CN109377561B - Conformal geometry-based digital-analog surface grid generation method - Google Patents
Conformal geometry-based digital-analog surface grid generation method Download PDFInfo
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Abstract
The invention provides a method for generating a digital-analog surface grid based on conformal geometry, which aims to simplify the problem of generating a three-dimensional model surface grid to a two-dimensional plane for processing. Wherein the method comprises the following steps: selecting a conformal mapping function from the three-dimensional model to a two-dimensional parameter area according to topology information of the three-dimensional model, then conducting finite element mesh subdivision on the parameter area, generating triangle units or quadrilateral units in mesh generation of the parameter area, mapping the mesh conformal inverse of the parameter area into the mesh of the space area according to the mapping function from the parameter area to the space area, and keeping the topological relation among nodes unchanged, wherein the mesh of the space area is the surface mesh of the required three-dimensional model. Compared with the mesh subdivision directly carried out on the three-dimensional digital-analog surface, the mesh subdivision operation carried out on the parameterized region is simpler, the calculation complexity is lower, and the three-dimensional model surface generated by the method has high mesh quality and good precision.
Description
Technical Field
The invention relates to the field of conformal geometry calculation technology and three-dimensional image processing, in particular to a method for generating a digital-analog surface grid based on conformal geometry in three-dimensional image processing.
Background
With the rapid development of computer technology, computer Aided Design (CAD) technology and Computational Fluid Dynamics (CFD), the number of three-dimensional models in the fields of engineering drawing, video games, scientific research, cultural relic restoration, biomedical and the like has been increasing explosively. A large number of digital-to-analog processing software and digital-to-analog files are used in different fields, which provides a good development opportunity for a three-dimensional model, and meanwhile, the complexity of practical problems in various application fields such as aerospace and the like also provides serious challenges for computational fluid dynamics, and one of the important problems is how to automatically generate grids around complex shapes. It is described that grid generation takes approximately 60% to 70% of the total computation cycle, whereas surface descriptions of complex topography and surface grid generation typically take 70% to 80% of the grid generation time. It follows that surface descriptions of complex contours and surface mesh generation are one of the "bottleneck" problems that prevent continuous longitudinal development of three-dimensional models.
The method is characterized in that the method is used for calculating conformal geometry, is a cross subject, is rooted in the field of pure mathematics, mainly researches conformal structures on curved surfaces, searches for association and balance between topology and geometry, and digs potential and internal characteristics of the curved surfaces to better understand the curved surfaces for graph analysis and processing. Conformal geometry plays an important role in digital geometry processing, in particular lays a theoretical foundation for surface parameterization, and provides a strict algorithm. The core for computing the conformal geometry is a conformal structure, and the curved surface with the conformal structure is a Riemann surface. There are many ways of studying conformal structures, complex analysis, algebraic geometry and differential geometry. Recently, with the development of three-dimensional scanning technology, the improvement of computing power and the further development of mathematical theory, computing conformal geometric theory and algorithm have been generalized from planar areas to metric curved surfaces with arbitrary topology.
