JPH05165929A - Method for designing paper pattern - Google Patents

Abstract

PURPOSE:To shorten calculation time by decreasing the number of times to repeat calculation for designing a paper pattern by evaluating a trial plane based on the change of an area ratio between the trial plane and a curved plane or the rate of the area ratio. CONSTITUTION:A curved plane 18 to be developed is divided into meshes, namely, into elements and a trial plane 22 provided with meshes corresponding to these respective meshes is estimated. Then, the trial plane 22 is deformed so that the shape of the meshes on the trial plane 22 can be equal to the shape of the meshes on the curved plane 18 corresponding to these meshes. Then, it is evaluated whether the deformed trial plane 22 satisfies the set conditions or not and based on the trial plane 22 satisfying the conditions, the paper pattern is designed. For example, it is evaluated whether the deformed trial plane 22 satisfies the set conditions or not based on the change rate of the area ratio between the trial plane 22 and the curved plane 18 or the rate of the area ratio. Thus, since the shapes of the developed plane and the curved plane 18 are made directly correspondent, the number of times for calculation can be decreased and the calculating time can be shortened.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a pattern design method,
More specifically, the present invention relates to a pattern design method for automatically designing a pattern from design data using a computer.

[0002]

2. Description of the Related Art As a conventional pattern designing method based on which the present invention is based, a paper collection of the Japan Society of Mechanical Engineers (C edition): Vol. 54, No. 498 (Sho 63-2)
"Plane expansion problem of free-form surface by finite element method" and 55
Volume 511 (1989-3) "Plane expansion problem of free-form surface by finite element method (2nd report, when there are various design constraints)", as applied to shoes and aircraft fuselage etc. There is.

In this method, the curved surface is converted as follows.
Perform a flat development to design one pattern. Deploy first
Divide a curved surface into a large number of triangular meshes (elements)
It Then, the triangles corresponding to these triangle meshes
Assume a quadrilateral trial plane with a mesh. next,
The shape of the triangular mesh on the quadrilateral trial plane is on the curved surface.
Force equal to the shape of the corresponding triangular mesh of
In the trial plane to deform the trial plane,
Design a paper pattern by evaluating the trial plane of. This trial plane
The finite element method, which is a mechanical method, is used for evaluation.
Strain energy U caused by the displacement of each node on the trial plane
The strain energy UT is calculated to be 1 × 10. ^-Ten~^-15When
Deformation and evaluation of the trial plane until reaching a value close to 0.
The value is repeating. For this reason, a pattern with a complicated curved surface shape
When designing the
There was a problem that time increased.

Here, the strain energy UT is the x-axis direction, y
Axial stress is σ _x , σ _y , and shear stress is τ _xy.
, x and y axis strains are ε _x and ε _y , respectively, shear strain is γ _xy , and the area of the triangular mesh of the trial plane is Ap, which is given by the following equation (1).

[0005]

[Equation 1]

As described above, in the conventional technique, since the strain energy inside the trial plane is used as a method for evaluating the shape of the trial plane, there is a direct one-to-one correspondence between the shape of the trial plane and the strain energy. Since there is no correspondence of, when designing a pattern by expanding a complicated curved surface into a plane,
Inevitably, it is necessary to reduce the strain energy convergence condition, which requires a considerable amount of time for calculation, which is a practical problem.

The present invention has been made to solve the above problems, and it is possible to shorten the calculation time for evaluating a trial plane and develop a curved surface of a complicated shape in a short time to design a pattern paper. It is an object of the present invention to provide a pattern design method that can be performed.

[0008]

In order to achieve the above object, the present invention divides a curved surface to be developed into meshes, and assumes a trial plane provided with meshes corresponding to each of the meshes. The trial plane is deformed so that the shape of the upper mesh and the shape of the mesh on the curved surface corresponding to the mesh become equal, and it is evaluated whether the trial plane after the deformation satisfies the setting conditions, and the trial that satisfies the setting conditions In a pattern design method for designing a pattern based on a plane, it is possible to evaluate whether the trial plane after deformation satisfies a setting condition based on a rate of change in an area ratio between a trial plane and a curved surface or a ratio of the area ratio. Characterize.

[0009]

In the present invention, the curved surface to be developed is divided into meshes, that is, elements, and a trial plane having a mesh corresponding to each mesh is assumed. Next, the trial plane is deformed so that the shape of the mesh on the trial plane and the shape of the mesh on the curved surface corresponding to this mesh become equal. Then, it is evaluated whether or not the deformed trial plane satisfies the setting condition, and the pattern is designed based on the trial plane satisfying the setting condition. In this case, in the present invention, it is evaluated whether or not the trial plane after deformation satisfies the setting condition based on the rate of change in the area ratio between the trial plane and the curved surface or the ratio of the area ratio.

The rate of change Dr of the area ratio is such that As is the area of the mesh of the curved surface, Ap is the area of the mesh of the trial plane, and n.
Is the number of meshes, it is given by the following equation (2).

