CN202838443U - System for drawing curved surface by calculating all real roots and multiple numbers of zero-dimensional trigonometric polynomial system - Google Patents

Abstract

The utility model relates to a system for drawing a curved surface by calculating all real roots and multiple numbers of a zero-dimensional trigonometric polynomial system. The system comprises an implicit expression curved surface equation generating device, a curved surface feature point generating device, a curved surface connection relationship diagram generating device, a curved surface drawing device and a display device, wherein the curved surface feature point generating device acquires 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; the curved surface connection relationship diagram generating device acquires a connection relationship diagram through determining the connection relationship among the feature points; the curved surface drawing device draws a mesh curved surface on the basis of the connection relationship diagram directly or draw the smoother curved surface through other light naturalizing methods; and the display device displays the obtained curved surface image.

Description

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

Technical field

The utility model relates to the Computerized three-dimensional modeling technique, particularly relates to a kind of by calculating the system of zero dimension trigonometric polynomial all real roots of 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 can correctly show curved surface, can be used for also showing that the engineering on the curved surface is used application such as finite element analysis, Computerized three-dimensional modeling, computer-aided design (CAD), object three-dimensional reconstruction.

How drawing out the definite curved surface of given implicit equation is a rather popular in recent years problem.Most popular is exactly Marching Cube method, given space constantly is subdivided into less cube, until the geometry of each small cubes mean camber can both be determined (referring to LORENSEN, W.E., AND CLINE, H.E.1987.Marching cubes:a high resolution 3d surface construction algorithm.In Proc.SIGGRAPH 1987, ACM Press.) thus .Plantinga and Vegter have proposed the normal change condition provides better GRIDDING WITH WEIGHTED AVERAGE (referring to 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. proposed based on the algorithm of Morse theory (referring to 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 is introduced the geometry that an argument is determined curved surface.These algorithms all are pure numerical methods, and directly to curved surface meshing, speed still all can only be processed curve or curved surface without singular point, to the geometric object of singular point is arranged, may draw out wrong figure.

In recent years some scholar proposes the sign magnitude hybrid algorithm, with the cylindric algebra method Euclidean space is divided into the cylindricality cell first, makes given curved surface reversion not in each cell, thereby obtains a kind of new method of calculating the surface geometry structure.The basic step of these class methods all is to determine first geometry and the annexation of curved surface by decomposition space, and then with non-strange part gridding (referring to 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 method, decomposition how to carry out the space just seems particularly important, and spatial decomposition generally is by definite singular point of a surface, and the annexation of curved surface at the 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 connecting, and namely the tuple of this singular point is exactly the tuple of corresponding real root.

Therefore, find the solution zero dimension trigonometric polynomial system, namely the trigonometric polynomial system at limited zero point is a basic problem in the fields such as computational science, engineering.Find the solution zero dimension trigonometric polynomial system with the method for numerical value and generally can cause the 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 zero dimension cam system real root of isolating (referring to 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, July19-21,94-107,2005.), but their method does not have the stop technology condition, can not process repeated root.The usefulness interval algorithms such as Collins and Descartes method are calculated (referring to 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 have also proposed a kind of algorithm that calculates based on eliminant (referring to 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.

The utility model content

Based on the deficiencies in the prior art, it is a kind of by calculating the system of zero dimension trigonometric polynomial all real roots of system and tuple march iso-surface patch thereof that the utility model provides, all real roots and the tuple unique point and the tuple thereof that obtain curved surface thereof of this system by calculating zero dimension trigonometric polynomial system, thereby determine that annexation obtains curved surface annexation figure between the unique point, draw out at last accurately surface chart picture of space structure, its system as shown in Figure 1.

A kind of by calculating the system of zero dimension trigonometric polynomial all real roots of system and tuple march iso-surface patch thereof, it is characterized in that comprising:

Implicit surface equation generating apparatus;

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

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

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

Display device shows and draws the surface chart picture that obtains.

At present, many fields of object three-dimensional modeling that need all relate to the problem of how to draw the implicit surface image.The technical scheme that the utility model proposes, adopt the unique point that at first obtains curved surface by calculating zero dimension trigonometric polynomial system, then by determining that annexation obtains curved surface annexation figure between the unique point, draw at last the method for curved surface by annexation figure, avoid the pure values method to draw the engineering roadblock that has the singular point implicit surface to make mistakes, can guarantee to draw out accurately solid surface image of space structure.The utility model operation is clear and definite, and robust as a result can be used for the volume rendering of implicit surface.

Description of drawings

Fig. 1 system chart of the present utility model;

Fig. 2 the technical solution of the utility model process flow diagram;

Fig. 3 determines curved surface features point process flow diagram;

Fig. 4 calculates all real root process flow diagrams of zero dimension trigonometric polynomial system;

Fig. 5 polynomial system pretreatment process figure;

Fig. 6 estimates the upper bound process flow diagram of root;

Determine set schematic diagram in limit among Fig. 7 curved surface annexation figure;

Determine face set schematic diagram among Fig. 8 curved surface annexation figure;

Fig. 9-11 the utility model design sketch.

Embodiment

A kind of by calculating the system of zero dimension trigonometric polynomial all real roots of system and tuple march iso-surface patch thereof, at first, determine the unique point of curved surface; Then determine the connection of unique point place curved surface; At last, draw out curved surface after.Idiographic flow is referring to Fig. 2.