The kinds of the grid generation algorithms are various, and the various algorithms are roughly summarized as follows according to the characteristics of the geometric analysis domain of the algorithms: mapping, leading edge propulsion (Advancing Front Method), delaunay triangulation, topology decomposition and geometric decomposition. The mapping method appears in the 70 th century and is the earliest grid generation method, which is to define a proper mapping function according to a parameter equation of a body boundary, map a target area into a parameter space to form a regular parameter domain, conduct grid subdivision on the regular parameter domain, inversely map grids (two-dimensional square and three-dimensional cube) of the parameter domain back to the original irregular area, thereby generating actual physical space grids. The front edge propulsion method is used for automatically generating the triangular grids of the plane area, has good effects and is popularized to the generation of the two-dimensional self-adaptive grids and the generation of the three-dimensional tetrahedral grids of any shape area, the grids can be generated only by giving the area boundary, the generated grids have repeatability, and in addition, the front edge propulsion method has the advantages of wide adaptability, high unit quality and the like, and the defects are mainly that the algorithm convergence cannot be guaranteed because of no strong theoretical basis. Delaunay triangulation is one of the most popular and universal full-automatic grid generation methods at present, the maximum and minimum angle characteristics and the circumscribing characteristics ensure that cells with higher quality can be generated, and along with the continuous development of Delaunay technology, a plurality of effective algorithms are developed at present, and can be generally divided into the following three main categories: a method of calculating Voronoi diagram represented by Bowyer and Green, sibsons; an air-to-circle method represented by Watson; diagonal switching algorithm represented by Lawson. The method for directly calculating the Voronoi diagram is generally complex, the required memory is large, the calculation efficiency is low, and the algorithm is rarely adopted at present; the Lawson algorithm is particularly suitable for two-dimensional Delaunay triangulation, does not have degradation phenomenon, is applicable to constraint conditions, has high calculation efficiency, and cannot be directly popularized to three-dimensional conditions; the Watson algorithm is simple in concept, easy to program and implement, and can implement constraint triangularization, so that the application frequency is the widest, but the method has the defect that sheet units with small internal angles can be formed. The topology decomposition method is initially applied to the two-dimensional plane problem, and now has been popularized to the three-dimensional space curved surface, and is essentially that on the basis of guaranteeing the continuity of the topology structure, triangle units are cut one at a time, so that the triangle units are as close to an equilateral triangle as possible, regardless of the specific shape of a body, after an initial grid is formed in a region to be split, the unit cells are gradually refined by means of half-segmentation, center segmentation, diagonal replacement and the like. The principle of the topological decomposition method is simple, and the implementation is convenient, but the topological decomposition method only starts from the topological factors of the body and is excessively dependent on the topological structure of the body without considering the geometrical factors, so that cells with good quality are difficult to obtain, and the three-dimensional body containing curved surfaces is difficult to process. The geometric decomposition method is a recursive half-segmentation method, which takes the geometric characteristics of the body into account more, and is a method for simultaneously generating nodes and cells. The geometric decomposition method has the advantages that the unit is divided from the geometric factors of the area to be divided, and the partial result with good shape can be obtained each time, so the method can generally form the unit cell with high quality, and has the defects that the complex area and the irregular shape are firstly divided into simple subareas, the manual intervention is relatively more, the execution efficiency is low, in addition, the unit shape which is only taken out each time by the layer-by-layer cyclic division method is good, but the final rest part cannot be ensured to be reasonable, namely the blank area is difficult to solve.
In view of the above, it is important to combine the conformal geometry technology to build a set of general surface grid generation methods for any complex digital model.
Disclosure of Invention
The invention provides a method for generating a digital-analog surface grid based on conformal geometry, which solves the problem of generating a three-dimensional model surface grid by using a conformal geometry technology.
The invention adopts the technical scheme that: a method for generating a digital-analog surface grid based on conformal geometry comprises the following steps:
(1) Judging whether the three-dimensional model surface can be oriented or not according to the input three-dimensional model data, and calculating the number of the deficiency of the curved surface and the number of the boundaries;
(2) Selecting a proper conformal mapping function according to the calculated topological structure, and conformally mapping the three-dimensional modular surface to one of a spherical surface, an European plane and a two-dimensional hyperbolic space according to a curved surface single-valued theorem;
(3) Conformally mapping the three-dimensional model to a two-dimensional parameter domain according to the selected conformally mapping function;
(4) Performing finite element mesh subdivision on a parameter domain according to the conformal mapping result, and generating a triangle unit mesh or a quadrilateral unit mesh according to the requirement;
(5) And reversely mapping the re-dissected grid back to the three-dimensional space domain according to the angle maintained in the conformal mapping process, and generating a three-dimensional model with the surface grid.
In step 2, the three-dimensional model is understood to be a connected curved surface for processing, and the topological structure of the curved surface is determined by whether the curved surface can be oriented, the number of ring handles (deficiency) and the number of boundaries.
In step 3, conformally transforming the segmented curved surface into one of three standard spaces according to the topology structure determined in step 2: sphere, european plane and two-dimensional hyperbolic space.
In step 4, the meshing has two important steps: grid nodes are reasonably distributed in the area to be split, namely points are distributed; the grid nodes are effectively connected to form triangular or quadrilateral grid cells, i.e., cell generation, wherein the triangular grid generation employs Delaunay triangulation.
