CN101149839A - Recurrence surface construction method based on triangle domain L curved surface and w curved surface - Google Patents

Abstract

The recursive curve-constitution method based on L curve and W curve on triangular field controls the non-stable of a computer and increases the reality and real-time property of a picture contour through the curve design of L curve and W curve based on triangular field. The basic idea is that: determine L curve in triangular field to strengthen restriction conditions to determine W curve, then constitute the curve constitution arithmetic in CAD system geometry storeroom through W curve to realize the curve splicing and smooth figuring based on triangular field.

Description

Recursive surface construction method based on L surface and W surface on triangular domain

Technical Field

The invention belongs to the technical field of computer aided design and manufacturing, and particularly relates to an implementation method of computer aided geometric design on a curved surface curve structure.

Background

Computer aided geometric design (CGAD) is a new discipline closely related to production practices and continuously developed along with the needs of the production practices, and the application range of the CGAD not only relates to three major manufacturing industries of aviation, shipbuilding and automobiles, but also relates to the technical fields of architectural design, bioengineering, medical diagnosis, aerospace materials, electronic engineering, robots, design of models of clothes, shoes and hats, and the like. With the development of computer graphics, the method is also widely applied to the fields of computer vision, topography and landform, military combat simulation, animation manufacturing, multimedia technology and the like. Therefore, as the requirements of computer graphic display on reality, instantaneity and interactivity are increasingly enhanced, as the trend that geometric design objects are closer to diversity, specificity and topological structure complexity is increasingly obvious, the requirements on appearance design are higher, and the curved surface modeling technology is greatly developed.

In profile design, a curvilinear surface cannot be described by a quadratic equation, often only a rough shape is given or only the fact that the curvilinear surface passes through a few spatial point columns is known, and a special method is needed for construction, and for a long time, parametric curvilinear surface representation is always the main tool for describing the geometric shape. The existing CAGD adopts a curve surface in a recursion form, is more suitable for being carried out in a computer compared with an initially defined formula of the curve surface, and can conveniently calculate and generate a required graph in actual modeling so that the instability of calculation is controlled. Both the conventional recursive L surface and the conventional recursive W surface are based on rectangular regions. And the effect of the L-shaped curved surface and the W-shaped curved surface developed on the basis of the triangular area on the surface modeling design is more efficient due to the subdivision property of the triangular area compared with the rectangular area. The invention provides a design of an L curved surface and a W curved surface based on a recursion form on a triangular region in curved surface modeling.

Disclosure of Invention

The invention aims to improve the realization effect of the prior art and provides a method for designing a recursive curved surface structure. The instability of a computer is controlled through the surface design based on the L surface and the W surface of the triangular domain, and the reality and the real-time performance of the figure appearance are improved.

In order to realize the purpose of the invention, the technical scheme is as follows:

the invention provides a method applied to recursive curved surface construction, which is characterized in that L curved surface construction and W curved surface construction of a triangular domain are emphasized, and Bernstein-Bezier curved surfaces are taken as examples in combination with other curved surface construction methods, so that the construction of the recursive curved surface is obtained.

1) Firstly, defining a general recursive form surface representation method on a triangular parameter domain: defining the symbols: j is a unit of ₀ ＝(1，0，0)，J ₁ ＝(0，1，0)，J ₂ ＝(0，0，1)，

Let { P _a | a | = n } is the vertex of the feature net on the triangular domain,

L _a ^T (x)，M _a ^T (x)，N _a ^T (x) For the deployment function at the corresponding node on the triangular domain, the specific requirements are as follows:

L _a ^T (x) Should satisfy 1. Ltoreq. A ₀ ≤n，0≤a ₁ ，a ₂ ≤n；

M _a ^T (x) Should satisfy 1. Ltoreq. A ₀ ≤n，0≤a ₁ ，a ₂ ≤n；

N _a ^T (x) Should satisfy 1. Ltoreq. A ₀ ≤n，0≤a ₁ ，a ₂ ≤n；

And isr＝1，Λ，n，|a|＝n-r。

The method for defining the general recursive curved surface on the triangular parameter domain comprises the following steps:

wherein

If a _i -1 < 0 or > 0, then

i＝0，1，2；

2) Defining an n-th order L surface over a triangular field

Setting:

(and satisfy

r＝1，2，Λ n，|a|＝n-r)

Defining the n-th order L surface on the triangular domain as follows according to 1): the n recursion curves must satisfy

Then is an n-order L-shaped surface on the triangular domain;

3) Defining W surface

If the n-th order L-shaped surface is continuously satisfied

r＝1，2，Λn；|a|＝n-r

Then is a W curved surface of n times;

4) Conversion to Bernstein-Bezier curved surface

Order:

then, get the Bernstein-Bezier curved surface

Wherein

(i+j+k＝n，u+v+w＝1)，P _ijk Is a node of the triangular domain network;

5) Bezier curved surface continuous splicing on triangular domain

If we give two pieces of Bezier surface n times in the triangular domain:

then the condition that Q (u, v, w) and P (u, v, w) achieve geometric continuity on the splice line is:

and then sequentially connecting all the divided triangular domain curved surfaces to obtain the required curved surface structure.

The invention has the advantages that:

1) The L curved surface and the W curved surface based on the triangular domain are fully utilized to construct the recursive curved surface. Different curves can be obtained by making specific limiting conditions on the blending polynomial in the L \ W curve. Compared with the initially defined formula of the curve, the method is more suitable for being carried out in a computer, and the required graph can be conveniently calculated and generated in the actual modeling.

2) The definition of the L-shaped surface based on the triangular domain is realized. The L curved surface is defined, namely the W curved surface can be deduced on the basis of the L curved surface, so that a foundation is provided for obtaining a Bezier curved surface or a curved surface construction algorithm in other CAD system geometric libraries. For two-dimensional and three-dimensional graphs, the conversion of geometric elements (straight lines, quadratic curves, spline curves, rotating surfaces, ruled surfaces and the like) in different CAD system geometric libraries is required to be realized, and a uniform data storage format of the geometric elements of the graphs is established.

3) The definition of the W surface based on the triangular domain is realized. On the basis of the defined W surface, the conversion from the W surface to the Bezier surface (or the surface construction algorithm in other CAD system geometry libraries) can be conveniently carried out, so that the Bezier surface can be obtained, and the realization data is smoothed. And then, the curved surface modeling is constructed, so that the reality of the figure appearance is improved.

Drawings

FIG. 1 is a schematic diagram of a curved surface in a triangular parameter domain;

FIG. 2 is a basic flow diagram of a recursive surface construction method;

FIG. 3 is a basic flow chart of a triangular domain L surface construction method;

FIG. 4 is a basic flow diagram of the construction of L surface parameters in the triangle domain.

Detailed Description

The invention is further elucidated with reference to the drawing.

The basic construction process of the recursive surface construction method based on W surface and L surface on triangle domain of the present invention is shown in fig. 2, and the following specific embodiments of the present invention are described according to specific examples.

The first is to construct a triangle field with size n, i.e. construct a triangle net with size n. The method of construction is shown in figure 1. A triangular parameter domain of 3 vertices with n =3 as shown in fig. 1 (a), and (b) a triangular control point array representing 10 nodes in this triangular parameter domain.

And then constructing a triangular domain curved surface according to the data:

first is the L-surface that constructs the triangular domain. The construction flow of the L-shaped curved surface is shown in FIG. 3: FIG. 3 The purpose of steps 100 to 101 is to define P from the vertices of the feature network over the triangular domain _a ： {P _a I a | = n }, where n is a defined triangular mesh of size n, a is composed of one ternary variable,

denotes the node position of the node, as indicated by the subscript shown in fig. 1 (b), | a | = a ₀ +a ₁ +a ₂ ＝n。P _a I.e., the node represented as the triangular domain, such as the node shown in fig. 1 (b).

Step 102 is according to the formula:

defining an initial iteration condition d _a ⁰ (x) In that respect The purpose of step 103 is to construct the deployment function L of the corresponding nodes of the triangular domain _a ^T (x)，M _a ^T (x)，N _a ^T (x) I.e. the coefficients of the triangular surface configuration. The specific coefficient construction flow is shown in fig. 4. Steps 104 to 107 are iterative processes that are based on a formula

And the value of the meter m is taken from 1, and the iteration is finished for n times.

