CN108986020A - A kind of three-dimension curved surface approximate expansion at plane adaptive approach - Google Patents
A kind of three-dimension curved surface approximate expansion at plane adaptive approach Download PDFInfo
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Abstract
The invention discloses a kind of three-dimension curved surface approximate expansions into the adaptive approach of plane, including carries out gridding to curved surface, obtains the grid intersection point collection of curved surface；Doubly curved surfaces are preset into plane, obtain corresponding grid intersection point collection in plane；Establish the nonlinear equation for making Doubly curved surfaces at plane and keeping topological structure；The point set in expansion plane is sought using triangulation method, makes nonlinear equation have minimum value by iteration initial value of plane point set, obtains corresponding plane coordinate point.The present invention solves figure of the three-dimension curved surface object in Image Acquisition and preprocessing process and planarizes problem, the processes such as appearance detection and analysis especially suitable for three-dimension curved surface object.
Description
Technical field
The present invention relates to automatic fields, and in particular to a kind of three-dimension curved surface approximate expansion at plane adaptive approach.
Background technique
It is complex-curved using more and more extensive with the development of technology with people to the needs of aesthetic appearance.But it is multiple
The development and realization of miscellaneous curved surface, complex-curved manufacturing technology are one of crucial.It is complicated in automated manufacturing field
The processing of curved surface can just be obtained by one piece of block plane by bending machining.The expansion problems for studying curved surface, to realization computer
Computer aided geometric design has realistic meaning.
Summary of the invention
In order to overcome shortcoming and deficiency of the existing technology, the present invention provides a kind of three-dimension curved surface approximate expansion into plane
Adaptive approach.
The present invention is joined so that the minimum principle of the distance between front and back mesh point and its neighborhood mesh point difference is unfolded using curved surface
Number equation carries out gridding to curved surface, and abstract curved surface transforms into the mathematical expression formula of plane, certainly with curved surface second fundamental form
The length corresponding relationship for adapting to determine weight factor and expansion front and back, the calculation method of iteration initial value is solved by triangulation method,
To be maximally maintained the topological structure of curved surface.The present invention is also suitable for carrying out the three dimensional surface data of target object flat
Smoothization processing, obtains its two-dimentional physical size information on the image.
The present invention adopts the following technical scheme:
A kind of three-dimension curved surface approximate expansion includes the following steps: at the adaptive approach of plane
S1 curved surface meshing:
The parametric equation r (u, v) of curved surface=(x (u, v), y (u, v), z (u, v)), enables u=u0, obtain one on curved surface
Curve r (u0, v), enable v=v0, obtain curve r (u, a v on curved surface0), the u0、v0For constant, wherein two curves
Intersecting point coordinate is r (u0, v0), change u, the value of v obtains the curved surface being spliced by quadrilateral mesh, further obtains grid
The coordinate of intersection point, with setIt indicates；
S2 presets curved surface and transforms into plane, and the grid intersection point coordinate set that curved surface is corresponded in plane is
The topological structure for keeping Doubly curved surfaces plane, then enable the grid intersection point P of curved surfaceiWith consecutive points PjThe vector of composition isAndTransform into corresponding grid intersection point Q after planeiWith consecutive points QjThe vector of composition isAndThe process that then surface mesh intersection point transforms into corresponding grid intersection point after plane is to make nonlinear equationThere is minimum value；
S3 determines weight factor g according to the curvature of curved surfaceij；
S4 seeks the iteration initial value of nonlinear equation using triangulation method, then substitutes into nonlinear equation, determines plane net
The coordinate of lattice intersection point.
By the value sequence of u in the S1, grid intersection point is ranked up, the value of u is then fixed, then traverses taking for v
Value, is ranked up grid intersection point coordinate, the grid intersection point coordinate after sequence is stored in set
I indicates the serial number of intersection point, and n is natural number.
