CN112231901B - Offset calculation method of T spline surface - Google Patents
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Abstract
The invention provides a bias calculation method of a T spline surface, which comprises the following steps: step 1, inputting an initial T spline surface, an offset distance d and an offset error epsilon; step 2, extracting Bei Jier from the initial T spline surface, and dispersing the initial T spline surface into Bei Jier units; step 3, calculating the number of sampling points of each boundary Bei Jier unit so as to obtain global parameter coordinates of all the sampling points; step 4, performing offset calculation on each sampling point to obtain points on a result curved surface; and 5, carrying out proper parameterization on the biased sampling points, and carrying out back calculation on the control vertex to obtain the T spline representation of the result curved surface. The T spline data structure adopted by the invention supports the T mesh with singular points, has strong universality, enables the result surface to support various complicated subsequent operations, and has scientific method, good manufacturability and wide popularization and application value.
Description
Technical Field
The invention provides a bias calculation method of a T spline surface, which relates to a bias method of a T spline surface and belongs to the technical field of Computer Aided Design (CAD) and Computer Aided Manufacturing (CAM). The invention relates to a T-spline (T-splines) which is a free-form surface representation method proposed in the document T-splines and T-NURCCs by Thomas W.Sederberg et al in 2003, and belongs to the technical field of Computer Aided Geometric Design (CAGD) and surface modeling.
Background
In the existing commercial computer aided design software, the three-dimensional geometric modeling system generally expresses a curved surface based on a Non-uniform rational B-spline (NURBS) method. However, the NURBS based on the rectangular topology has inherent limitations, such as the problem of splicing gaps among multiple NURBS curved surfaces, the problem of redundancy of control vertexes, and the like, which are gradually highlighted in the development process of the intelligent design of products nowadays. The T spline theory proposed in 2003 breaks the limitation of the NURBS rectangular topology, and has the following three advantages: (1) local thinning: control mesh encryption can be carried out on a part of the T spline surface according to requirements, and control vertexes do not need to be introduced in order to meet topological constraint in a whole row and column manner; (2) seamless splicing: the problem of gaps and overlapping exists at the joint of the curved surfaces when the NURBS model is cut. By introducing a T point at the junction, a plurality of NURBS curved surfaces or T spline curved surfaces containing different node vectors can be spliced into a single seamless watertight T spline curved surface; and (3) data compression: the NURBS surface containing the redundant control vertex can be simplified into a T spline surface with a T point, and data compression and simplification are realized. Therefore, compared with the NURBS method, the T spline method has remarkable advantages in the expression of complex three-dimensional models.
The definition of the offset surface is a set of loci of points that are at a constant distance d from the original surface along the normal direction of the surface. The offset calculation of the curved surface has wide application in computer aided design and computer aided manufacturing, namely CAD/CAM, such as the generation of tool path of a numerical control machine tool, the planning of a robot motion path, and a parameterization offset command which is very commonly used in CAD modeling operation and is used for generating a solid model, and the like, in most complex model modeling processes, especially in a modern B-rep technology system which is used as the basis of a CAD system, the curved surface offset is the operation which expands the curved surface modeling function into the solid modeling function and has strong applicability, can quickly provide a topologically approximately symmetrical surface for the solid model generation in engineering modeling application, and can quickly generate and copy a plurality of similar shape characteristics, thereby greatly improving the modeling efficiency of the complex model, therefore, the function which is necessary when the CAD system is constructed facing practical application, and the CAD technology is also an indispensable link for pushing the CAD technology from a theoretical level to practical application; in CAM application, the tool path generation based on the surface offset is the core of CAM algorithm. The offset calculation of the surface is not a simple task, however, for several reasons: (1) The offset distance may not exist, because some points on the curved surface have more than one normal direction; (2) Generally, the type of the biased curved surface is different from the original curved surface; (3) The shape of the curved surface after offset is not only related to the shape of the original curved surface, but also related to factors such as offset distance, offset direction, local curvature and the like; (4) Self-intersection, partial deletion, etc. may occur after surface offset. Piegl et al studied the calculation method of the NURBS curved surface offset in the literature [ Computing offsets of NURBS curves and surfaces ], which first calculated the number of sampling points from the perspective of data point sampling, and performed offset processing on the sampling points, then performed node removal within a given error, and finally interpolated the resulting curved surface into a NURBS curved surface. Compared with NURBS, the T spline has outstanding superiority in the field of complex surface modeling, so that the bias calculation method of the T spline surface is provided, the bias operation of the T spline surface is realized, and the method has outstanding application value to CAD/CAM.
