KR101739224B1 - Method for calculating distance between a given surface and a point in a space filled with unstructured mesh - Google Patents

Abstract

The present invention provides a method for calculating the shortest distance from one arbitrary point to a specifically defined surface on a two dimensional plane filled with unstructured meshes, comprising the steps of: storing an original level-set value (_0 = ); defining a narrow band with respect to a cut element formed of the two dimensional plane; performing an initialization operation such that = where is the number of narrow band layers, h is a characteristic length of the cut element, is a width of the narrow band, and = 2h; calculating a cross point, X_s, between an interface segment and a corner of the cut element using _0; determining the shortest distance, d_x, from an arbitrary point, X_n, to the interface segment within the narrow band of the cut element; and updating the level-set value of X_n (_n = min{d_x,_n}). The present invention can perform more rapid and definite calculation.

Description

A method for calculating a shortest distance from an arbitrary point in a space filled with an unmodified grating system to a surface defined in a certain way.

The present invention relates to a technique for calculating a shortest distance from an arbitrary point in a two-dimensional or three-dimensional space filled with an unmodified grating system to a surface defined in a specific manner, and relates to an image processing field including animation, To a method of calculating the shortest distance from a certain point in a 2-dimensional or 3-dimensional space filled with an unstructured grid system that is widely applicable to a surface defined in a specific way.

Recently, computer graphics and computer animation techniques are widely used in various industrial fields.

For example, in computer animation, a face motion capturing technique is used to realistically express a change in human facial expression. To implement such a technique, it is necessary to calculate the distance between the surface elements constituting the face at a specific point (for example, a light source).

In addition, in the case of a fluid phenomenon implementation by CFD (Computational Fluid Dynamics), it is necessary for the simulation model to be reproduced to be closer to the actual one, It is necessary to calculate the shortest distance more quickly and accurately.

However, conventionally, there has been a problem that the process of calculating the shortest distance from an arbitrary point to a specific defined plane in a two-dimensional or three-dimensional space is slow in calculation speed and difficult to be implemented through the analysis of partial differential equations.

U.S. Pat. No. 7,633,429

SUMMARY OF THE INVENTION The present invention has been conceived to solve the problems as described above, and it is an object of the present invention to provide a method of calculating a shortest distance from an arbitrary point in a two- or three- And to provide a method of calculation.

The objects of the present invention are not limited to the above-mentioned objects, and other objects not mentioned can be clearly understood by those skilled in the art from the following description.

In order to achieve the above object, there is provided a method of calculating a shortest distance from an arbitrary point to a line defined in a two-dimensional plane filled with an unoriented grating system according to the present invention, Defining a narrow band for a cut element that is in a two-dimensional plane, storing a value of a narrow band (? ₀ =?), Defining a narrow band for a cut element in a two- initializing to φ = δ, where h is the characteristic length of the cut element, δ is the width of the narrow band and δ = 2ωh, φ is the number of layers, φ is the number of layers, ₀ , calculating an intersection X _s between an interface segment and an edge of the cut element, calculating a shortest distance d _x from the arbitrary point X _n in the narrow band of the cut element to the interface segment To And updating the level set value of X _n (φ _n = min {d _x , φ _n }).

(수학식 1)로 교차점 X_s를 계산할 수 있다. The cut elements are triangular, in calculating the intersection point X _s, and any two vertices of the x _1, x ₂ from the triangle, the level set value at the φ ₁ x _1, at a φ ₂ x ₂ Assuming a level set value,

The intersection X _s can be calculated by the following equation (1).

벡터의 스칼라 프로젝션(scalar projection)의 절대 값

을,

(수학식 2)로 계산할 수 있다.In determining the shortest distance d _x , if X _f is the foot of the perpendicular to the interface at X _n and X _sm is the center of the cut segment,

The absolute value of a scalar projection of a vector

of,

(2). ≪ / RTI >

이면,

(수학식 3)으로 최단 거리 d_x를 결정할 수 있다.

If so,

The shortest distance d _x can be determined by the following equation (3).

이 아니면, X_n으로부터 상기 인터페이스 세그먼트의 각 교차점까지의 거리 X_si를 계산하고, 계산된 값 중에서 상기 최단 거리 d_x로 계산하는,

(수학식 4)로 나타낼 수 있다.

, Calculating a distance X _si from X _n to each intersection point of the interface segment and calculating the shortest distance d _x among the calculated values,

(Equation 4).