In the S2, make nonlinear equationThere is minimum value, it is specifically right respectively
xi、yiDerivation obtains:
It finds out it and each mesh point coordinate Q in plane is unfoldedi(xi, yi) nonlinear equation is made to reach minimum value, wherein gijFor
Weight factor, ∑NIt indicates and point PiThe foot mark summation of consecutive points.
The curvature P of the S3 mean camberiWith consecutive points PjThe vector of compositionProjection in surface normal vector
δijSize indicate.
The projection δijIt is as follows:
And
Wherein,For normal vector, rv、ruSurface parameter equation r (u, v) is respectively indicated to v, u derivation.
The curvature meets following condition:
∑N gij=1,∑NIt indicates and point PiThe foot mark summation of consecutive points.
In the S4, the iteration initial value of nonlinear equation is sought using triangulation method, determines the coordinate of plane grid intersection point,
It is specific as follows:
The length that expansion each quadrilateral mesh of plane is determined according to the curvature of curved surface, determines its geometric properties；
Quadrangular mesh partition is found out another at triangle by known Atria edge lengths and two apex coordinates
A apex coordinate finds out the initial value Q of Nonlinear System of Equationsi(xi, yi)。
The length that expansion each quadrilateral mesh of plane is determined according to the curvature of curved surface, determines its geometric properties, has
Body are as follows:
Since any of center of surface, reselection and its adjacent two o'clock constitute a triangle Δ P1P2P3, according to
Curved surface corresponds to a triangle Δ Q of plane in the curvature situation of the point1Q2Q3, wherein it is knownIt is corresponded in plane, then The geometric properties of triangle in plane are determined.
The parametric equation of curved surface is first to be fitted again interpolation by the discrete point to curved surface to obtain.
The curved surface that quadrilateral mesh is spliced, quadrilateral mesh setting are uniform.
Beneficial effects of the present invention:
(1) present invention, will be bent to keep the minimum principle of distance change in the neighborhood of Doubly curved surfaces front and back between points
The problem of the problem of face is unfolded is converted to a solution Nonlinear System of Equations, is apparent from the process of certainly problem.
(2) present invention adaptively determines weight factor and expansion using the curved surface the second form in differential geometric theory
The corresponding relationship of front and back side length of element.
(3) present invention is also asked suitable for three-dimension curved surface object in Image Acquisition and the figure planarization preprocessing process
Topic.
Detailed description of the invention
Fig. 1 is work flow diagram of the invention；
Fig. 2 is the present embodiment surface parameter equation track schematic diagram；
Fig. 3 is the present embodiment curved surface quadrangle networking schematic diagram；
Fig. 4 is the schematic diagram that the present embodiment curved surface transforms into plane；
Fig. 5 is the present embodiment curved surface curvature schematic diagram；
Fig. 6 is the present embodiment surface mesh corresponding flat network diagram；
Fig. 7 is that the present embodiment solves iteration initial value schematic diagram.
Specific embodiment
Below with reference to examples and drawings, the present invention is described in further detail, but embodiments of the present invention are not
It is limited to this.
Embodiment
As shown in Figure 1, a kind of three-dimension curved surface approximate expansion includes the following steps: at the adaptive approach of plane
S1 curved surface meshing:
The parametric equation r (u, v) of curved surface=(x (u, v), y (u, v), z (u, v)), enables u=constant u0, obtain on curved surface
One curve r (u0, v), enable v=constant v0, obtain curve r (u, a v on curved surface0), wherein the intersection point of two curves is sat
Mark is r (u0, v0), it is smaller and uniformly for principle with grid, change u, the value of v obtains the song being spliced by quadrilateral mesh
Face obtains the coordinate of grid intersection point, with set It indicates.
The parametric equation of curved surface is to obtain the parameter of curved surface by being first fitted, then interpolation to some discrete points on curved surface
Method.