Disclosure of Invention
The object of the invention is:
the invention aims to provide a bias calculation method of a T spline surface, which is based on a T spline data structure with stronger universality and can adapt to various complex conditions in surface modeling.
The technical scheme is as follows:
the invention discloses a bias calculation method of a T spline surface, which comprises the following steps:
and 5, carrying out proper parameterization on the biased sampling points, and carrying out back calculation on the control vertex so as to obtain the T spline representation of the result curved surface.
The initial T spline surface in the step 1 is a bi-cubic T spline surface which is based on a T spline data structure supporting singular points, wherein the singular points refer to parameter points which have a valence of not 4 and are not T points in a parameter domain T grid; the basic topological elements of the T spline data structure comprise a surface patch, an edge, a half edge, a node, an anchor point and a Bei Jier surface patch; the specific details can be found in the literature [ An effective data structure for calculation of unstructured T-spline surfaces ];
the step 2 of "Bei Jier extraction" refers to a method for converting a T spline surface into a sliced surface Bei Jier; bei Jier (Bezier, french, bei Jier curved surface is invented by Bei Jier (Bezier) in 1982, has excellent properties of geometric invariance, affine invariance, convex hull property and the like, and is a common curved surface representation form;
wherein, the step 2 of extracting Bei Jier from the initial T spline surface and dispersing the extracted data into Bei Jier units comprises the following specific steps:
2.1, extending node lines of all low-price nodes in a parameter domain T grid, wherein for a bicubic T spline surface, the extending distance of each point is two node distances along a certain direction;
step 2.2, cutting the dough sheet penetrated by the extension line;
step 2.3, calculating a definition domain of a mixing function corresponding to each parameter point, wherein the definition domain of each mixing function can be obtained by parameter point coordinates and local node distance vectors, and the local node distance vector of each parameter point is obtained by extending two node distances from the parameter point to each direction;
step 2.4, if the definition domain in the step 2.3 can not completely cover a certain face in the T grid, the face needs to be segmented;
step 2.5, after the segmentation result of the parameter domain T grid is obtained, a corresponding Bei Jier extraction operator can be calculated, and the main idea is to linearly express a B spline basis function by using a Bernstein basis function by using a node insertion algorithm; bei Jier the specific solving process of the extraction operator can be referred to in the literature [ Isologeometric finite element data structure based on Bezier extraction of T-spheres ].
Wherein, in step 3, the step of "calculating the number of sampling points of each boundary Bei Jier unit so as to obtain the global parameter coordinates of all the sampling points" includes the following steps:
step 3.1, calculating the number of sampling points corresponding to Bei Jier units by using a formula (3-1) for all boundary Bei Jier units:
where ε is the error given in step 1, M 1 、M 2 And M 3 Corresponding to the boundaries of the three second-order partial derivatives of the Bei Jier patch;
step 3.2, for two groups of opposite sides of the parameter domain, respectively selecting the side with larger number of sampling points as a reference side, and establishing a reference point on the reference side according to the number of the sampling points of each unit calculated in the step 3.1;
and 3.3, introducing a ray penetrating through the whole parameter domain for each reference point, wherein the coordinates of the intersection points of all the rays are the global parameter coordinates of the sampling points.
Wherein, the step 4 of performing offset calculation on each sampling point to obtain a point on a result curved surface comprises the following specific steps:
for each sampling point, a unit normal vector of a corresponding position is calculated, and then a point on a result curved surface can be calculated by using a formula (3-2):
S 1 (u 0 ，v 0 )＝S(u 0 ，v 0 )+dN(u 0 ，v 0 ) (3-2)
wherein u 0 、v 0 Global parameter coordinates of the sampling points, d is the offset distance given in step 1, N (u) 0 ，v 0 ) Is the unit normal vector of the corresponding position.