Lower mask body is introduced crucial realization details:

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

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.Carry out following steps:

(1) determines the zero dimension trigonometric polynomial system that unique point satisfies.

Singular point is included in zero dimension trigonometric polynomial system

∑ 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)

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

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

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

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

Real root in;

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

∑ ₁＝{f ₃(c ₁，c ₂，x ₃)} (4)

Real root in.

(2) find the solution zero dimension trigonometric polynomial system, flow process as shown in Figure 4.

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

I. carry out pre-service, by linear transformation it is transformed to local general position, flow process as shown in Figure 5;

(a) calculate the polynomial system ∑ that first polynomial expression forms in the zero dimension trigonometric polynomial system ₁={ f ₁(x ₁) all real root Zero (∑s ₁);

(b) to i from 2 to n-1, make ∑ _i={ f ₁(x ₁), f ₂(x ₁, x ₂) ..., f _i(x ₁, x ₂..., x _i), calculate ∑ _iAll real roots;

Carry out following steps:

(i) calculate penultimate equation about x _I-1Isolation circle of real root:

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

(ii) calculate last equation about x _iThe upper bound R of root _i, flow process as shown in Figure 6:

If real number row (ξ ₁, ξ ₂..., ξ _I-1) comprise the interval row [a that it and end points are rational number by one ₁, b ₁] * [a ₂, b ₂] * ... * [a _I-1, b _I-1] expression, polynomial expression is split:

$f_i(x_1,x_2,\·\·\·,x_i)={f_i}^{+}-{f_i}^{-},---\left(6\right)$

Wherein

It is the minimum positive coefficient polynomial expression of monomial item number that satisfies following formula.

Definition upper bound polynomial expression

With the many following formulas of lower bound

${f_i}^u\left(x_i\right)={f_i}^{+}(b_1,\·\·\·b_{i-1},x_i)-{f_2}^{-}(a_1,\·\·\·a_{i-1},x_i),$ (7)

${f_i}^d\left(x_i\right)={f_i}^{+}(a_1,\·\·\·a_{i-1},x_i)-{f_2}^{-}(b_1,\·\·\·b_{i-1},x_i),$

If

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 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 can calculate Rd.F then _i(ξ ₁, ξ ₂..., ξ _I-1, x _i) the upper bound of root be

R＝max{Ru，Rd}。(10)

Calculate at last 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 (∑ _I-1).(11)

(iii) with linear transformation with the system of equations ∑ _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 ∑ _iDo linear transformation

Obtain new polynomial system

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

∑ ' _iBe positioned at local general position, its all real roots have different

Coordinate is with ∑ _iReal root corresponding one by one, and tuple is identical.

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

Calculating eliminant row

Real root and ∑ _iReal root corresponding one by one, and only differ from a linear transformation.

(v) ask the monotropic first polynomial equation of calculating

Real root

(vi) by linear transformation

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

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

With monobasic real root η _iPromote back the real root (ξ of full scale equation group ₁, ξ ₂..., ξ _i).

(c) make i=n, executable operations (b) (i) (ii), calculates r _N-1And R _n

(d) to ∑ _nDo linear transformation

Obtain new polynomial system

∑ ' _nBe positioned at local general position, its all real roots have different

Coordinate is with ∑ _nReal root corresponding one by one, and tuple is identical.

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

Calculating eliminant row

Real root and ∑ _nReal root corresponding one by one, only differ from a linear transformation, and tuple is identical.

III. ask and calculate monotropic first polynomial equation

All real roots

And tuple.

IV. by linear transformation

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

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

With monobasic real root η _nPromote back the real root (ξ of full scale equation group ₁, ξ ₂..., ξ _n).

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

Zero (∑ _n)={ (ξ ₁, ξ ₂..., ξ _n) ∈ R ⁿ| f ₁(ξ ₁)=...=f _n(ξ ₁..., ξ _n)=0} (20) thus the computation process of finishing.

2. generate curved surface annexation figure.

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

Point is integrated into the first step and obtains.

The limit set comprises the annexation of putting on independent singular point and the line of singularity, the annexation of generic point on point and the curved surface on the line of singularity.To an independent singular point, the number of the line of singularity that is connected with him can be determined by the number of calculating the point on his line of singularity; To the point on the line of singularity, with the number of point on the curved surface that he is connected, can determine by calculating on his curved surface number of generic point.As shown in Figure 7.

Face set is the connection of limit and curved surface in the set of limit.Can be connected to the same number of the point of two end points on one side by calculating determines.As shown in Figure 8.

3. drafting curved surface.

The face set that obtains in the step 2 has been a triangle gridding of curved surface, we can directly draw grid surface with him, perhaps draw more smooth curved surface by other SmoothNumerical TechniqueandIts methods on this basis, such as MATLAB commonly used, OPENGL, MATH, the software for drawing platforms such as MAYA all can carry out this 3 D rendering.Effect such as Fig. 9, Figure 10, shown in Figure 11, with respect to existing 3 D rendering technology, the technical solution of the utility model has been drawn out more accurately stereoeffect, has solved the singular point place and the adjacent domain network is drawn inaccurate technical matters.

Claims (1)

1. one kind by calculating the system of zero dimension trigonometric polynomial all real roots of system and tuple march iso-surface patch thereof, it is characterized in that comprising:

Implicit surface equation generating apparatus;

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

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

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

Display device shows and draws the surface chart picture that obtains.

Images