In step 5, the grid of the parameter domain is mapped into the grid of the space domain according to the mapping function from the parameter domain to the space domain, the topological relation among the nodes is kept unchanged, the three-dimensional transformation is changed into two-dimensional transformation, the angle is kept, and the local area is changed. The change rate of the local surface element and the average curvature of the curved surface completely maintain all geometric information of the original curved surface, and the original curved surface is completely reconstructed according to the change rate and the average curvature of the surface element.
The three-dimensional models can be regarded as connected curved surfaces, and parameterization mapping of the three-dimensional models can be regarded as parameterization mapping of the curved surfaces. The whole three-dimensional model surface grid generation process is equivalent to solving a smooth bijection from a curved surface space domain to a plane parameter domain mathematically.
Compared with the prior art, the invention has the advantages that: compared with the mesh dissection directly performed on the three-dimensional digital-analog surface, the mesh dissection performed on the parameterized region is simpler in operation, lower in calculation complexity and easier to implement in algorithm. Through the steps, the three-dimensional model surface grid generation problem is simplified into a smooth bijection solving the three-dimensional space domain to two-dimensional parameter domain and a two-dimensional parameter domain grid generation problem. The three-dimensional model generated by the step has high surface grid quality and good precision.
Drawings
FIG. 1 is a schematic diagram of an implementation of digital-to-analog surface mesh generation based on conformal geometry;
FIG. 2 is a schematic diagram of one embodiment according to the present invention;
fig. 3 is a schematic diagram of a finite element mesh based subdivision process.
Detailed Description
The invention is described below with reference to the drawings and the detailed description. Wherein fig. 1 depicts a specific process for generating a digital-to-analog surface mesh based on a conformal geometry. Fig. 3 depicts a finite element mesh subdivision implementation procedure.
The specific implementation steps are as follows:
(1) According to the given three-dimensional model, it is converted into an off or obj format that can be stored in half of the data structure. Firstly, boundary recognition is carried out on a model in a half-side data structure, the edge with only one adjacent surface is taken as a boundary edge in the traversing process, and if a closed loop exists between the boundary edges, the boundary can be taken as a boundary; and then carrying out the deficiency identification on the curved surface, and taking the maximum number of mutually disjoint simple closed curves which can be drawn on the curved surface and are not used for dividing the curved surface as the number of the deficiency of the curved surface of the model. And counting the number of boundaries and the number of deficiency grids of the curved surface, and storing the number of boundaries and the number of deficiency grids as topology information of the curved surface for the next step.
For a surface with a deficiency of 0, its conformal structure can be calculated from the harmonic mapping. Assume two 0-deficit-grid curved surfaces M 1 ，M 2 ,h:M 1 →M 2 Representing a mapping of degree 1 for the surface, and then minimizing the harmonic energy E (h),
the Laplace equation for h can be reduced to:
Δ PL h＝(Δ PL h 0 ,Δ PL h 1 ,Δ PL h 2 )
if h is harmonic, then delta PL The tangential component of h is 0. Defining a projection operator:
wherein the method comprises the steps ofThe tensor product, I, is the identity matrix. If and only if h satisfies:
h is harmonic, where n is M 2 Is defined by the normal to (d).
Additional constraints must be added to ensure that the process converges to a unique solution. Here the centroid of the surface is forced to lie at the origin:
is M 1 Is a part of the area of the substrate. This constraint can guarantee the uniqueness of the solution during rotation. Under this constraint, a partial differential equation is constructed:
at this time, the steady state solution of h is M 1 To M 2 Is a conformal mapping of (c). The equation can be effectively solved iteratively.
The method can be calculated by full pure differentiation for a curved surface with a deficiency of 1, is based on the Hodge theory, and is effective for closed or non-closed curved surfaces with any deficiency. The core idea of the method is to find a complete base in a completely pure 1 form, and the algorithm is approximately as follows:
and inputting a grid curved surface M.