Wherein, in step 5, the step of "carrying out proper parameterization on the biased sampling points and back-calculating the control vertex so as to obtain the T spline representation of the result surface" comprises the following specific steps:
step 5.1, assume that step 4 resulted in (n + 1) × (m + 1) data points { Q k，l Where k = 0-n, l = 0-m;
step 5.2, calculating the node vector U by adopting a centripetal parameterization method to obtain each l
Then obtaining u through a formula (3-3) k Obtaining a node vector U;
step 5.3, obtaining a node vector V by using a method similar to the step 5.2;
step 5.4, back-calculating the control vertex of the result curved surface; first using the node vector U and the parameter U k Curve interpolation for m +1 times is carried out: for l = 0-m, an interpolation is constructed at the point Q 0，l ，Q 1，l ，......，Q n，l From which a control point matrix R is derived i，l (ii) a Then using the node vector V and the parameter V l Curve interpolation is carried out for n +1 times: for i =0 to n, interpolation is constructed at the point R i，0 ，R i，1 ，……，R i，m Thereby obtaining a control point matrix P of the final result surface i，j ；
And 5.5, determining a node vector U, V, determining a parameter domain of the result surface, solving a control vertex of the result surface, traversing each unit of the parameter domain, and storing a corresponding result in the form of a T spline data structure in the step 1 to obtain the T spline representation of the result surface.
(III) advantages
1. The invention can realize the offset calculation of the T spline surface within a given error range, the result surface is also expressed by the T spline surface, and the unification of the expression modes of the front and rear offset surfaces is beneficial to the subsequent processing of the result surface;
2. the T-spline data structure adopted by the invention supports the T-grid with singular points, and has strong universality, so that the result curved surface can support various complicated subsequent operations.
3. The method of the invention is scientific, has good manufacturability and has wide popularization and application value.
Drawings
FIG. 1 is a flow chart of offset calculation for a T-spline surface.
FIG. 2 is a schematic view of a ball nose mill machining a surface of a part.
FIG. 3 is a parameter domain T-grid of an initial T-spline surface in an example.
FIG. 4 shows the result of Bei Jier extraction of the parameter domain T mesh of the initial T-spline surface in the example.
The numbers, symbols and codes in the figures are explained as follows:
Ω in fig. 2 represents a machined surface of the ball nose mill; c represents the sphere center of the ball end mill cutter head, namely the cutter location point; r represents the radius of the ball end mill cutter head; p represents a contact point of the ball end mill and the processing surface at a certain time; n is a unit normal vector corresponding to the P point;
v1, V2, and V3 in fig. 3 and 4 are numbers of nodes corresponding to the parameter domain T grid; s1, S2, S3, S4, S5, S6, S7 and f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12, f13, f14 are numbers of corresponding planes in the parameter domain T grid.
Detailed Description
The invention provides a bias calculation method of a T spline surface, which can calculate the bias surface of the T spline surface within a given error range, and the result surface is expressed by the same T spline as the original surface. The method can be applied to the generation of numerical control machining cutter track, particularly for a spherical milling cutter, the relation between the machining surface and the cutter track is completely the same as the relation between the front curved surface and the rear curved surface of the offset, and the radius R of the spherical milling cutter is the offset distance d. The technical solution of the present invention will be clearly and completely described below with reference to an example of applying the present invention to the generation of a path of a ball milling cutter tool and the accompanying drawings.
The invention discloses a bias calculation method of a T spline surface, wherein a calculation flow chart is shown in figure 1, and the method specifically comprises the following processes:
1) And inputting an initial T spline surface, an offset distance d and an offset error epsilon. A schematic diagram of this example is shown in fig. 2; for the example, the initial T-spline surface represents the surface shape Ω to be processed by the ball milling cutter, and the parameter domain T mesh corresponding to the T-spline surface is shown in fig. 3; the offset distance represents the radius R of the ball milling cutter, and the offset distance d =10mm; the offset error represents the error between the track of the cutter point C of the spherical milling cutter and the theoretical track, and the offset error epsilon =0.1mm.
2) The initial T-spline surface input in (1) is subjected to Bei Jier extraction, and is discretized into Bei Jier cells. This section is described in detail below with reference to the example of fig. 3:
firstly, node lines of low-price points are extended, and for a bicubic T-spline surface, the extending distance of each point is the distance between two nodes in a certain direction. For example, for point V1, its parameter point is extended by two nodal distances in two directions, so as to divide the surfaces S5 and S6 into two sub-surfaces; for point V2, the parameter point is extended by two nodal distances, so that the surfaces S4, S5, S6, S7 are divided into two sub-surfaces.