(수학식 5)로 정의되는 사인(sign) 함수(function)라 할 때, 모든 임의의 점 X_n에서의 레벨 셋의 사인(sign)을 φ= φ.sgn(φ₀)로 나타낼 수 있다. sgn,

The sign of the level set at any arbitrary point X _n can be denoted by phi = phi.sgn (phi ₀ ), where phi is the sign function defined by the expression (5).

In the shortest distance calculation method from the arbitrary one point to the defined surface in the three-dimensional space filled with the unaligned grating system of the present invention, an original level-set value is stored for (φ ₀ = φ) step, defining a narrow band (narrow band) for the cutting element (cut element) that is a three-dimensional space, and ω is the number of layers (layer) of the narrow band, initialize to φ = δ, where h is the characteristic length of the cut element, δ is the width of the narrow band and δ = 2ωh, φ ₀ is used to initialize the interface segment calculating an intersection X _s of the interface segment with an edge of the cut element X _s , calculating a radius r _sc of the circle including the geometric center X _sc of all of the interface segment planes and all intersection points, Inside the band Comprises the step _{(φ n = min {d x} , φ n}) of step and updating (update) the level set values of the X _n to determine the minimum distance d _x to said interface segment from any point X _n of do.

(수학식 1)로 교차점 X_s를 계산할 수 있다. Wherein the cut element is a tetrahedron, and wherein in calculating the intersection X _s , x ₁ , x ₂ are any two vertexes of the tetrahedron, φ ₁ is a level set value at x ₁ , φ ₂ Is a level set value at x ₂ ,

The intersection X _s can be calculated by the following equation (1).

로 나타낼 수 있다. In the step of calculating the geometric center X _sc and the radius r _sc of the circle, the radius r _sc of the circle is the maximum distance from the geometric center X _sc to each vertex X _si of the interface segment

.

으로 계산하며, X_f를 X_n에서 인터페이스(interface)에 내린 수선의 발(foot of perpendicular)이라고 할 때, 수선의 발 X_f에서 기하 중심 X_sc까지의 거리 d_fsc를 피타고라스 정리(Pythagorean Theorem)를 이용하여 구하면,

(수학식 6)이다. In determining the minimum distance d _x, the n orthogonal distance d _xf of the plane including up to the interface from the segment, X _n when said unit normal vector (unit vector normal) of said interface segment, and

The calculation and, X _f a to the waterline down to the interface (interface) in the X _n when called (foot of perpendicular), Pythagoras distances d _fsc in the repair to X _f to the geometric center X _sc cleanup (Pythagorean Theorem) , ≪ / RTI >

(Equation 6).

In determining the minimum distance d _x, d is the _fsc ≥r If _sc, when the said edge of the edge _i interface _{segment, d x = min {(Xn} , edge i)}.

(수학식 7)을 이용하여 수선의 발 X_f를 구하는 단계, 상기 수선의 발이 상기 컷 엘리먼트의 내부에 있는지 여부를 확인하는 단계, 상기 수선의 발이 상기 컷 엘리먼트의 외부에 있으면, d_x=min{(X_n, edge_i)}로 구하는 단계 및 상기 수선의 발이 상기 컷 엘리먼트의 내부에 있으면, d_x=d_xf로 구하는 단계를 포함하여 이루어질 수 있다. In one embodiment of the present invention it is not a d _fsc ≥r _sc,

If obtaining a to X _f of the repair using the (expression 7), the method comprising the foot of the perpendicular determine whether the inside of the cutting element, the foot of the perpendicular to the outside of the cutting element, d _x = min (X _n , edge _i ), and if the foot of the waterline is inside the cut element, calculating d _x = d _xf .

(수학식 8)을 이용하여 프로젝티드 사인드 영역(projected signed area)을 계산하고, n_max=│n_x│이면, A_2D=A_yz이고, n_max=│n_y│이면, A_2D=A_zx이고, n_max=│n_z│이면, A_2D=A_xy이고, 이러한 프로젝티드 사인드 영역을 일차원 어레이(array)인 “Area2D”로 저장하고, 상기 “Area2D” 어레이의 모든 성분의 사인(sign)을 비교하고, 모든 성분이 같은 사인이면 상기 수선의 발이 상기 컷 엘리먼트의 내부에 있는 것으로 간주하고, 그렇지 않으면 상기 수선의 발이 상기 컷 엘리먼트의 외부에 있는 것으로 간주할 수 있다.In the step of confirming whether or not the foot of the waterline is inside the cut element, with respect to each triangle constituting the face of the tetrahedron,

When using the (Expression 8) and calculates the projectile suited sine de area (projected area signed), n _max = │n │ _x, and A _2D = A _yz, if _{n max = │n y │, A} 2D = A _zx , and n _max = | n _z |, then A _2D = A _xy and stores the projected syndrome area as a one-dimensional array " Area2D ", and signs of all components of the " Area2D & the sign of the waterline is considered to be inside the cut element if all the components are the same sine and otherwise the foot of the waterline is considered to be outside the cut element.

