points of section are set up, to various special arch work( Energy curve sets up the coordinates matrix C parameterized under local coordinate system O1-UVW, and global coordinate is obtained by formula (5) coordinate transform The bent coordinates matrix D of specific function under O-XYZ；

Under local coordinate system O1-UVW, the method that orthodontic bow-wire is bent with reference to staff regards specific function song as some Straightway, arc section and helical row combination, its expression formula is respectively G, H, K, then the bent expression formula of specific function is F= F { G1, G2, H1, K1 ... }, then expression formula is converted into coordinates matrix C, it is that the coordinate of characteristic point is deposited in a matrix in order；

The bent position i (i=0 ..., 12) of insertion specific function, represents according to the tooth position order of patient data, position i with I-th easement curve section is corresponding, insertion specific function bent type j (j=0,1 ...), j=0 under default conditions, represents and inserts What is entered is the easement curve section based on Bezeir curves, i.e. D₀=P (t), expression represents a kind of bent type of specific function respectively, Its coordinates matrix is D_j, make j=1 represent open vertical loop, j=2 represents T-shaped song, therefore D_j(j=1,2 ...) represent specific function The Mathematical Modeling database of curve, insertion specific function is bent to represent on i-th position of transition curve section that i-th transition is bent Line segment expression formula P_iT () is substituted for the coordinates matrix D of the specific function song of jth type_j。

Beneficial effects of the present invention are：

1st, dental arch curve is divided into by bracket straightway and easement curve section two parts using the segmentation method of discretization, is met The characteristics of manually bending, each several part curve is easy to accurate expression, so as to realize the accurate expression of integrated curved.

2nd, using three rank Bezeir curve representations easement curves section, shape table reaches accurate and is easy to parameter adjustment, by setting Put e_i、f_iNumerical value change Bezeir curve control points P₁、P₂, and then the convex hull shape of Bezeir curves is adjusted, transition can be made bent Line segment more presses close to the actual conditions of various teeth, reaches the purpose of the personalized expression of orthodontic bow-wire curve.

3rd, the position adjustment of bracket straightway is mainly flat in orthodontic bow-wire First ray plane by bracket straightway Position adjustment is realized in the rotation of mobile and around bracket straightway two end points of row, and easy to adjust, logic is simple, and operand is small.

4th, suitable parameterized model is established to the bent location and shape of specific function, is to insert special in specified location Fixed specific function is bent to provide necessary model basis.

5th, a kind of personalized orthodontic bow-wire Parameter Expression method proposed by the present invention is covered including bracket straightway, mistake Cross that curved section, specific function are bent in interior whole orthodontic bow-wire parts, and reddendo signula singulis portion shape adjustment Cheng Jinhang expression, can accurately express the shape facility of orthodontic bow-wire, and easy to adjust, be being digitized into for orthodontic bow-wire Shape and automation shaping provide necessary support.

Brief description of the drawings

For ease of explanation, the present invention is described in detail by following specific implementations and accompanying drawing.

The shaping arch wire schematic diagram that Fig. 1 is obtained for the inventive method；

Fig. 2 is arch wire curve of the present invention adjustment process expression schematic diagram；

Fig. 3 is special arch local coordinate system of the invention and global coordinate relative position relation schematic diagram；

Fig. 4 is the bent Parameter Expression schematic diagram of specific function of the present invention；

Fig. 5 is that the present invention inserts the bent and T-shaped bent orthodontic bow-wire curve synoptic diagram of open vertical loop, Ω.

Specific embodiment

It is specific below by what is shown in accompanying drawing to make the object, technical solutions and advantages of the present invention of greater clarity Embodiment describes the present invention.However, it should be understood that these descriptions are merely illustrative, and it is not intended to limit model of the invention Enclose.Additionally, in the following description, the description to known features and technology is eliminated, it is of the invention to avoid unnecessarily obscuring Concept.