All the parameter points then need to be traversed to compute the domain of the mixing function associated with the parameter points. Taking node V1 as an example, local nodal distance vectors in two parameter directions of the T-spline surface are calculated according to a definition mode of the local nodal distance vector of the T-spline surface, and then a definition domain of a mixing function related to parameter points can be obtained according to the parameter point coordinates and the local nodal distance vectors. If the domain cannot completely cover a face in the T-grid, the face needs to be sliced. As shown in fig. 3, the domain of the blending function corresponding to the parameter point V1 cannot completely cover the patch S1, and therefore S1 needs to be segmented.
After the steps, each surface sheet of the T-spline surface is Bei Jier units and can be represented by a Bei Jier surface. The Bei Jier extraction process for a T-spline surface can be understood as the process of solving the Bei Jier extraction operator. Bei Jier extraction operator converts a T spline control vertex into a control vertex on a tile Bei Jier cell. The key for solving the extraction operator of each Bei Jier unit is how to obtain Bei Jier extraction vectors corresponding to all T spline mixing functions related to the unit, and the extraction vectors of the T spline mixing functions can be obtained by direct products of the extraction vectors in two directions, so that the Bei Jier extraction problem of the T splines is converted into the extraction vector solving problem of the univariate B spline basis functions. The univariate B-spline basis function and the Bernstein basis function have a mapping relationship as shown in the formula (5-1):
wherein e denotes the cell name, a is the subscript of the basis function, e a Is a unit vector with a component of 1 at a and the rest of 0,c e The extracted vector c is the extracted vector of the univariate B-spline basis function which is finally required, so that the extracted vector c can be obtained by a formula as long as the node vector before the node is inserted and the node needing to be inserted are obtained e . The whole extraction process of the T-spline surface Bei Jier is completed.
3) Fig. 34 shows the result of Bei Jier extraction of the parameter domain T grid, and for all the boundary Bei Jier cells shown in fig. 4, the number of sampling points corresponding to Bei Jier cells is calculated using the formula (5-2):
where ε is a given error, M 1 、M 2 And M 3 Corresponding to the boundaries of the three second partial derivatives of the Bei Jier patch, respectively. Then, for two groups of opposite sides of the parameter domain, the sides with larger number of sampling points are respectively selected as reference sides, reference points are established on the reference sides, finally, a ray penetrating through the whole parameter domain is introduced for each reference point, and the intersection point of all the rays is the global parameter coordinate of the sampling points.
4) The method for carrying out offset calculation on each sampling point is shown in the formula (5-3):
S 1 (u 0 ，v 0 )＝S(u 0 ，v 0 )+dN(u 0 ，v 0 ) (5-3)
wherein u is o 、v 0 Global parameter coordinates of the sampling points, d is the offset distance given in step 1, N (u) 0 ，v 0 ) Is the unit normal vector of the corresponding position.
5) Thus, (n + 1) × (m + 1) data points { Q } have been obtained k，l } k＝0～n，l＝0～m Next, a T-spline surface needs to be constructed that interpolates these data points.
Firstly, the parameter domain T mesh of the resulting surface needs to be determined, that is, the node vector U and the node vector V in two directions are determined, the calculation method is described in detail below by taking the determination of the node vector U as an example, and the node vector V can be obtained by using a similar method. The first step is to obtain for each l by using a centripetal parameterization method
The second step is to obtain u through the formula (5-4) k To obtain u k A node vector U is obtained.
The calculation method of centripetal parameterization is shown in a formula (5-5):
wherein | Q k，l -Q k-1，l I represents a point Q k-1，l To Q k，l A chord length of
And secondly, reversely calculating the control vertex of the result surface. First using the node vector U and the parameter U k Curve interpolation for m +1 times is carried out: for l = 0-m, an interpolation is constructed at the point Q 0，l ，Q 1，l ，……，Q n，l From which a control point matrix R is obtained i，l (ii) a Then using the node vector V and the parameter V l And (3) performing n +1 times of curve interpolation: for i =0 to n, interpolation is constructed at the point R i，0 ，R i，1 ，......，R i，m Thereby obtaining a control point matrix P of the final result surface i，j 。
After the node vector U, V is determined, the parameter domain of the result curved surface is determined, the control vertex of the result curved surface is also solved, then each unit of the parameter domain is traversed, the corresponding result is stored in the form of the T spline data structure adopted by the invention, the T spline representation of the result curved surface is obtained, and the result curved surface is the solved cutter track of the spherical milling cutter, namely the track of the cutter location point C. The whole process of the present invention is completed from this point on.
