CN102609987B - Method and system for drawing curved surface by calculating all real roots and multiple numbers of zero-dimensional trigonometric polynomial system - Google Patents

Abstract

The invention relates to a method and system for drawing a curved surface by calculating all real roots and multiple numbers of a zero-dimensional trigonometric polynomial system. The method comprises the following steps of: determining feature points of the curved surface, and obtaining the feature points and multiple numbers of the curved surface by calculating all real roots and multiple numbers of the zero-dimensional trigonometric polynomial system; generating a connection relationship diagram of the curved surface, and obtaining the connection relationship diagram by determining the connection relationship among the feature points; and drawing the curved surface, and directly drawing a mesh surface on the basis of the connection relationship diagram or drawing the smoother curved surface through other light naturalizing methods.

Description

A kind of method and system by calculating the zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof

Technical field

The present invention relates to Computerized three-dimensional modeling technique, particularly relating to a kind of method and system by calculating the zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof.

Background technology

Determine geometry and the annexation of given algebraic surface, and be an important topic of computer graphics and Geometric Modeling with grid approximate representation curved surface.Curved surface meshing correctly can show curved surface, also can be used for showing the engineer applied on curved surface, the application such as such as finite element analysis, Computerized three-dimensional modeling, computer-aided design (CAD), object three-dimensional reconstruction.

How drawing out the curved surface that given implicit equation determines is a rather popular in recent years problem.Most popular is exactly Marching Cube method, given space is constantly subdivided into less cube, until the geometry of each small cubes mean camber can both be determined (see LORENSEN, W.E., AND CLINE, H.E.1987.Marching cubes:a high resolution 3d surface construction algorithm.In Proc.SIGGRAPH 1987, ACM Press.) .Plantinga and Vegter propose normal change condition thus provide better GRIDDING WITH WEIGHTED AVERAGE (see PLANTINGA, S., AND VEGTER, G.2004.Isotopic implicit surface meshing.In Proc.Symp.on Geometry Processing, 251-260.PLANTINGA, S., AND VEGTER, G.2007.Isotopic meshing of implicit surfaces.The Visual Computer 23, 45-58.) .Hart etc. propose algorithm based on Morse theory (see HART, J.C.1998.Morse theory for implicit surface modeling.In Mathematical Visualization, Springer-verlag, H.Hege and K.Polthier, Eds., 257-268.), this algorithm introduces an argument to determine the geometry of curved surface.These algorithms are all pure numerical methods, and directly to curved surface meshing, speed, but the curve that all can only process without singular point or curved surface, to the geometric object having singular point, may draw out the figure of mistake.

In recent years some scholar proposes sign magnitude hybrid algorithm, first by cylindric algebra method, Euclidean space is divided into cylindricality cell, makes given curved surface not reversion in each cell, thus obtains a kind of new method calculating surface geometry structure.The basic step of these class methods is all first by geometry and the annexation of decomposition space determination curved surface, and then by non-strange Partial Mesh (see D.S.Arnon, G.Collins, and S.Mccallum, 1988.Cylindrical algebraic decomposition, iii:an adjacency algorithm for three-dimensional space.J.Symbolic Comput.5, 1, 2, 163-187.J.S.Cheng, X.S.Gao, and M.Li, 2005.Determine the topology of real algebraic surfaces.In Mathematics of Surfaces, Springer-Verlag, 121-146.S.Mccallum, and, G.E.Collins, 2002.Local box adjacency algorithms for cylindrical algebraic decompositions.J.Symbolic Comput.33, 321-342.B.Mourrain, and J.P.Tecourt, 2005.Computing the topology of real algebraic surfaces.In MEGA electronic proceedings.).In this approach, the decomposition how carrying out space just seems particularly important, and spatial decomposition is generally by determining singular point of a surface, and the annexation of curved surface at singular point place obtains.Singular point of a surface is the real root of a zero dimension trigonometric polynomial system, the branch amount that singular point is connected to, the i.e. tuple of this singular point, is exactly corresponding real multiplicity of a root.