As shown in Figure 1, Figure 2, shown in Fig. 3, Fig. 4, Fig. 5, this specific embodiment uses following technical scheme：A kind of personalization is just Abnormal arch wire Parameter Expression method, it is characterised in that：The process that implements of methods described is：

Step one：The foundation of patient's tooth position coordinate

By taking common 14 teeth of the upper jaw as an example, preform orthodontic bow-wire dental arch curve is carried out into sliding-model control, be divided into 14 bracket straightways and 13 transition linkage sections, if the spatial line segment that two benchmark point coordinates on every tooth are formed is institute In the bracket straightway of tooth position, if connect two neighboring bracket straightway is easement curve section, the FDI teeth adopted international standards Position writing-method, its corresponding bracket straightway is represented with the tooth positional representation of every tooth, is represented with the tooth position of two adjacent teeth Method represents the easement curve section in the middle of it, represents that the tooth position order for importing patient data is 18 using FDI record of tooth position methods ..., 11,21,…,28；If i is expressed as i-th tooth (i=0,1 ..., 13) of the patient according to tooth position order, so 18 correspondences of tooth position I=0, two point coordinates are (x on tooth position 18₀,y₀,z₀) and (x₁,y₁,z₁), 21 corresponding i=7 of tooth position, according to tooth position order Two point coordinates on i-th tooth are (x_2i, y_2i, z_2i) and (x_2i+1, y_2i+1, z_2i+1)；

Step 2：The expression of offset

Offset refers to the distance for being set in the extension of bracket straightway both sides, and bracket straightway Mathematical Modeling is based on orthodontic arch Silk datum mark is set up, and orthodontic bow-wire uses expression way familiar to doctor in the offset that bracket both sides are added：In i-th bracket Straightway adds nearly midpoint offset a_iWith remote midpoint offset b_i, its Mathematical Modeling is by formula (1) expression

In formula：Nearly midpoint offset a_iWith remote midpoint offset b_iNumerical values recited represent the extension of bracket straightway two Apart from size, (X_2i, Y_2i, Z_2i) represent the starting point and ending point of straightway after addition offset on i-th tooth；

Step 3：The expression of easement curve section

The shape of easement curve section can be adjusted by the size of Set scale parameter, in the easement curve of orthodontic bow-wire curve The selection of section part has 43 rank Bezeir curves at control point, and its expression formula can be reduced to：

In formula：P₀、P₁、P₂、P₃Represent 34 control points of rank Bezeir curves, t ∈ [0,1]；

Two adjacent bracket straightways determine an easement curve section, control point P₀、P₃It is the starting point of easement curve section And terminating point, while being also the end points of bracket straightway after addition offset.Two adjacent bracket straightways are thrown in X/Y plane X, y value of movie queen's intersection point are brought into every section of space line of line segment, obtain their coordinates in Z-direction, the x, y, z tried to achieve Value just constitutes an intermediate point (x, y, z), and the intermediate point is connected with corresponding bracket straightway end points again, and the line segment for being connected is pressed After being distributed according to certain proportion, the coordinate points of its distributing point are two control point P₁、P₂；

It is (X to try to achieve intersection point of the straight line where adjacent brackets straightway under X/Y plane projection₀, Y₀)；

Shown in two space line general equations such as formula (3) of bracket straightway：

By intersecting point coordinate (X₀, Y₀) substitute into formula (3) in the middle of two point coordinates be (X₀, Y₀, Z₁) and (X₀, Y₀, Z₂).So Four control points of easement curve section are：

Wherein：e_i、f_iIt is scale parameter, by setting e_i、f_iNumerical value change Bezeir curve control points P₁、P₂；

Step 4：The conversion of every section of arch wire local coordinate system and global coordinate system

First, the patient's upper jaw data message for being provided according to orthodontist, sets up dental arch curve global coordinate O-XYZ, Then with global coordinate origin of coordinates O as origin, it is O1-UVW to set up local mark, reference axis successively with global coordinate coordinate Overlapping of axles, then determines dental arch curve target insertion point position, here by any linkage section left and right sides end points P₂、P₃Coordinate Try to achieve, finally, local coordinate system O1-UVW is converted into target location through coordinate transform, so as to realize the second sequence specific function The digital expression of bowl line.P₁(x₁, y₁, z₁)、P₂(x₂, y₂, z₂)、P₃(x₃, y₃, z₃)、P₄(x₄, y₄, z₄) it is given bracket Point coordinates.