The above description is only a preferred embodiment of the present invention, and should not be taken as limiting the invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (4)
1. A bias calculation method of a T spline surface is applied to a spherical milling cutter, and is characterized in that: it comprises the following steps:
step 1, inputting an initial T spline surface, an offset distance d and an offset error epsilon; the initial T spline surface represents the surface shape omega to be processed by the spherical milling cutter, and the parameter domain T mesh corresponding to the T spline surface; the offset distance d represents the radius R of the spherical milling cutter, and the offset error epsilon represents the error between the track of the cutter point C of the spherical milling cutter and the theoretical track;
step 2, extracting Bei Jier from the initial T spline surface, and dispersing the initial T spline surface into Bei Jier units;
step 3, calculating the number of sampling points of each boundary Bei Jier unit so as to obtain global parameter coordinates of all the sampling points;
step 4, performing offset calculation on each sampling point to obtain points on a result curved surface;
step 5, parameterizing the biased sampling points, and back-calculating a control vertex to obtain a T spline representation of a result curved surface; wherein,
step 5.1, set step 4 to obtain (n + 1) × (m + 1) data points { Q k，l Where k = 0-n, l = 0-m;
step 5.2, calculating the node vector U by adopting a centripetal parameterization method to obtain each l
Then obtaining u through a formula (3-3) k Obtaining a node vector U;
step 5.3, a node vector V in the other direction of the parameter domain can be obtained by using the method in the step 5.2;
step 5.4, back-calculating the control vertex of the result curved surface; first using the node vector U and the parameter U k Curve interpolation for m +1 times is carried out: for l = 0-m, an interpolation is constructed at the point Q 0，l ，Q 1，l ，……，Q n，l Thereby obtaining a control point matrix R; then using the node vector V and the parameter V l And (3) performing n +1 times of curve interpolation: for i = 0-n, an interpolation is constructed at the point R i，0 ，R i，1 ，……，R i，m Thereby obtaining a control point matrix P of the final result surface;
and 5.5, determining a parameter domain of the result curved surface after determining the node vector U, V, solving a control vertex of the result curved surface, traversing each unit of the parameter domain, storing a corresponding result into a data structure of an initial T spline curved surface, and obtaining a T spline representation of the result curved surface, wherein the result curved surface is the solved cutter track of the spherical milling cutter, namely the track of the cutter locus C.
2. The offset calculation method for a T-spline surface according to claim 1, characterized in that: the step 2 specifically comprises:
2.1, extending node lines of all nodes in a parameter domain T grid, wherein for a bicubic T spline surface, the extending distance of each point is two node distances along one direction;
step 2.2, cutting the dough sheet penetrated by the extension line;
step 2.3, calculating a definition domain of a mixing function corresponding to each parameter point, wherein the definition domain of each mixing function can be obtained by parameter point coordinates and local node distance vectors, and the local node distance vector of each parameter point is obtained by extending two node distances from the parameter point to each direction;
step 2.4, if the definition domain in the step 2.3 can not completely cover one surface in the T grid, the surface needs to be segmented;
and 2.5, after the segmentation result of the parameter domain T grid is obtained, calculating a corresponding Bei Jier extraction operator.
3. The offset calculation method of a T-spline surface according to claim 1, characterized in that: in step 3, specifically:
step 3.1, calculating the number of sampling points corresponding to Bei Jier units by using a formula (3-1) for all boundary Bei Jier units:
wherein M is 1 、M 2 And M 3 Corresponding to the boundaries of the three second-order partial derivatives of the Bei Jier patch;
3.2, for two groups of opposite sides of the T-spline surface parameter domain, respectively selecting the sides with larger number of sampling points as reference sides, and establishing reference points on the reference sides according to the number of the sampling points of each unit calculated in the step 3.1;
and 3.3, introducing a ray penetrating through the whole parameter domain for each reference point, wherein the coordinates of the intersection points of all the rays are the global parameter coordinates of the sampling points.
4. The offset calculation method of a T-spline surface according to claim 1, characterized in that: in step 4, for each sampling point, the unit normal vector of the corresponding position is calculated, and then the point on the result curved surface can be calculated by using the formula (3-2):
S 1 (u 0 ，v 0 )＝S(u 0 ,v 0 )+dN(u 0 ,v 0 ) (3-2)
wherein u is 0 、v 0 Respectively, the global parameter coordinates of the sample points, N (u) 0 ，v 0 ) Is the unit normal vector of the corresponding position.
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