Outputting a full pure differential basis form
Calculate the homology group b= { e 1 ,e 2 ,…,e 2g }；
Computing the homomorphic group base omega= { w on dual 1 ,w 2 ,…,w 2g }；
Each w i Dispersing to form ζ of blend 1 i ；
Calculate each ζ i Is expressed as i . Structured all-pure 1 form
For any deficiency of surfaces, the conformal structure of the surface can be calculated by discrete Ricci flow, and the specific steps are as follows:
assuming smooth mapping between given surfaces with metrics(S 1 ,g 1 )→(S 2 ,g 2 ) Locally, non-linearly mappedIt can be approximated by a linear mapping between the tangential planes, a so-called derivative mapping:
various distortions are introduced, and these distortions need to be measured using a Riemann metric. Linear mapping->The classification of (2) may take advantage of the above-described linear mapping classification ideas, for example: if for any point p E S 1 ，/>Are equidistant (or conformal, area-preserving, quasi-conformal), then as a whole +.>Is equidistant (or conformal, area-preserving, quasi-conformal).
The conformal metrics are calculated and embedded as follows:
the differential operator on length, angle, area and surface S can be calculated by the Riemann metric. During the parameterization, the Riemann metric is changed conformally, as are other related quantities besides angle. Until the measure g changes toThen the gaussian curvature will become:
where Δ is the laplacian of the original metric g; the geodesic curvature at this time will become:
from the Gauss-Bonnet equation, the total curvature is independent of the choice of the Riemann metric, which can prove that there is a unique conformal metricSo that the target curvature is constant and the geodetic curvature is zero.
The parameterization by curvature control is essentially to find the appropriate conformal measure e 2u g, making the induced curvature consistent with the preset target curvature.
In a smooth case, the conformal deformation maps an infinitely small circle to an infinitely small circle and maintains the intersection angle between circles, which can ensure the conformality and the conformality of the conformal mappingAngle retention. In discrete cases, a radius gamma is set at each vertex of the mesh by the introduction of the circle packing method i Different circles intersect each other.
In the Euclidean background, the conformal deformation is realized by changing the radius under the condition of ensuring that the intersection angle of the circles is unchangedThe purpose of parameterization is achieved. The discrete Ricci curvature flow has a form similar to the smooth case Ricci flow equation:
in the spherical or hyperbolic geometry context, when the Ricci curvature flow is used to calculate the spherical or hyperbolic metric, the process is the same as in the euclidean case, as long as the corresponding spherical or hyperbolic geometry is used when calculating the geometric quantity.
Let the hyperbolic circle (C, R) be an euclidean circle (C, R) with C as the center and R as the radius. The method meets the following conditions:
wherein u= (e r -1)/(e r +1)。
Next, the first face [ v ] is first applied 0 ,v 1 ,v 2 ]The parameters are respectively as follows:
surface f adjacent to the first surface ijk Embedded therein. Let vertex v i ,v j Already in the hyperbolic plane, then the third vertex v k Namely v is i ,v j Is the intersection of two circles of vertices. In the same way, other are as follows ijk Adjacent faces are pressed into the stackStacks, embedded in order in the hyperbolic plane. The embedding mode in the hyperbolic space is basically the same as that of the Euclidean space. Similarly, the embedding method under spherical geometry is the same as under Euclidean space and hyperbolic space.