Local coordinate system O1-UVW changes to target location under global coordinate O-XYZ, need to be revolved by three coordinate transforms Turn-translation-rotation.Coordinate system O1-UVW is first by around own axes O1W axle rotation alphas angle (transformation matrix R₁) to coordinate system O1'-U'V'W', coordinate system O1'-U'V'W' are by translation transformation (transformation matrix R₂) to coordinate system O1 "-U " V " W ", coordinate system O1 "-U " V " W " are around own axes O1 " V " rotation β angular transformation (transformation matrix R₃) to target location O1 " '-U " ' V " ' W " '.Assuming that Specific function arch dental arch curve local coordinate system drag equation matrix is C, and it is converted into target location under global coordinate Matrix equation be D, then：

D=R₂×(C×R₁)×R₃ (5)

In formula：

Step 5：The mathematical expression of the position adjustment of bracket straightway

According to the distance of adjustment translation, the anglec of rotation and rotation mode realize bracket straightway in the sequence of orthodontic bow-wire first Position adjustment in the bent plane of row, the coordinate of the bracket straightway after gained position adjustment under global coordinate O-XYZ is by formula (5) Represent, dragging groove straightway local coordinate system O1 "-U " V " W " in, the translation of bracket straightway and the Mathematical Modeling rotated around Q2 It is A, the translation of bracket straightway and the Mathematical Modeling rotated around Q1 are A ', as shown in formula (6),

In formula：Q represents the bracket straightway coordinate points equation under local coordinate system, and h represents that bracket straightway is moved in parallel Distance, ω represents the angle of rotation, and l represents that the mould of Q1Q2 is long；

Step 6：The bent expression of specific function

Specific function song Parameter Expression method is coordinate system where the patient teeth's data message provided with orthodontist O-XYZ is global coordinate, with the local coordinate system O1-UVW that easement curve two end points of section are set up, to various special arch work( Energy curve sets up the coordinates matrix C parameterized under local coordinate system O1-UVW, and global coordinate is obtained by formula (5) coordinate transform The bent coordinates matrix D of specific function under O-XYZ；

Under local coordinate system O1-UVW, the method that orthodontic bow-wire is bent with reference to staff regards specific function song as some Straightway, arc section and helical row combination, its expression formula is respectively G, H, K, then the bent expression formula of specific function is F= F { G1, G2, H1, K1 ... }, then expression formula is converted into coordinates matrix C, it is that the coordinate of characteristic point is deposited in a matrix in order；

The bent position i (i=0 ..., 12) of insertion specific function, represents according to the tooth position order of patient data, position i with I-th easement curve section is corresponding, insertion specific function bent type j (j=0,1 ...), j=0 under default conditions, represents and inserts What is entered is the easement curve section based on Bezeir curves, i.e. D₀=P (t), expression represents a kind of bent type of specific function respectively, Its coordinates matrix is D_j, make j=1 represent open vertical loop, j=2 represents T-shaped song, therefore D_j(j=1,2 ...) represent specific function The Mathematical Modeling database of curve, insertion specific function is bent to represent on i-th position of transition curve section that i-th transition is bent Line segment expression formula P_iT () is substituted for the coordinates matrix D of the specific function song of jth type_j。

General principle of the invention and principal character and advantages of the present invention has been shown and described above.The technology of the industry Personnel it should be appreciated that the present invention is not limited to the above embodiments, simply explanation described in above-described embodiment and specification this The principle of invention, without departing from the spirit and scope of the present invention, various changes and modifications of the present invention are possible, these changes Change and improvement all fall within the protetion scope of the claimed invention.The claimed scope of the invention by appending claims and its Equivalent thereof.