As shown in Fig. 2, when u is constant, available longitudinal curve, when v is constant, an available transverse direction
Curve, two curves must have an intersection point, and coordinate is r (u0, v0).If enabling u=u1, u2, u3 ... according to certain rule, v is enabled
=v1, v2, v3 ... must then have intersection point r (u1, v1)、r(u1, v2)、r(u1, v3)、r(u2, v1)、r(u2, v2)、r(u2, v3)、r
(u3, v1)、r(u3, v2)、r(u3, v3)…….By the value of the fixed u of value sequence of u, then the value of v is traversed, when having traversed u
Value the grid intersection point on curved surface then can be obtained.Such as fixed u=u1, value v1, v2, v3, v4 ... of v are then traversed,
R (u then can be obtained1, v1)、r(u1, v2)、r(u1, v3)、r(u1, v4) ..., then fixed u=u2, then traverse v value v1,
R (u then can be obtained in v2, v3, v4 ...2, v1)、r(u2, v2)、r(u2, v3)、r(u2, v4) ..., the coordinate of these intersection points is had
Sequence it is stored in setIn.
S2 mathematicization curved surface transforms into the process of plane: default curved surface is transformed into plane, corresponding quadrilateral mesh in plane
Putting coordinate isTo keep the topological structure before and after Doubly curved surfaces as far as possible, P is enablediWith consecutive points Pj
The vector of composition isAndEnable QiWith consecutive points QjThe vector of composition isAndThen
The process, which can be attributed to, makes Nonlinear System of EquationsThere is minimum value.
As shown in figure 3, plane is divided into several quadrilateral mesh, and mesh point coordinate is it is known that ∑NIndicate to point
PiThe foot mark summation of adjacent point.For example, the value of N is 2,7,6 as i=1, as i=8, the value of N is 2,7,11,
12、13、9、4、3。
It is assumed that curved surface is transformed into plane, the coordinate for corresponding to mesh point is Qi(xi, yi)。
Keep Nonlinear System of Equations value minimum, then respectively to xi、yiDerivation obtains:
P can be usedij 2Instead of Qij 2Simplify and calculate, above formula becomes
The solution of Nonlinear System of Equations is converted into solve the plane coordinates Q for setting up above formulai(xi, yi)。
S3 determines weight factor g according to the curvature of curved surfaceij, second fundamental form of a surface shows the curvature of curved surface.
Weight factor g is determined according to the curvature of curved surfaceij, curvature is bigger, weight factor gijSmaller, curvature is smaller, weight factor gijMore
Greatly；I and j indicates the serial number of intersection point.
As shown in figure 5, the curvature of curved surface can use PiWith consecutive points PjThe vector of compositionThrowing in surface normal vector
Shadow δijSize indicate.ΔijIt is bigger, show that curved surface is bigger in the bending of this point, δijIt is smaller, show curved surface in the bending of this point
It is smaller.
Andrv、ruSurface parameter equation r (u, v) is respectively indicated to v, u derivation.It will
Point PiCoordinate substitutes intoCurved surface then can be obtained in point PiNormal vector
Wherein, weight factor meets two conditions:
S4 seeks the iteration initial value of nonlinear equation using triangulation method, determines the coordinate of plane grid intersection point.
As shown in fig. 6, reselection and its adjacent two o'clock constitute a triangle Δ since any of center of surface
P1P2P3, a triangle Δ Q of plane is corresponded in the curvature situation of the point according to curved surface1Q2Q3.It is known thatIt is corresponded in plane, thenIn this way, the geometric properties of the triangle in plane are determined.
It according to above-mentioned steps, is started to spread out from the center of curved surface, for other some coordinate Q of plane trianglei(xi,
yi) solution, can be according to the following steps:
As shown in Figure 7, it is known that two o'clock T1(x1, y1)、T2(x2, y2) coordinate and they arrive another point T (x, y) distance d1、
d2, for the coordinate of point T, can solve using the following method.
Its unit vector is
Wherein
IfIt isObtained vector after rotating counterclockwise 90 degree, then
VectorAboutWithIt is decomposed:
Wherein, α is indicatedIt arrivesCorner,
Simultaneously
So
According to the cosine law it is found that
Then
There are two values by the sin α acquired herein, cast out negative value, retain positive value.