The basic idea of the direct variation method is as follows: the mapping from the triangular mesh to the planar region is represented by a piecewise linear mapping,M.fwdarw.D. Fixing a triangle, laying it on a plane, using v i ,v j ,v k To represent plane coordinates of the vertices byTo represent the plane coordinates of the vertex image, the linear mapping matrix can be written directly:
if e ijk At 0, the linear mapping restricted to this triangle is a conformal transform. The gap between the entire mapping and the equidistant transformation is defined as:
here s ijk Representing the area of the corresponding triangle. This energy is then optimized with a nonlinear optimization method and the result is taken as an approximation of the global conformal transformation:
if the objective is to seek a guaranteed area transform, then:
e ijk :＝(λ 1 λ 2 -1) 2 ＝(det(A ijk )-1) 2
if the aim is to equally transform, the local energy is defined:
e ijk :＝(λ 1 -1) 2 +(λ 2 -1) 2 ＝(λ 1 +λ 2 ) 2 -2(λ 1 +λ 2 )-2λ 1 λ 2 +2
equivalently:
e ijk :＝tr(A ijk ) 2 -2tr(A ijk )-2det(A ijk )+2
the following are specific steps of the discrete Ricci curvature flow algorithm:
1. by minimizing the energy equation:
initial side length and radius are respectively assigned to the sides and the vertexes of the grid, and initial values can be assigned:
2. the cosine theorem is used to calculate the Riemann metric, i.e., the side length, of the current grid:
3. all interior angles of the grid are calculated:
4. computing a discrete gaussian curvature K for each vertex i ：
5. The vertex radius gamma is updated using the following formula i ：
Wherein epsilon is the step size;
6. unitizing the radius at the mesh vertices to have a product of 1:
7. checking the current curvature K i With a target curvatureThe difference is selected, wherein the difference is the largest:
if the error is already less than the threshold value, the calculation may be stopped, otherwise the second step is called back. The whole convergence process takes a short time. Selecting a vertex v i Error curveIs an exponential curve.
(2) The two main steps of mesh dissection are point placement and cell generation, respectively. Nodes of the grid can be divided into two types of boundary nodes and internal nodes, and the following two node point distribution methods are adopted:
1. boundary node generation method
If the given area boundary is given in the form of an equation of a straight line and a curve, setting the subdivision size of the triangle unit as d, and equally dividing the area boundary by d, so that the coordinates of boundary nodes can be calculated; if the region has internal constraints, its internal feature constraints can be handled as follows: if the node is the point, directly counting the point as a boundary node into a node array; if the ring is an unsealed ring formed by straight line segments, circular arcs, curves and the like, the ring can be regarded as a directional curve, and is abstracted into a closed ring with zero area to be treated as an inner boundary ring.
2. Method for generating internal node
For the generation of internal nodes, on the basis of the traditional parallel line point distribution, a point distribution method that two groups of equidistant parallel lines are crossed to form a 60-degree angle is adopted, regular triangles can be generally formed between points distributed by the method, and the quality of grids can be greatly improved.
The concrete point distribution method is as follows:
first, find the maximum X of the abscissa and ordinate of the region max 、Y max Minimum value X min 、Y min The method comprises the steps of carrying out a first treatment on the surface of the Points are distributed in the minimum matrix containing box of the plane area, and Y=Y max And y=y min These two lines are extreme scan lines between which equidistant transverse scan lines can be generated, the number of which is:
M＝int[(Y max -Y min )/(length×sin60°)+0.5]
the equation for the scan line is:
Y＝Y i ＝Y min +(Y max -Y min )×i/M i＝0,1,2…M
the number of the points on the scanning line is as follows:
N＝int[(X max -X min )/length+0.5]
the coordinates of the point X on the scan line are:
wherein i=0, 1,2, … M; j=0, 1,2, … N; k is a natural number.
The distance between two adjacent points on the scanning line is length, the distance between the adjacent scanning lines is length multiplied by 60 degrees, and length/2 is the distance for staggering the two adjacent points longitudinally, so that the internal nodes can easily form an equilateral triangle.
After the nodes are distributed according to the method, nodes outside the outer boundary and nodes inside the inner boundary are deleted, nodes close to the boundary are deleted to ensure the quality of mesh subdivision, and the rest nodes are internal nodes.
The planar area triangular mesh is generated by adopting a Delaunay triangular subdivision method. Let S be E d The finite set of points in space, delaunay triangulation of the set of points S is defined as the simplex unit complex D (S) satisfying the following conditions:
the set of 0-simplex components of complex D (S) (i.e., the set of all vertices of D (S)) is a subset of S;
the complex bottom space is the convex hull CH (S) of the point set S;
any one d-simplex delta T E D (S), |t|=k+1, satisfying: any q.epsilon.S-T, q is delta T Is connected with the outside of the ball.
After triangular grids are generated in the two-dimensional parameter domain, the grids in the parameter domain are conformally mapped into grids in the space domain according to the mapping function from the parameter domain to the space domain, and the topological relation among nodes is kept unchanged, so that the obtained grids in the space domain are the required curved triangular grids.