Wherein the specific construction flow of step 103 is shown in fig. 4. Firstly construct L _a ^T (x) Let us order

Wherein a and b are L _a ^T (x) The construction factor of (1). Inputting n and J obtained in step 101 in FIG. 3 ₀ ，J ₀ Defined as a vector: j. the design is a square ₀ = (1,0,0). Step 200 sets counter r to 1, let | a | = n-r. Determining the first qualified coefficients a, b, i.e. the condition is fulfilled, is performed in step 201

r =1,2, Λ n; | a | = n-r to determine a, b satisfied by the first round. Then the counter r is incremented by 1 until r = n, resulting in a, b satisfying the n set of equations. Step 205, according to the formula

To obtain L _a ^T (x) In that respect Inputting n and J in the same flow ₁ ，J ₁ Defined as a vector: j. the design is a square ₁ = (0,1,0) construction M _a ^T (x) Inputting n and J ₂ ，J ₂ Defined as a vector: j is a unit of ₁ = (0,0,1) construct N _a ^T (x) In that respect Thus, three fitting functions for constructing the L-shaped curved surface are obtained.

Then, determining the deployment function of the W curved surface on the basis of the L curved surface. W curved surface satisfies the formular =1,2, Λ n; a | = n-r andr =1, Λ, n, | a | = n-r. In the L-shaped curved surface structureL of _a ^T (x)，M _a ^T (x)，N _a ^T (x) In the above-mentioned conditions, L after three strengthening conditions are obtained _a ^T (x)，M _a ^T (x)，N _a ^T (x) As a result of the construction factor is step 102 in fig. 3.

Then, according to the well-defined W surface, a surface construction algorithm in a CAD system geometric library is constructed, wherein a Bezier algorithm is selected as an example, and the specific definition method is shown in the technical scheme step 4); then, the curved surface splicing based on the triangular domain is carried out, and the specific implementation method is shown in step 5) of the technical scheme. By this we have completed the construction of a recursive surface.

Claims (1)

1. A recursive surface construction method based on L surface and W surface in triangle domain is characterized in that the method comprises the following main steps:

1) Firstly, defining a general recursive form surface representation method on a triangular parameter domain:

defining the symbol: j. the design is a square ₀ ＝(1，0，0)，J ₁ ＝(0，1，0)，J ₂ ＝(0，0，1)，

Let { P _a | a | = n } is the vertex of the feature net on the triangular domain,

L _a ^T ((x)，M _a ^T (x)，N _a ^T (x) The specific requirements for the deployment function at the corresponding node on the triangular domain are as follows:

L _a ^T (x) Should satisfy 1. Ltoreq. A ₀ ≤n，0≤a ₁ ，a ₂ ≤n；

M _a ^T (x) Should satisfy 1. Ltoreq. A ₀ ≤n，0≤a ₁ ，a ₂ ≤n；

N _a ^T (x) Should satisfy 1. Ltoreq. A ₀ ≤n，0≤a ₁ ，a ₂ ≤n；

And is

The method for defining the general recursive form surface on the triangular parameter domain comprises the following steps:

wherein if a _i -1 < 0 or > 0, then let

2) Defining an n-th order L surface over a triangular field

Setting:

(and satisfy

r＝1，2，Λn，|a|＝n-r)

Defining the n-th order L surface on the triangular domain as follows according to 1): the n recursion curves must satisfy

Then is an n-order L-shaped surface on the triangular domain;

3) Defining W surface

If the L-shaped surface continues to satisfy the requirement of n times

Then is a W curved surface of n times;

4) Conversion to Bernstein-Bezier curved surface

Order:

then, get the Bernstein-Bezier curved surface

WhereinP _ijk Is a node of the triangular domain network;

5) Bezier curved surface continuous splicing on triangular domain

If we give two pieces of Bezier surface n times in the triangular domain:

then the condition that Q (u, v, w) and P (u, v, w) achieve geometric continuity on the splice line is:

and then sequentially connecting all the divided triangular domain curved surfaces to obtain the required curved surface structure.

Images