The present invention during three-dimension curved surface is unfolded, be unfolded front and back mesh point and its neighborhood mesh point between away from
The minimum principle of deviation carries out gridding to curved surface using surface parameter equation, and the mathematical expression that abstract curved surface transforms into plane is public
Formula adaptively determines the grid before and after weight factor and Doubly curved surfaces with the curved surface second fundamental form in differential geometric theory
The corresponding relationship of distance between point, the calculation method for solving iteration initial value is determined by triangulation method, to protect to the greatest extent
Hold the topological structure of curved surface.The present invention solves figure of the three-dimension curved surface object in Image Acquisition and preprocessing process and planarizes
Problem, the processes such as appearance detection and analysis especially suitable for three-dimension curved surface object.
The above embodiment is a preferred embodiment of the present invention, but embodiments of the present invention are not by the embodiment
Limitation, other any changes, modifications, substitutions, combinations, simplifications made without departing from the spirit and principles of the present invention,
It should be equivalent substitute mode, be included within the scope of the present invention.
Claims (10)
1. a kind of three-dimension curved surface approximate expansion is at the adaptive approach of plane, which comprises the steps of:
S1 curved surface meshing:
The parametric equation r (u, v) of curved surface=(x (u, v), y (u, v), z (u, v)), enables u=u0, obtain a curve r on curved surface
(u0, v), enable v=v0, obtain curve r (u, a v on curved surface0), the u0、v0For constant, wherein the intersection point of two curves
Coordinate is r (u0, v0), change u, the value of v obtains the curved surface being spliced by quadrilateral mesh, further obtains grid intersection point
Coordinate, with setIt indicates；
S2 presets curved surface and transforms into plane, and the grid intersection point coordinate set that curved surface is corresponded in plane is
The topological structure for keeping Doubly curved surfaces plane, then enable the grid intersection point P of curved surfaceiWith consecutive points PjThe vector of composition isAndTransform into corresponding grid intersection point Q after planeiWith consecutive points QjThe vector of composition isAndThe process that then surface mesh intersection point transforms into corresponding grid intersection point after plane is to make nonlinear equationThere is minimum value；
S3 determines weight factor g according to the curvature of curved surfaceij；
S4 seeks the iteration initial value of nonlinear equation using triangulation method, substitutes into nonlinear equation, determines plane grid intersection point
Coordinate.
2. adaptive approach according to claim 1, which is characterized in that by the value sequence of u in the S1, hand over grid
Point is ranked up, and then fixes the value of u, then traverses the value of v, is ranked up to grid intersection point coordinate, the grid after sequence
Intersecting point coordinate is stored in setI indicates the serial number of intersection point, and n is natural number.
3. adaptive approach according to claim 1, which is characterized in that in the S2, make nonlinear equationThere is minimum value, specifically respectively to xi、yiDerivation obtains:
It finds out it and each mesh point coordinate Q in plane is unfoldedi(xi, yi) nonlinear equation is made to reach minimum value, wherein gijFor power because
Son, ∑NIt indicates and point PiThe foot mark summation of consecutive points.
4. adaptive approach according to claim 1, which is characterized in that the curvature P of the S3 mean camberiWith it is adjacent
Point PjThe vector of compositionProjection δ in surface normal vectorijSize indicate.
5. adaptive approach according to claim 4, which is characterized in that the projection δijIt is as follows:
And
Wherein,For normal vector, rv、ruSurface parameter equation r (u, v) is respectively indicated to v, u derivation.
6. adaptive approach according to claim 4, which is characterized in that the curvature meets following condition:
∑N, gij=1,∑NIt indicates and point PiThe foot mark summation of consecutive points.