Therefore, solving zero dimension trigonometric polynomial system, i.e. the trigonometric polynomial system at limited zero point, is a basic problem in the field such as computational science, engineering.Solving zero dimension trigonometric polynomial system by the method for numerical value generally can cause result incorrect because of the error of numeric representation, and can not calculate tuple.The people such as Lu propose a kind of method of isolating zero dimension cam system real root (see Z.Lu, B.He, Y.Luo and L.Pan, An algorithm of real root isolation for polynomial systems, in Proceedings of International Workshop on Symbolic-Numeric Computation, Xi ' an, China, Julyl9-21,94-107,2005.), but their method does not have stop technology condition, can not process repeated root.Interval algorithm and the Descartes method such as Collins calculate (see G.E.Collins, J.R.Johnson, and W.Krandick, Interval arithmetic in cylindrical algebraic decom-position, J.Symb.Comput., 34:145-157,2002.).Xia and Yang it is also proposed a kind of algorithm calculated based on eliminant (see B.Xia and L.Yang, An algorithm for isolating the real solutions of semi-algebraic systems, J.Symb.Comput., 34:461-477,2002.), but their method can lose efficacy in some cases.

Summary of the invention

Based on the deficiencies in the prior art, the invention provides a kind of method and system by calculating the zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof, the method obtains unique point and the tuple thereof of curved surface by all real roots and tuple thereof calculating zero dimension trigonometric polynomial system, thus determine that between unique point, annexation obtains curved surface annexation figure, finally draw out space structure surface chart picture accurately.

By calculating a method for the zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof, comprise the following steps:

1) unique point of curved surface is determined;

Unique point comprises the independent singular point of curved surface, the point on the line of singularity, the non critical point contiguous with singular point.Perform following steps:

(1) the zero dimension trigonometric polynomial system that unique point meets is determined;

(2) zero dimension trigonometric polynomial system is solved; Comprise the following steps:

I. pre-service is carried out to polynomial system, is transformed to local general position by linear transformation, comprise the following steps:

A () calculates isolation circle of real root;

B () estimates the upper bound of root, perform following steps:

I polynomial expression is split into the minimum positive coefficient polynomial expression sum of two monomial item numbers by ();

(ii) upper bound polynomial expression and the many following formulas of lower bound is calculated;

(iii) upper bound of upper bound polynomial expression and lower bound root of polynomial is calculated;

(iv) maximal value in the upper bound of upper bound polynomial expression and lower bound root of polynomial is got.

C () determines linear transformation, polynomial system is transformed to local general position;

II. project in the one-dimensional space by all real roots, projecting by calculating eliminant, obtaining a monotropic first equation, this projection keeps multiplicity of a root constant;

III. all real roots and tuple thereof that calculate monotropic first polynomial equation is asked.

IV. by linear transformation, unitary real root is promoted back the real root of full scale equation group.

2) curved surface annexation Fig. 1 is generated

The annexation figure of curved surface comprises three parts: unique point (some set), the annexation (limit set) between unique point, the annexation (face is gathered) on the triangular plate that unique point is formed and limit.

3) curved surface is drawn.

Step 2) in the face set that obtains be a triangle gridding of curved surface, we directly can draw grid surface with him, or on this basis by curved surface that the drafting of other SmoothNumerical TechniqueandIts methods is more smooth.

Correspondingly, the present invention proposes and devise a kind of system by calculating the zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof.

At present, many fields of object three-dimensional modeling that needs all relate to the problem of how to draw implicit surface image.The technical scheme that the present invention proposes, adopt the unique point first obtaining curved surface by calculating zero dimension trigonometric polynomial system, then by determining that between unique point, annexation obtains curved surface annexation figure, the method of curved surface is drawn finally by annexation figure, avoid pure values method and draw the engineering roadblock having singular point implicit surface to make mistakes, can ensure to draw out space structure solid surface image accurately.The inventive method is clear and definite, and result robust, may be used for the volume rendering of implicit surface.

Accompanying drawing explanation

Fig. 1 technical scheme process flow diagram of the present invention;

Fig. 2 determination curved surface features point process flow diagram;

Fig. 3 calculates all real root process flow diagrams of zero dimension trigonometric polynomial system;

Fig. 4 polynomial system pretreatment process figure;

Fig. 5 estimates the upper bound process flow diagram of root;

Limit set schematic diagram is determined in Fig. 6 curved surface annexation figure;

Face set schematic diagram is determined in Fig. 7 curved surface annexation figure;

Fig. 8-10 3 D rendering design sketch;

Figure 11 system chart of the present invention.