Claims (1)

1. a kind of personalized orthodontic bow-wire Parameter Expression method, it is characterised in that：The process that implements of methods described is：

Step one：The foundation of patient's tooth position coordinate

By taking common 14 teeth of the upper jaw as an example, preform orthodontic bow-wire dental arch curve is carried out into sliding-model control, be divided into 14 Bracket straightway and 13 transition linkage sections, if the spatial line segment that two benchmark point coordinates on every tooth are formed is place tooth The bracket straightway of position, if connect two neighboring bracket straightway is easement curve section, the FDI teeth position note adopted international standards Record method, its corresponding bracket straightway is represented with the tooth positional representation of every tooth, with the tooth positional representation table of two adjacent teeth Show the easement curve section in the middle of it, represent that the tooth position order for importing patient data is 18 using FDI record of tooth position methods ..., 11, 21,…,28；If i is expressed as i-th tooth (i=0,1 ..., 13) of the patient according to tooth position order, so 18 corresponding i of tooth position =0, two point coordinates are (x on tooth position 18₀,y₀,z₀) and (x₁,y₁,z₁), 21 corresponding i=7 of tooth position, according to tooth position order i-th Two point coordinates on individual tooth are (x_2i, y_2i, z_2i) and (x_2i+1, y_2i+1, z_2i+1)；

Step 2：The expression of offset

Offset refers to the distance for being set in the extension of bracket straightway both sides, and bracket straightway Mathematical Modeling is based on orthodontic bow-wire base Set up on schedule, orthodontic bow-wire uses expression way familiar to doctor in the offset that bracket both sides are added：In i-th bracket straight line The nearly midpoint offset a of Duan Tianjia_iWith remote midpoint offset b_i, its Mathematical Modeling is by formula (1) expression

$\left\{\begin{array}{c}\left(l_x,l_y,l_z\right)=\frac{\left(x_{2i+1}-x_{2i},y_{2i+1}-y_{2i},z_{2i+1}-z_{2i}\right)}{\sqrt{\left(x_{2i+1}-x_{2i}\right)^2+\left(y_{2i+1}-y_{2i}\right)^2+\left(z_{2i+1}-z_{2i}\right)^2}}\\ i=0,\mathrm{...},6\\ \left[\begin{array}{c}\left(X_{2i},Y_{2i},Z_{2i}\right)\\ \left(X_{2i+1},Y_{2i+1},Z_{2i+1}\right)\end{array}\right]=\left[\begin{array}{c}\left(x_{2i}-l_x\·b_i,y_{2i}-l_y\·b_i,z_{2i}-l_z\·b_i\right)\\ \left(x_{2i+1}+l_x\·a_i,y_{2i+1}+l_y\·a_i,z_{2i+1}+l_z\·a_i\right)\end{array}\right]\\ i=7,\mathrm{...},13\\ \left[\begin{array}{c}\left(X_{2i},Y_{2i},Z_{2i}\right)\\ \left(X_{2i+1},Y_{2i+1},Z_{2i+1}\right)\end{array}\right]=\left[\begin{array}{c}\left(x_{2i}-l_x\·a_i,y_{2i}-l_y\·a_i,z_{2i}-l_z\·a_i\right)\\ \left(x_{2i+1}+l_x\·b_i,y_{2i+1}+l_y\·b_i,z_{2i+1}+l_z\·b_i\right)\end{array}\right]\end{array}\right.---\left(1\right)$

In formula：Nearly midpoint offset a_iWith remote midpoint offset b_iNumerical values recited represent the distances of the extension of bracket straightway two Size, (X_2i, Y_2i, Z_2i) represent the starting point and ending point of straightway after addition offset on i-th tooth；

Step 3：The expression of easement curve section

The shape of easement curve section can be adjusted by the size of Set scale parameter, in the easement curve section portion of orthodontic bow-wire curve Component selections have 43 rank Bezeir curves at control point, and its expression formula can be reduced to：

$\begin{array}{c}P\left(t\right)=P_0\·{(1-t)}^3+3\·P_1\·t\·{(1-t)}^2\\ +3\·P_2\·t^2\·\left(1-t\right)+P_3\·t^3\end{array}---\left(2\right)$

In formula：P₀、P₁、P₂、P₃Represent 34 control points of rank Bezeir curves, t ∈ [0,1]；