7. adaptive approach according to claim 1, which is characterized in that in the S4, ask non-linear using triangulation method
The iteration initial value of equation determines the coordinate of plane grid intersection point, specific as follows:
The length that expansion each quadrilateral mesh of plane is determined according to the curvature of curved surface, determines its geometric properties；
Quadrangular mesh partition is found out into another top by known Atria edge lengths and two apex coordinates at triangle
Point coordinate, that is, find out the initial value Q of Nonlinear System of Equationsi(xi, yi)。
8. adaptive approach according to claim 7, which is characterized in that described to determine that expansion is flat according to the curvature of curved surface
The length of each quadrilateral mesh in face, determines its geometric properties, specifically:
Since any of center of surface, reselection and its adjacent two o'clock constitute a triangle Δ P1P2P3, according to curved surface
A triangle Δ Q of plane is corresponded in the curvature situation of the point1Q2Q3, wherein it is knownIt is corresponded in plane, then The geometric properties of triangle in plane are determined.
9. adaptive approach according to claim 1, which is characterized in that the parametric equation of curved surface be by curved surface from
Scatterplot is first fitted again interpolation and is obtained.
10. adaptive approach according to claim 1, which is characterized in that the curved surface that quadrilateral mesh is spliced, four sides
The setting of shape grid is uniform.
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN110246225A (en) * | 2019-04-15 | 2019-09-17 | 上海交通大学医学院附属第九人民医院 | Tongue cancer operative defect tongue body method for reconstructing, system, readable storage medium storing program for executing and equipment |
| CN112347589A (en) * | 2020-11-26 | 2021-02-09 | 南京玻璃纤维研究设计院有限公司 | Space geometry unfolding method and device and electronic equipment |
| CN114818224A (en) * | 2022-05-27 | 2022-07-29 | 中国空气动力研究与发展中心计算空气动力研究所 | Structural grid generation method, device, equipment and storage medium |
| CN116597109A (en) * | 2023-01-13 | 2023-08-15 | 东莘电磁科技(成都)有限公司 | Complex three-dimensional curved surface co-grid generation method |
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2018
- 2018-07-17 CN CN201810784407.6A patent/CN108986020A/en active Pending
Cited By (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110246225A (en) * | 2019-04-15 | 2019-09-17 | 上海交通大学医学院附属第九人民医院 | Tongue cancer operative defect tongue body method for reconstructing, system, readable storage medium storing program for executing and equipment |
| CN110246225B (en) * | 2019-04-15 | 2023-04-21 | 上海交通大学医学院附属第九人民医院 | Method, system, readable storage medium and apparatus for reconstructing a defective tongue after tongue cancer surgery |
| CN112347589A (en) * | 2020-11-26 | 2021-02-09 | 南京玻璃纤维研究设计院有限公司 | Space geometry unfolding method and device and electronic equipment |
| CN114818224A (en) * | 2022-05-27 | 2022-07-29 | 中国空气动力研究与发展中心计算空气动力研究所 | Structural grid generation method, device, equipment and storage medium |
| CN114818224B (en) * | 2022-05-27 | 2023-05-23 | 中国空气动力研究与发展中心计算空气动力研究所 | Structural grid generation method, device, equipment and storage medium |
| CN116597109A (en) * | 2023-01-13 | 2023-08-15 | 东莘电磁科技(成都)有限公司 | Complex three-dimensional curved surface co-grid generation method |
| CN116597109B (en) * | 2023-01-13 | 2023-12-22 | 东莘电磁科技(成都)有限公司 | Complex three-dimensional curved surface co-grid generation method |
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Effective date of registration: 20200508 Address after: Nansha District Avenue South Ring of 511458 cities in Guangdong province Guangzhou City, No. 25 Hua Da Guangzhou production and Research Institute Applicant after: SOUTH CHINA University OF TECHNOLOGY Applicant after: Chengdu Saiman Technology Co.,Ltd. Address before: Nansha District Avenue South Ring of 511458 cities in Guangdong province Guangzhou City, No. 25 Hua Da Guangzhou production and Research Institute Applicant before: SOUTH CHINA UNIVERSITY OF TECHNOLOGY |
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