Embodiment

By calculating a method and system for the zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof, first, determine the unique point of curved surface; Then the connection of unique point place curved surface is determined; Finally, curved surface is drawn out after.Idiographic flow is see Fig. 1.

Lower mask body introduction key realize details:

1. determine curved surface f (x ₁, x ₂, x ₃the unique point of)=0, flow process as shown in Figure 2.

Unique point comprises the independent singular point of curved surface, the point on the line of singularity, the non critical point contiguous with singular point.Perform following steps:

(1) the zero dimension trigonometric polynomial system that unique point meets is determined.

Singular point is included in zero dimension trigonometric polynomial system

Σ ₃＝{f ₁(x ₁)，f ₂(x ₁，x ₂)，f ₃(x ₁，x ₂，x ₃)} (1)

Real root in, wherein

f ₃(x ₁，x ₂，x ₃)＝f(x ₁，x ₂，x ₃)

$f_2(x_1,x_2)=\mathrm{SquareFreePart}\left({\mathrm{Resul}\tan t}_{x3}(f_3,\frac{\∂f_3}{\∂x_3})\right);---\left(2\right)$

$f_1\left(x_1\right)=\mathrm{SquareFreePart}\left({\mathrm{Resul}\tan t}_{x2}(f_2,\frac{\∂f_2}{\∂x_2})\right)$

Point on the line of singularity is included in zero dimension cam system

Σ ₂＝{f ₂(c ₁，x ₂)，f ₃(c ₁，x ₂，x ₃)} (3)

Real root in;

The non critical point of singular point vicinity is included in zero dimension cam system

Σ ₁＝{f ₃(c ₁，c ₂，x ₃)} (4)

Real root in.

(2) solve zero dimension trigonometric polynomial system, flow process as shown in Figure 3.

In the process calculating real root, we comprise this real root with one all the time, and end points is that approximate representation real root is carried out in the interval of rational number, and the calculating wherein relating to real root all uses interval to calculate.To solution zero dimension trigonometric polynomial system Σ _n={ f ₁(x ₁), f ₂(x ₁, x ₂) ..., f _n(x ₁, x ₂..., x _n), perform following steps:

I. carry out pre-service, transformed to local general position by linear transformation, flow process as shown in Figure 4;

A () calculates the polynomial system Σ of first polynomial expression composition in zero dimension trigonometric polynomial system ₁={ f ₁(x ₁) all real root Zero (Σ ₁);

B () from 2 to n-1, makes Σ to i _i={ f ₁(x ₁), f ₂(x ₁, x ₂) ..., f _i(x ₁, x ₂..., x _i), calculate Σ _iall real roots;

Perform following steps:

I () calculates most penultimate equation about x _i-1isolation circle of real root:

$r_{i-1}=\frac{1}{2}\underset{({\mathrm{\ξ}}_1,{\mathrm{\ξ}}_2,...,{\mathrm{\ξ}}_{i-2})\∈\mathrm{Zero}\left({\mathrm{\Σ}}_{i-2}\right)}{\min }\left\{\right|{\mathrm{\θ}}_1-{\mathrm{\θ}}_2|:{\mathrm{\θ}}_1,{\mathrm{\θ}}_2\∈\mathrm{Zero}\left(f_{i-1}({\mathrm{\ξ}}_1,{\mathrm{\ξ}}_2,...,{\mathrm{\ξ}}_{i-2},x_{i-1})\right),{\mathrm{\θ}}_1\≠{\mathrm{\θ}}_2\}.---\left(5\right)$

(ii) last equation is calculated about x _ithe upper bound R of root _i, flow process as shown in Figure 5:

If real number row (ξ ₁, ξ ₂..., ξ _i-1) comprise it by one and end points is the interval row [a of rational number ₁, b ₁] × [a ₂, b ₂] × ... × [a _i-1, b _i-1] represent, polynomial expression is split:

f _i(x ₁，x ₂，…，x _i)＝f _i ⁺-f _i ^-， (6)

Wherein f _i ⁺, f _i ^-∈ Z ⁺[x ₁, x ₂..., x _i], be meet the minimum positive coefficient polynomial expression of the monomial item number of above formula.