Two adjacent bracket straightways determine an easement curve section, control point P₀、P₃It is starting point and the end of easement curve section Stop, while being also the end points of bracket straightway after addition offset, two adjacent bracket straightways are after X/Y plane projection X, y value of intersection point are brought into every section of space line of line segment, obtain their coordinates in Z-direction, and the x, y, z value tried to achieve is just One intermediate point (x, y, z) of composition, the intermediate point is connected with corresponding bracket straightway end points again, and the line segment for being connected is according to one After certainty ratio distribution, the coordinate points of its distributing point are two control point P₁、P₂；

It is (X to try to achieve intersection point of the straight line where adjacent brackets straightway under X/Y plane projection₀, Y₀)；

Shown in two space line general equations such as formula (3) of bracket straightway：

$\left\{\begin{array}{c}\frac{X-X_{2i}}{X_{2i+1}-X_{2i}}=\frac{Y-Y_{2i}}{Y_{2i+1}-Y_{2i}}=\frac{Z-Z_{2i}}{Z_{2i+1}-Z_{2i}}\\ \frac{X-X_{2i+2}}{X_{2i+3}-X_{2i+2}}=\frac{Y-Y_{2i+2}}{Y_{2i+3}-Y_{2i+2}}=\frac{Z-Z_{2i+2}}{Z_{2i+3}-Z_{2i+2}}\end{array}\right.---\left(3\right)$

By intersecting point coordinate (X₀, Y₀) substitute into formula (3) in the middle of two point coordinates be (X₀, Y₀, Z₁) and (X₀, Y₀, Z₂), so transition Four control points of curved section are：

$\left\{\begin{array}{c}{P}_0=\left(X_{2i+1},Y_{2i+1},Z_{2i+1}\right)\\ P_1=(X_{2i+1}+e_i\·\left(X_0-X_{2i+1}\right),Y_{2i+1}+e_i\·\left(Y_0-Y_{2i+1}\right),\\ Z_{2i+1}+e_i\·\left(Z_0-Z_{2i+1}\right))\\ P_2=(X_{2i+2}-f_i\·\left(X_{2i+2}-X_0\right),Y_{i+2}-f_i\·\left(Y_{2i+2}-Y_0\right),\\ Z_{2i+2}-f_i\·\left(Z_1-Z_{2i+2}\right))\\ P_3=\left(X_{2i+2},Y_{2i+2},Z_{2i+2}\right)\end{array}\right.---\left(4\right)$

Wherein：e_i、f_iIt is scale parameter, by setting e_i、f_iNumerical value change Bezeir curve control points P₁、P₂；

Step 4：The conversion of every section of arch wire local coordinate system and global coordinate system

First, the patient's upper jaw data message for being provided according to orthodontist, sets up dental arch curve global coordinate O-XYZ, then With global coordinate origin of coordinates O as origin, it is O1-UVW to set up local mark, reference axis successively with global coordinate reference axis weight Close, dental arch curve target insertion point position is then determined, here by any linkage section left and right sides end points P₂、P₃Coordinate is tried to achieve, Finally, local coordinate system O1-UVW is converted into target location through coordinate transform, so as to realize the second sequence specific function arch The digital expression of curve, P₁(x₁, y₁, z₁)、P₂(x₂, y₂, z₂)、P₃(x₃, y₃, z₃)、P₄(x₄, y₄, z₄) it is given bracket point seat Mark；

Local coordinate system O1-UVW changes to target location under global coordinate O-XYZ, need to by three coordinate transforms, rotate- Translation-rotation, coordinate system O1-UVW is first by around own axes O1W axle rotation alphas angle (transformation matrix R₁) to coordinate system O1'- U'V'W', coordinate system O1'-U'V'W' are by translation transformation (transformation matrix R₂) to coordinate system O1 "-U " V " W ", coordinate system O1 "- U " V " W " are around own axes O1 " V " rotation β angular transformation (transformation matrix R₃) to target location O1 " '-U " ' V " ' W " ', it is assumed that it is special Function arch dental arch curve local coordinate system drag equation matrix is C, its square for being converted into target location under global coordinate Battle array equation is D, then：

D=R₂×(C×R₁)×R₃ (5)