(수학식 5)로 정의되는 사인(sign) 함수(function)라 할 때, 모든 임의의 점 X_n에서의 레벨 셋의 사인(sign)을 φ= φ.sgn(φ₀)로 나타낼 수 있다. sgn,

The sign of the level set at any arbitrary point X _n can be denoted by phi = phi.sgn (phi ₀ ), where phi is the sign function defined by the expression (5).

According to the present invention, it is possible to calculate faster and more accurately than basic methods by providing a method of calculating the shortest distance from a certain point in a two-dimensional or three-dimensional space filled with an unoriented grating system to a specifically defined surface There is an effect to make.

In addition, according to the present invention, it is expected to be very useful in image processing techniques including animation, and it is expected to be a useful tool in the analysis of computational fluid dynamics problems found in many industrial fields such as mechanical, chemical, and civil engineering .

1 is a diagram showing an arbitrary element including an interface of an unmodulated grating system.
FIG. 2 is a schematic diagram for calculating a shortest distance from an arbitrary point to a boundary line in a two-dimensional plane according to an embodiment of the present invention. FIG.
3 is a schematic diagram for calculating the shortest distance from an arbitrary point to an interface in a three-dimensional space according to an embodiment of the present invention.
FIG. 4 is a flowchart illustrating a method of calculating a shortest distance from a certain point to a plane defined in a two-dimensional plane filled with an unoriented grating system according to an embodiment of the present invention.
FIG. 5 is a flowchart illustrating a method of calculating a shortest distance from a certain point to a line defined in a three-dimensional space filled with an unaligned grid system according to an embodiment of the present invention.

While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that the invention is not intended to be limited to the particular embodiments, but includes all modifications, equivalents, and alternatives falling within the spirit and scope of the invention.

The terminology used in this application is used only to describe a specific embodiment and is not intended to limit the invention. The singular expressions include plural expressions unless the context clearly dictates otherwise. In the present application, the terms "comprises" or "having" and the like are used to specify that there is a feature, a number, a step, an operation, an element, a component or a combination thereof described in the specification, But do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, or combinations thereof.

Unless defined otherwise, all terms used herein, including technical or scientific terms, have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. Terms such as those defined in commonly used dictionaries are to be interpreted as having a meaning consistent with the contextual meaning of the related art and are to be interpreted in an ideal or overly formal sense unless expressly defined in the present application Do not.

In the following description of the present invention with reference to the accompanying drawings, the same components are denoted by the same reference numerals regardless of the reference numerals, and redundant explanations thereof will be omitted. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings. In the following description, well-known functions or constructions are not described in detail since they would obscure the invention in unnecessary detail.

In the present invention, a principal performing the shortest distance calculation method from a certain point to a surface defined in a two-dimensional plane or a three-dimensional space filled with an unaligned grid system may be a control unit for controlling the entire computer, And a central processing unit (CPU) for processing a series of programs. That is, the shortest distance calculation method of the present invention is composed of an algorithm or logic, which is a kind of software, and the software algorithm can be executed in the control unit of the computer or the central processing unit.

1 is a diagram showing an arbitrary element including an interface of an unmodulated grating system.

Referring to FIG. 1, an interface 10, a cut element 110, and an interface segment 120 are shown in an unmodulated grating system.

The triangulated points 20 are the nodes of the first layer N1 and the vertex 30 is the nodes of the second layer N2.

FIG. 2 is a schematic diagram for calculating the shortest distance from any point to a boundary in a two-dimensional plane according to an embodiment of the present invention, and FIG. 4 is a graph Fig. 3 is a flowchart showing a method of calculating a shortest distance from a certain point to a line defined in a two-dimensional plane. Fig.

Referring to FIGS. 2 and 4, a method of calculating a shortest distance from a certain point to a line defined in a two-dimensional plane filled with an unmodulated grating system will be described.

First, an original level-set value is stored (? ₀ =?) (S401).

Then, a narrow band is defined for a cut element that is a line on a two-dimensional plane (S403).