Definition upper bound polynomial f _i ^uwith lower bound polynomial f _i ^d(x _i):

f _i ^u(x _i)＝f _i ⁺(b ₁，…，b _i-1，x _i)-f _i ^-(a ₁，…，a _i-1，x _i)，

(7)

f _i ^d(x _i)＝f _i ⁺(a ₁，…，a _i-1，x _i)-f _i ^-(b ₁，…，b _i-1，x _i)，

If f _i ^u(x _i) shape is

${f_i}^u\left(x_i\right)=c_d{x}_i^d+c_{d-1}{x}_i^{d-1}+...+c_0,---\left(8\right)$

Calculate f _i ^u(x _i) the upper bound of root

$\mathrm{Ru}=1+\underset{0\≤k\≤d-1}{\max }\left|\frac{c_k}{c_d}\right|,---\left(9\right)$

Similarly Rd can be calculated.Then f _i(ξ ₁, ξ ₂..., ξ _i-1, x _i) the upper bound of root be

R＝max{Ru，Rd}。(10)

Finally calculate f _i(x ₁, x ₂..., x _i) about x _ithe upper bound of root

R _i=max{R:R is f _i(ξ ₁, ξ ₂..., ξ _i-1, x _i) upper bound of root, (ξ ₁, ξ ₂..., ξ _i-1) ∈ Zero (Σ _i-1).(11)

(iii) with linear transformation by system of equations Σ _i={ f ₁(x ₁), f ₂(x ₁, x ₂) ..., f _i(x ₁, x ₂..., x _i) transform to local general position:

To Σ _ido linear transformation

Obtain new polynomial system

${\mathrm{\Σ}}_i^{\′}=\{f_1(X_i^i-\frac{r_1}{R_2}{X}_{i-1}^i),...,f_i(X_i^i-\frac{r_1}{R_2}{X}_{i-1}^i,...,X_2^i-\frac{r_{i-1}}{R_i}{X}_1^i,X_1^i)\},---\left(13\right)$

Σ ' _ibe positioned at local general position, its all real roots have different coordinate, with Σ _ireal root one_to_one corresponding, and tuple is identical.

(iv) all real roots are projected in the one-dimensional space, obtain a monotropic first equation:

Calculating eliminant arranges

real root and Σ _ireal root one_to_one corresponding, and only differ from a linear transformation.

V () is asked and is calculated monotropic first polynomial equation real root

(vi) linear transformation is passed through

${\mathrm{\ξ}}_1=\left(\underset{j=1}{\overset{i-1}{\mathrm{\Π}}}\frac{R_{j+1}}{r_j}\right)({\mathrm{\η}}_i-{\mathrm{\η}}_{i-1}),---\left(15\right)$

η _i∈Zero(T _i ⁱ)， ${\mathrm{\&eta;}}_{i-1}\&Element;\mathrm{Zero}\left(T_{i-1}^{i-1}\right),$ $|{\mathrm{\&eta;}}_i-{\mathrm{\&eta;}}_{i-1}|<\left(\underset{j=1}{\overset{i-2}{\mathrm{\&Pi;}}}\frac{r_j}{R_{j+1}}\right)r_{i-1}$

By unitary real root η _ipromote back the real root (ξ of full scale equation group ₁, ξ ₂..., ξ ₁).

C () makes i=n, executable operations (b) (i), and (ii), calculates r _n-1and R _n;

D () is to Σ _ndo linear transformation

Obtain new polynomial system

Σ ' _nbe positioned at local general position, its all real roots have different coordinate, with Σ _nreal root one_to_one corresponding, and tuple is identical.

II. all real roots are projected in the one-dimensional space, obtain a monotropic first equation;

Calculating eliminant arranges

real root and Σ _nreal root one_to_one corresponding, only differ from a linear transformation, and tuple is identical.

III. ask and calculate monotropic first polynomial equation all real roots and tuple.

IV. linear transformation is passed through

${\mathrm{\&xi;}}_n=\left(\underset{j=1}{\overset{n-1}{\mathrm{\&Pi;}}}\frac{R_{j+1}}{r_j}\right)({\mathrm{\&eta;}}_n-{\mathrm{\&eta;}}_{n-1}),---\left(19\right)$

${\mathrm{\&eta;}}_n\&Element;\mathrm{Zero}\left(T_n^n\right),$ ${\mathrm{\&eta;}}_{n-1}\&Element;\mathrm{Zero}\left(T_{n-1}^{n-1}\right),$ $|{\mathrm{\&eta;}}_n-{\mathrm{\&eta;}}_{n-1}|<\left(\underset{j=1}{\overset{n-2}{\mathrm{\&Pi;}}}\frac{r_j}{R_{j+1}}\right)r_{n-1}$

By unitary real root η _npromote back the real root (ξ of full scale equation group ₁, ξ ₂..., ξ _n).