Claims (1)
1. A method for generating a digital-analog surface grid based on conformal geometry is characterized by comprising the following steps: the method comprises the following steps:
judging whether the surface of the three-dimensional model can be oriented according to the input three-dimensional model data, and calculating the number of the deficiency of the curved surface and the number of the boundaries;
selecting a proper conformal mapping function according to the calculated topological structure, and conformally mapping the three-dimensional modular surface to one of a spherical surface, an European plane and a two-dimensional hyperbolic space according to a curved surface single-valued theorem;
step (3), conformally mapping the three-dimensional model to a two-dimensional parameter domain according to the selected conformally mapping function;
step (4), performing finite element mesh subdivision on a parameter domain according to a conformal mapping result, and generating a triangle unit mesh or a quadrilateral unit mesh according to requirements;
step (5), reversely mapping the grid subjected to the re-dissection back into a three-dimensional space domain according to the angle maintained in the conformal mapping process, and generating a three-dimensional model with the surface grid;
in the step (1), conformal transformation from the three-dimensional space region to the two-dimensional parameter region can keep the curved surface angle unchanged, and mesh subdivision on the planar parameter region can greatly reduce the calculation amount of interpolation;
in the step (1), the three-dimensional model is understood to be a communicated curved surface for processing, and the topological structure of the curved surface is determined by whether the curved surface can be oriented or not, and the number of ring handles, namely the number of deficiency grids and the number of boundaries;
in step (3), conformally transforming the curved surface to one of three standard spaces according to the three-dimensional model topology structure determined in step (2): sphere, european plane and two-dimensional hyperbolic space;
in step (4), the mesh subdivision has two important steps: according to a known grid topological structure, reasonably distributing grid nodes in a region to be split; effectively connecting the grid nodes to form a triangular or quadrilateral unit grid, wherein the generation of the triangular grid follows the Delaunay rule;
in the step (5), inversely mapping the grid subjected to parameter domain subdivision according to the conformal mapping function selected in the step (2), keeping the topological relation among nodes unchanged, completely keeping all geometric information of the original curved surface by the change rate of local surface elements and the average curvature of the curved surface, and completely reconstructing the original digital-analog curved surface according to the change rate and the average curvature of the surface elements;
the three-dimensional model can be regarded as a communicated curved surface, conformal parameterization of the three-dimensional model surface can be regarded as parameterization mapping of the curved surface, and the whole three-dimensional model surface grid generation process is mathematically equivalent to solving a smooth bijection from a curved surface space domain to a plane parameter domain;
the method comprises the following specific implementation steps:
(1) According to the given three-dimensional model, converting the model into an off or obj format which can be stored in a half-side data structure, firstly, carrying out boundary recognition on the model in the half-side data structure, taking the edge with only one adjacent surface as a boundary edge in the traversing process, and if a closed loop exists between the boundary edges, taking the edge as a boundary; then, performing the deficiency identification on the curved surface, taking the maximum number of mutually disjoint simple closed curves which can be drawn on the curved surface and are not divided by the curved surface as the number of the deficiency of the curved surface of the model, counting the number of boundaries and the number of the deficiency of the curved surface, and storing the number of boundaries and the number of the deficiency as the topology information of the curved surface for the next step;
for a 0-deficit surface, the conformal structure can be calculated from the harmonic mapping, assuming two 0-deficit mesh surfaces M 1 ，M 2 ,h:M 1 →M 2 Representing a mapping of degree 1 for the surface, and then minimizing the harmonic energy E (h),
the Laplace equation for h can be reduced to:
Δ PL h＝(Δ PL h 0 ,Δ PL h 1 ,Δ PL h 2 )
if h is harmonic, then delta PL The tangential component of h is 0, defining the projection operator:
wherein the method comprises the steps ofFor tensor product, I is the identity matrix if and only if h satisfies:
h is harmonic, where n is M 2 Is a normal to (2);
adding additional constraints forces the centroid of the surface to lie at the origin:
is M 1 Under this constraint, constructing a partial differential equation:
the steady state solution of h is M 1 To M 2 Is a conformal mapping of (2);
for a curved surface with a deficiency of 1, the algorithm is calculated through full pure differentiation and based on the Hodge theory, and is as follows:
the mesh surface M is input,
outputting a full pure differential basis form
Calculate the homology group b= { e 1 ,e 2 ,…,e 2g }；
Computing the homomorphic group base omega= { w on dual 1 ,w 2 ,…,w 2g }；
Each w i Dispersing to form ζ of blend 1 i ；
Calculate each ζ i Is expressed as i Constructing an all-pure 1 form
For any deficiency of surfaces, the conformal structure of the surfaces is calculated by discrete Ricci flow, and the specific steps are as follows:
assuming smooth mapping between given surfaces with metrics(S 1 ,g 1 )→(S 2 ,g 2 ) Locally, non-linear mapping ++>With its first order approximation, the linear mapping between the tangential planes approximates, i.e., leads to a mapping:
various distortions are brought about, the measurement of which requires the use of a Riemann metric for any point p.epsilon.S 1 ，/>Are equidistant, or are conformal, are of a certain area, are of a quasi-conformal shape, are of an overall +.>Equidistant, or conformal, area-keeping, quasi-conformal;
the conformal metrics are calculated and embedded as follows:
calculating the length, angle, area and differential operator on curved surface S by Riemann metric, during parameterization, riemann metric is conformally changed, and other relevant quantities except angle are correspondingly changed until metric g is changed toThen the gaussian curvature will become:
where Δ is the laplacian of the original metric g; the geodesic curvature at this time will become:
from the Gauss-Bonnet equation, the total curvature is independent of the choice of the Riemann metric, which can prove that there is a unique conformal metricSo that the target curvature is constant and the geodetic curvature is zero;
parameterization by curvature control, i.e. finding the appropriate conformal measure e 2u g, making the induced curvature consistent with the preset target curvature;
in the smooth case, the conformal deformation maps an infinitely small circle to an infinitely small circle and maintains the intersection angle between the circles, in the discrete case, a radius gamma is set at each vertex of the mesh by the introduction of the circle packing method i Different circles are intersected with each other;
in the Euclidean background, the conformal deformation is realized by changing the radius under the condition of ensuring that the intersection angle of the circles is unchangedFor parameterization purposes, the discrete Ricci curvature flow has a form similar to the smooth case Ricci flow equation:
in the spherical or hyperbolic geometric context, when using Ricci curvature flow to calculate spherical or hyperbolic metrics, the process is the same as in the euclidean case, when calculating geometric quantities, using corresponding spherical or hyperbolic geometries;
let the hyperbolic circle (C, R) be an euclidean circle (C, R) with C as the center and R as the radius, satisfy:
wherein u= (e r -1)/(e r +1)；
Next, the first face [ v ] is first applied 0 ,v 1 ,v 2 ]The parameters are respectively as follows:
surface f adjacent to the first surface ijk Embedded therein, assume vertex v i ,v j Already in the hyperbolic plane, the third vertex v k Namely v is i ,v j Is the intersection of two circles of vertices, and in the same manner, the other is equal to f ijk Adjacent surfaces are pressed into stacks and are sequentially embedded into hyperbolic planes, the embedding mode in the hyperbolic space is the same as the embedding mode in the Euclidean space, and the embedding method under spherical geometry is the same as the embedding mode in the Euclidean space and the hyperbolic space;
the direct variation method thinking is as follows: the mapping from the triangular mesh to the planar region is represented by a piecewise linear mapping,M-D, fixing a triangle, laying it on a plane, v i ,v j ,v k To represent the plane coordinates of the vertices, with +.>To represent the plane coordinates of the vertex image, the linear mapping matrix can be written directly:
if e ijk 0, then the linear mapping constrained to this triangle is a guaranty angleThe gap between the transform, the entire map and the equidistant transform is defined as:
here s ijk Representing the area of the corresponding triangle, then optimizing this energy using a nonlinear optimization method, and taking the result as an approximation of the global conformal transformation:
if the objective is to seek a guaranteed area transform, then:
e ijk :＝(λ 1 λ 2 -1) 2 ＝(det(A ijk )-1) 2
if the aim is to equally transform, the local energy is defined:
e ijk :＝(λ 1 -1) 2 +(λ 2 -1) 2 ＝(λ 1 +λ 2 ) 2 -2(λ 1 +λ 2 )-2λ 1 λ 2 +2
equivalently:
e ijk :＝tr(A ijk ) 2 -2tr(A ijk )-2det(A ijk )+2
the following are specific steps of the discrete Ricci curvature flow algorithm:
1. by minimizing the energy equation:
initial side length and radius are respectively assigned to the sides and the vertexes of the grid, and initial values can be assigned:
2. the cosine theorem is used to calculate the Riemann metric, i.e., the side length, of the current grid:
3. all interior angles of the grid are calculated:
4. computing a discrete gaussian curvature K for each vertex i ：
5. The vertex radius gamma is updated using the following formula i ：
Wherein epsilon is the step size;
6. unitizing the radius at the mesh vertices to have a product of 1:
7. checking the current curvature K i With a target curvatureThe difference is selected, wherein the difference is the largest:
when the error is less than the thresholdValue, stopping calculation, otherwise jumping back to the second step, and selecting a vertex v i Error curveIs an exponential curve;
(2) The two main steps of mesh subdivision are point distribution and cell generation respectively, and nodes of the mesh can be divided into two types of boundary nodes and internal nodes, wherein the following point distribution methods of the two nodes are as follows:
1. boundary node generation method
If the given area boundary is given in the form of an equation of a straight line and a curve, setting the subdivision size of the triangle unit as d, and equally dividing the area boundary by d, so that the coordinates of boundary nodes can be calculated; if the region has internal constraints, its internal feature constraints can be handled as follows: if the node is the point, directly counting the point as a boundary node into a node array; if the ring is an unsealed ring formed by straight line segments, circular arcs, curves and the like, the unsealed ring is regarded as a directional curve, and is abstracted into a closed ring with zero area to be treated as an inner boundary ring;
2. method for generating internal node
For the generation of internal nodes, on the basis of the traditional parallel line point distribution, a point distribution method that two groups of equidistant parallel lines are crossed to form an angle of 60 degrees is adopted;
the concrete point distribution method is as follows:
first, find the maximum X of the abscissa and ordinate of the region max 、Y max Minimum value X min 、Y min The method comprises the steps of carrying out a first treatment on the surface of the Points are distributed in the minimum matrix containing box of the plane area, and Y=Y max And y=y min These two lines are extreme scan lines between which equidistant transverse scan lines can be generated, the number of which is:
M＝int[(Y max -Y min )/(length×sin60°)+0.5]
the equation for the scan line is:
Y＝Y i ＝Y min +(Y max -Y min )×i/M i＝0,1,2…M
the number of the points on the scanning line is as follows:
N＝int[(X max -X min )/length+0.5]
the coordinates of the point X on the scan line are:
wherein i=0, 1,2, … M; j=0, 1,2, … N; k is a natural number;
the distance between two adjacent points on the scanning line is length, the distance between the adjacent scanning lines is length multiplied by 60 degrees, and length/2 is the distance for staggering the two adjacent points longitudinally, so that the internal nodes can easily form an equilateral triangle;
after the nodes are distributed according to the method, nodes outside the outer boundary and nodes inside the inner boundary are deleted, nodes close to the boundary are removed to ensure the quality of mesh subdivision, and the rest nodes are internal nodes;
the plane area triangular grid is generated by adopting Delaunay triangular subdivision method, and S is assumed to be E d The finite set of points in space, delaunay triangulation of the set of points S is defined as the simplex unit complex D (S) satisfying the following conditions:
the set of 0-simplex components of complex D (S) (i.e., the set of all vertices of D (S)) is a subset of S;
the complex bottom space is the convex hull CH (S) of the point set S;
any one d-simplex delta T E D (S), |t|=k+1, satisfying: any q.epsilon.S-T, q is delta T Is connected with the outside of the ball;
after triangular grids are generated in the two-dimensional parameter domain, the grids in the parameter domain are conformally mapped into grids in the space domain according to the mapping function from the parameter domain to the space domain, and the topological relation among nodes is kept unchanged, so that the obtained grids in the space domain are the required curved triangular grids.
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