If ω is the number of layers in the narrow band, h is the characteristic length of the cut element, δ is the width of the narrow band, and δ = 2ωh, then φ = δ (S405).

Then, an intersection X _s between the interface segment and the edge of the cut element is calculated using? ₀ (S407).

Then, the shortest distance d _x from the arbitrary point X _n in the narrow band of the cut element to the interface segment is determined (S409).

Then, the level set value of X _n is updated (φ _n = min {d _x , φ _n }) (S411).

In the embodiment of FIG. 2, the cut element is implemented as a triangle.

At this time, in step S407 of calculating the intersection X _s , x ₁ , x ₂ are arbitrary two vertexes of a triangle, φ ₁ is a level set value at x ₁ , φ ₂ is a level set value at x ₂ , The intersection X _s can be calculated by the following equation.

벡터의 스칼라 프로젝션(scalar projection)의 절대 값

을 다음 수학식과 같이 계산할 수 있다. When the shortest distance d _x is a foot of perpendicular that X _f is lowered from X _n to an interface in step S 409 and X _sm is the center point of the cut segment,

The absolute value of a scalar projection of a vector

Can be calculated as shown in the following equation.

이면, 다음 수학식으로 최단 거리 d_x를 결정할 수 있다.

, The shortest distance d _x can be determined by the following equation.

이 아니면, X_n으로부터 상기 인터페이스 세그먼트의 각 교차점까지의 거리 X_si를 계산하고, 계산된 값 중에서 최단 거리 d_x로 계산하는 것인 다음 수학식으로 나타낼 수 있다. But,

Or this can be from X _n represented by the following equation to calculate the distance X _si to each intersection of said interface segment, and the shortest distance from among the calculated values calculated by _x d.

In the present invention, we define sign as follows.

The sign of the level set at any arbitrary point X _n can be denoted by φ = φ.sgn (φ ₀ ), given a sign function defined by equation (5).

FIG. 3 is a schematic diagram for calculating a shortest distance from an arbitrary point to an interface in a three-dimensional space according to an embodiment of the present invention, and FIG. 5 is a graph Fig. 3 is a flowchart showing a method of calculating a shortest distance from a certain point to a surface defined in a three-dimensional space; Fig.

3 and 5, a method of calculating the shortest distance from a certain point to a plane defined in a three-dimensional space filled with an unoriented grating system will be described.

First, an original level-set value is stored (? ₀ =?) (S501).

Then, a narrow band is defined for a cut element in a three-dimensional space (S503).

If ω is the number of layers in the narrow band, h is the characteristic length of the cut element, δ is the width of the narrow band, and δ = 2ωh, then φ = δ Initialize (S505).

Then, an intersection X _s between the interface segment and the edge of the cut element is calculated using? ₀ (S507).

Then, the geometric center X _sc of the interface segment plane and the radius r _sc of the circle including all the intersection points are calculated (S509).

Then, the shortest distance d _x from the arbitrary point X _n in the narrow band of the cut element to the interface segment is determined (S511).

Then, the level set value of X _n is updated (φ _n = min {d _x , φ _n }) (S 513).

(수학식 1)로 교차점 X_s를 계산할 수 있다. And Example-cut elements are tetrahedra (tetrahedron) in Figure 3, wherein in step (S507) for calculating a cross point X _s, x _1, and any two vertices of from tetrahedral to x _2, the φ ₁ in x ₁ and a level set value, when the φ ₂ to as level set value at x _2,

The intersection X _s can be calculated by the following equation (1).

로 나타낼 수 있다. In step S509 of calculating the geometric center X _sc and the radius r _sc of the circle, the radius r _sc of the circle is the maximum distance from the geometric center X _sc to each vertex X _si of the interface segment

.

으로 계산한다. In step (S511) for determining a minimum distance d _x, n for the orthogonal distance of the plane to the d _xf, including an interface segment from, X _n when said unit normal vector (unit vector normal) of said interface segment, and

.

Then, let X _{f be the} foot of perpendicular from X _n to the interface, the distance d _fsc from the foot X _f of the waterline to the geometric center X _sc is calculated using the Pythagorean Theorem The following equation is obtained.

Referring to Figure 3 (a), in step (S511) for determining a minimum distance d _x from the present invention, d is _fsc ≥r _sc, when the said edge of the edge _i interface segment, d _x = min {(Xn , edge _i )}.

Referring to Figure 3 (b), in the present invention is not a d _sc ≥r _fsc, first using the following equation to calculate the X _f of repair.