So just obtain Σ _n={ f ₁(x ₁), f ₂(x ₁, x ₂) ..., f _n(x ₁, x ₂..., x _n) all real roots:

Zero(Σ _n)＝{(ξ ₁，ξ ₂，…，ξ _n)∈R ⁿ|f ₁(ξ ₁)＝…＝f _n(ξ ₁，…，ξ _n)＝0} (20)

Thus complete computation process.

2. generate curved surface annexation figure.

The annexation figure of curved surface comprises three parts: unique point (some set), the annexation (limit set) between unique point, the annexation (face is gathered) on the triangular plate that unique point is formed and limit.

Point is integrated into the first step and obtains.

Limit set comprises the annexation that independent singular point and the line of singularity are put, point and the general annexation put on curved surface on the line of singularity.To an independent singular point, the number of the line of singularity be connected with him, can be determined by the number calculating the point on his line of singularity adjacent; To the point on a line of singularity, the number of the point on the curved surface be connected with him, can be determined by the number calculating general point on his curved surface adjacent.As shown in Figure 6.

Face set is the connection of limit and curved surface in the set of limit.Can determine by calculating the number be connected to the point of two end points in one side.As shown in Figure 7.

3. draw curved surface.

The face set obtained in step 2 has been a triangle gridding of curved surface, we directly can draw grid surface with him, or draw more smooth curved surface by other SmoothNumerical TechniqueandIts methods on this basis, such as conventional MATLAB, OPENGL, the software for drawing platforms such as MATH, MAYA all can carry out this 3 D rendering.Effect is as shown in Fig. 8, Fig. 9, Figure 10, and relative to existing 3 D rendering technology, technical scheme of the present invention delineates stereoeffect more accurately, solves singular point place and adjacent domain network draws inaccurate technical matters.

Correspondingly, the present invention have also been devised the system based on the method above by the calculating zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof, and as shown in figure 11, native system comprises:

Implicit surface equation generating apparatus;

Curved surface features dot generation device, it obtains unique point and the tuple thereof of curved surface by the calculating zero dimension all real roots of trigonometric polynomial system and tuple thereof;

Curved surface connection layout generating apparatus, it is by determining that the annexation between unique point obtains annexation figure;

Surface-rendering device, it directly draws grid surface or draws more smooth curved surface by other SmoothNumerical TechniqueandIts methods on the basis of annexation figure;

Display device, the surface chart picture obtained is drawn in display.

Claims (5)

1., by calculating a method for the zero dimension all real roots of trigonometric polynomial system and tuple march iso-surface patch thereof, comprise the following steps:

1) unique point of curved surface is determined; Unique point comprises the independent singular point of curved surface, the point on the line of singularity, the non critical point contiguous with singular point; It comprises the steps:

(1) the zero dimension trigonometric polynomial system that unique point meets is determined;

(2) all real roots and the tuple thereof of zero dimension trigonometric polynomial system is calculated; It comprises the following steps:

I. to zero dimension trigonometric polynomial system Σ _n={ f ₁(x ₁), f ₂(x ₁, x ₂) ..., f _n(x ₁, x ₂..., x _n) carry out pre-service, transformed to local general position by linear transformation;

II. project in the one-dimensional space by all real roots, projecting by calculating eliminant, obtaining a monotropic first equation, this projection keeps multiplicity of a root constant;

III. all real roots and the tuple thereof of monotropic first polynomial equation is calculated;

IV. by linear transformation, unitary real root is promoted back the real root of full scale equation group;

2) curved surface annexation figure is generated

The annexation figure of curved surface comprises three parts: some set, i.e. step 1) determine the unique point set of curved surface; Limit is gathered, the annexation namely between unique point; Face is gathered, i.e. the triangular plate of unique point formation and the annexation on limit;

3) curved surface is drawn

Step 2) in the face set that obtains be a triangle gridding of curved surface, directly draw grid surface with this, or draw more smooth curved surface by other SmoothNumerical TechniqueandIts methods on this basis.