In formula：

$\begin{array}{ccc}{R}_1=\left[\begin{array}{cccc}cos\mathrm{\α}& -sin\mathrm{\α}& 0& 0\\ sin\mathrm{\α}& \cos \mathrm{\α}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]& R_2=\left[\begin{array}{cccc}1& 0& 0& x\\ 0& 1& 0& y\\ 0& 0& 1& z\\ 0& 0& 0& 1\end{array}\right]& R_3=\left[\begin{array}{cccc}cos\mathrm{\β}& 0& sin\mathrm{\β}& 0\\ 0& 1& 0& 0\\ -sin\mathrm{\β}& 0& \cos \mathrm{\β}& 0\\ 0& 0& 0& 1\end{array}\right]\end{array}$

$cos\mathrm{\α}=\frac{x_3-x_2}{\sqrt{{(x_3-x_2)}^2+{(y_3-y_2)}^2}},cos\mathrm{\β}=\frac{x_3-x_2}{\sqrt{{(x_3-x_2)}^2+{(z_3-z_2)}^2}},$

$x=\frac{x_2+x_3}{2},y=\frac{y_2+y_3}{2},z=\frac{z_2+z_3}{2},$

$sin\mathrm{\α}=\frac{y_3-y_2}{\sqrt{{(x_3-x_2)}^2+{(y_3-y_2)}^2}},sin\mathrm{\β}=\frac{z_3-z_2}{\sqrt{{(x_3-x_2)}^2+{(z_3-z_2)}^2}}.$

Step 5：The mathematical expression of the position adjustment of bracket straightway

According to the distance of adjustment translation, the anglec of rotation and rotation mode realize that bracket straightway is bent in orthodontic bow-wire First ray Position adjustment in plane, the coordinate of the bracket straightway after gained position adjustment under global coordinate O-XYZ is by formula (5) table Show, dragging groove straightway local coordinate system O1 "-U " V " W " in, the translation of bracket straightway and the Mathematical Modeling rotated around Q2 are A, the translation of bracket straightway and the Mathematical Modeling rotated around Q1 are A ', as shown in formula (6),

$A=\left[\begin{array}{cc}l\·(1-cos\mathrm{\ω})& l\\ h+l\·sin\mathrm{\ω}& h\\ 0& 0\\ 1& 1\end{array}\right],A^{\′}=\left[\begin{array}{cc}0& l\·cos\mathrm{\ω}\\ h& h+l\·cos\mathrm{\ω}\\ 0& 0\\ 1& 1\end{array}\right]---\left(6\right)$

In formula：Q represents the bracket straightway coordinate points equation under local coordinate system, h represent that bracket straightway moves in parallel away from From ω represents the angle of rotation, and l represents that the mould of Q1Q2 is long；

Step 6：The bent expression of specific function

Specific function song Parameter Expression method is coordinate system O-XYZ where the patient teeth's data message provided with orthodontist It is global coordinate, it is bent to various special arch functions with the local coordinate system O1-UVW that easement curve two end points of section are set up Line sets up the coordinates matrix C parameterized under local coordinate system O1-UVW, and global coordinate O- is obtained by formula (5) coordinate transform The bent coordinates matrix D of specific function under XYZ；

Under local coordinate system O1-UVW, the method that orthodontic bow-wire is bent with reference to staff regards specific function song as some straight Line segment, arc section and helical row combination, its expression formula are respectively G, H, K, then the bent expression formula of specific function is F=F { G1, G2, H1, K1 ... }, then expression formula is converted into coordinates matrix C, it is that the coordinate of characteristic point is deposited in a matrix in order；

The bent position i (i=0 ..., 12) of insertion specific function, represents according to the tooth position order of patient data, position i and i-th Easement curve section is corresponding, insertion specific function bent type j (j=0,1 ...), j=0 under default conditions, and represent insertion is Easement curve section based on Bezeir curves, i.e. D₀=P (t), expression represents a kind of bent type of specific function, its coordinate respectively Matrix is D_j, make j=1 represent open vertical loop, j=2 represents T-shaped song, therefore D_j(j=1,2 ...) represent specific function curve Mathematical Modeling database, insertion specific function is bent to be represented i-th easement curve segment table on i-th position of transition curve section Up to formula P_iT () is substituted for the coordinates matrix D of the specific function song of jth type_j。