Then, it is checked whether or not the feet of the waterline are inside the cut element.

Then, if the foot of the waterline is outside the cut element, d _x = min {(X _n , edge _i )}, and if the foot of the waterline is inside the cut element, then d _x = d _xf .

Hereinafter, a process for checking whether or not the foot of the waterline is inside the cut element will be described in detail.

First, for each triangle forming the face of the tetrahedron, the projected signed area is calculated using the following equation.

Then, if n _max = | n _x |, then A _2D = A _yz , and if n _max = | n _y | then A _2D = A _zx and if n _max = | n _z | then A _2D = A _xy .

Then, these projected signal areas are stored as a one-dimensional array " Area2D ", and the sign of all the components of the " Area2D " array is compared.

As a result of the comparison, if all the components are the same sine, the feet of the waterline are considered to be inside the cut element, otherwise, the feet of the waterline are considered to be outside the cut element.

(수학식 5)로 정의되는 사인(sign) 함수(function)라 할 때, 모든 임의의 점 X_n에서의 레벨 셋의 사인(sign)을 φ= φ.sgn(φ₀)로 나타낼 수 있다. In the present invention,

The sign of the level set at any arbitrary point X _n can be denoted by phi = phi.sgn (phi ₀ ), where phi is the sign function defined by the expression (5).

On the other hand, the method of calculating the shortest distance from a certain point to a plane (line) defined in a two-dimensional plane or a three-dimensional space filled with an unmodified grating system according to an embodiment of the present invention is a computer- It can be implemented as a computer-readable code on a medium. A computer-readable recording medium includes all kinds of recording apparatuses in which data that can be read by a computer system is stored.

For example, the computer-readable recording medium includes a ROM, a RAM, a CD-ROM, a magnetic tape, a hard disk, a floppy disk, a removable storage device, a nonvolatile memory, , And optical data storage devices.

In addition, the computer readable recording medium may be distributed and executed in a computer system connected to a computer communication network, and may be stored and executed as a code readable in a distributed manner.

While the present invention has been described with reference to several preferred embodiments, these embodiments are illustrative and not restrictive. It will be understood by those skilled in the art that various changes and modifications may be made therein without departing from the spirit of the invention and the scope of the appended claims.

110, 210 cut element,
120, 220 interface segment

Claims (5)

A method for calculating a shortest distance from a certain point to a line specifically defined in a two-dimensional plane filled with an unstructured grid system,
Storing an original Level-Set value (? ₀ =?);
Defining a narrow band for a cut element in a two-dimensional plane;
ω is the number of layers of the narrow band, h is the characteristic length of the cut element, δ is the width of the narrow band, and δ = 2ωh, φ = δ Initializing;
calculating an intersection X _s between an interface segment and an edge of the cut element using φ ₀ ;
Determining a shortest distance d _x from an arbitrary point X _n in the narrow band of the cut element to the interface segment; And
Wherein the step of updating the level set value of X _n comprises: ( _n = min {d _x , _n }}.
The method according to claim 1,
Wherein the cut element is triangular,
In calculating the intersection X _s ,
x ₁ and x ₂ are arbitrary two vertexes of a triangle, φ ₁ is a level set value at x ₁ , and φ ₂ is a level set value at x ₂ ,

(1)
To calculate a shortest distance X _s . 1. A method of calculating a shortest distance from a point to a plane defined in an arbitrary three-dimensional space filled with an unstructured grid system,
Storing an original Level-Set value (? ₀ =?);
Defining a narrow band for a cut element in a three-dimensional space;
ω is the number of layers of the narrow band, h is the characteristic length of the cut element, δ is the width of the narrow band, and δ = 2ωh, φ = δ Initializing;
calculating an intersection X _s between an interface segment and an edge of the cut element using φ ₀ ;
Calculating a radius r _sc of a circle including the geometric center X _sc and all intersections of the interface segment plane;
Determining a shortest distance d _x from an arbitrary point X _n in the narrow band of the cut element to the interface segment; And
Wherein the step of updating the level set value of X _n comprises: ( _n = min {d _x , _n }}. The method of claim 3,
The cut element is a tetrahedron,
In calculating the intersection X _s ,
when x ₁ and x ₂ are arbitrary two vertexes of a tetrahedron and φ ₁ is a level set value at x ₁ and φ ₂ is a level set value at x ₂ ,

(1)
To calculate a shortest distance X _s . A computer-readable recording medium having recorded thereon a program capable of executing a method according to any one of claims 1 to 4 by a computer.

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