2. method according to claim 1, wherein, comprises the pre-service of zero dimension trigonometric polynomial system:

1) the isolation circle r of real root is calculated _i, i=1,2 ... n-1;

2) upper bound R of root is estimated _i, i=2,3 ... n;

3) determine linear transformation, polynomial system is transformed to local general position.

3. method according to claim 2, wherein, estimate that the upper bound of root comprises:

If real number row (ξ ₁, ξ ₂..., ξ _i-1) comprise it by one and end points is the interval row [a of rational number ₁, b ₁] × [a ₂, b ₂] × ... × [a _i-1, b _i-1] represent, polynomial expression is split:

f _i(x ₁，x ₂，…，x _i)＝f _i ⁺-f _i ^-，

Wherein f _i ⁺, f _i ^-∈ Z ⁺[x ₁, x ₂..., x _i], be meet the minimum positive coefficient polynomial expression of the monomial item number of above formula;

Definition upper bound polynomial f _i ^uwith lower bound polynomial f _i ^d(x _i):

f _i ^u(x _i)＝f _i ⁺(b ₁，…，b _i-1，x _i)-f _i ^-(a ₁，…，a _i-1，x _i)，

f _i ^d(x _i)＝f _i ⁺(a ₁，…，a _i-1，x _i)-f _i ^-(b ₁，…，b _i-1，x _i)，

If f _i ^u(x _i) shape is

${f_i}^u\left(x_i\right)=c_d{x}_i^d+c_{d-1}{x}_i^{d-1}+...+c_0,$

Calculate f _i ^u(x _i) the upper bound of root

$\mathrm{Ru}=1+\underset{0\&le;k\&le;d-1}{\max }\left|\frac{c_k}{c_d}\right|,$

Fortune uses the same method and calculates the lower bound Rd of root; Then f _i(ξ ₁, ξ ₂..., ξ _i-1, x _i) the upper bound of root be

R＝max{Ru，Rd}

Finally calculate f _i(x ₁, x ₂..., x _i) about x _ithe upper bound of root

R _i=max{R:R is f _i(ξ ₁, ξ ₂..., ξ _i-1, x _i) upper bound of root, (ξ ₁, ξ ₂..., ξ _i-1) ∈ Zero (Σ _i-1).

4. method according to claim 2, wherein determines linear transformation, polynomial system is transformed to local general position and comprises:

To Σ _ndo linear transformation

Obtain new polynomial system

Σ ' _nbe positioned at local general position, its all real roots have different coordinate, with Σ _nreal root one_to_one corresponding, and tuple is identical.

5. method according to claim 1, wherein, is comprised the real root that unitary real root promotes back full scale equation group by linear transformation:

Pass through linear transformation

${\mathrm{\&xi;}}_1={\mathrm{\&eta;}}_1,{\mathrm{\&xi;}}_i=\left(\underset{j=1}{\overset{i-1}{\mathrm{\&Pi;}}}\frac{R_{j+1}}{r_j}\right)({\mathrm{\&eta;}}_i-{\mathrm{\&eta;}}_{i-1}),$

η _i∈Zero(T _i ⁱ)， ${\mathrm{\&eta;}}_{i-1}\&Element;\mathrm{Zero}\left(T_{i-1}^{i-1}\right),$ $|{\mathrm{\&eta;}}_i-{\mathrm{\&eta;}}_{i-1}|<\left(\underset{j=1}{\overset{i-2}{\mathrm{\&Pi;}}}\frac{r_j}{R_{j+1}}\right)r_{i-1},$ i＝2，3，…，n

By unitary real root η ₁, η ₂..., η _npromote back the real root (ξ of full scale equation group ₁, ξ ₂..., ξ _n); So just obtain Σ _n={ f ₁(x ₁), f ₂(x ₁, x ₂) ..., f _n(x ₁, x ₂..., x _n) all real roots:

Zero(Σ _n)＝{(ξ ₁，ξ ₂，…，ξ _n)∈R ⁿ|f ₁(ξ ₁)＝…＝f _n(ξ ₁，…，ξ _n)＝0}。