[0011]

[Equation 2]

However, t represents the number of deformations of the trial plane.
The present inventors investigated the relationship between the value of the rate of change Dr and the calculation time, and compared the developed shape with the conventional determination method using strain energy. The results are shown in Tables 1, 2, 3 and FIG. In addition, * 1 in Tables 1, 2, and 3 are evaluation results by the conventional strain energy evaluation method. According to these tables and figures, it is clear that if the trial plane is deformed and evaluated until Dr ≦ 1.0%, a pattern substantially equivalent to the conventional determination method can be obtained in half or less of the conventional time. Became.

[0013]

[Table 1]

[0014]

[Table 2]

[0015]

[Table 3]

By modifying the above equation (2), the following equation (3) is obtained.

[0017]

[Equation 3]

Therefore, instead of the rate of change Dr of the area ratio described above, the area of the curved surface with respect to the area of the trial plane deformed at the t-th time relative to the area ratio of the area of the trial plane deformed at the (t + 1) th time and the area of the curved surface. The trial plane may be evaluated depending on whether or not the ratio of the area ratio of is less than 1.01.

In the above, the ratio of the element of the curved surface, that is, the area of the mesh to the area of the trial plane, that is, the area of the mesh is used, but the reciprocal may be used.

[0020]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described in detail below with reference to the drawings. FIG. 3 shows hardware for adapting the pattern designing method of this embodiment. This hardware is a personal computer 10, a keyboard 12 for inputting various data, and a personal computer 1.
The printer 14 is provided with a display 14 and a printer 16 for displaying the result of processing at 0.

Next, the software of this embodiment will be described with reference to FIG. In step 30, the 3D design drawing determined by CAD or the like is taken in, and step 3
2 in 3 using the conversion software
Divide into a square mesh. FIG. 4 schematically shows an example in which the curved surface 18 is divided into a large number of triangular meshes 20. In this example, the curved surface 18 is divided into 2 (N−1) · (M−1) triangular meshes 20 by N points in the vertical direction and M points in the horizontal direction. In the next step 34, the divided triangular mesh is converted into three-dimensional point data.

In the next step 36, it is possible to form a quadrangle as an initial trial plane by regarding the periphery of the curved surface 18 as a shape composed of four contour lines based on the three-dimensional point-by-point data. If you can form a pentagon
Judge from the length of one contour line. That is, as shown in FIG. 5, it is considered that the circumference of the curved surface 18 is composed of four contour lines A, B, C, and D having lengths a, b, c, and d.
The contour line A, which is the longest side, is set as the bottom side as shown in FIG.
Next, the contour lines B and D on both sides of the contour line A are compared, and the shorter contour line, that is, the contour line B in the example of FIG. 5, is perpendicular to the end points of the contour line A as shown in FIG. Stand up. The length (a ² +) of the hypotenuse of the triangle formed by the contour lines A and B
b ² ) ^1/2 is compared with the sum c + d of the lengths of the remaining contour lines C and D. If (a ² + b ² ) ^1/2 ≧ c + d, a quadrangle cannot be formed as a trial plane. on the other hand,
When (a ² + b ² ) ^1/2 <c + d, a quadrangle can be formed as a trial plane. When it is determined in step 36 that a quadrangle can be formed as the trial plane, a quadrangle initial trial plane is created in step 38, and step 3
If it is determined in 6 that a quadrilateral trial plane cannot be formed, a pentagonal initial trial plane is created in step 40.
In this way, the perimeter of the initial trial plane matches the perimeter of the curved surface.

FIG. 7 shows a quadrilateral trial plane 22 corresponding to the curved surface 18 of FIG. This trial plane 22 is
2 (M-1). (N-1) triangular meshes 2 in FIG.
There are 2 (M−1) · (N−1) triangular meshes 24 corresponding to each 0. The hatched triangular mesh 26 located around the trial plane 22 indicates a dummy triangular mesh that is a triangle mesh having higher rigidity than the internal unhatched triangular mesh. These dummy triangular meshes 26 are necessary to make the perimeter of the pattern the same as the perimeter of the curved surface, and the details will be described later.

FIG. 8 shows a pentagonal trial plane 22 corresponding to the curved surface 18 of FIG. The pentagonal trial plane 22 is formed by bending the point f near the center of the contour line A, and forming the point f until a quadrangle is formed by the side E having a length e formed by bending the contour line B, C, and D. It is created by bending the two sides sandwiched.

In the next step 42, the same plane expansion calculation as the conventional one is performed based on the initial trial plane. In the next step 46, whether the trial plane created by judging whether the rate of change Dr of the area ratio of the above formula (2) is a predetermined value (for example, 1.0%) or less can be adopted as the reference pattern paper. To judge. If the rate of change of the area ratio exceeds the predetermined value, in step 44, pay attention to the triangular mesh on the trial plane and the triangular mesh on the curved surface corresponding to this triangle mesh, and determine the mesh shape on the trial plane. The trial plane is deformed so that the mesh shape on the curved surface corresponding to this mesh becomes equal. Since the method of this modification is the same as the conventional one, detailed description thereof will be omitted. Next, the plane expansion calculation in step 42 and the magnitude of the change rate Dr in step 46 described above are performed, and steps 42, 46, and 44 are repeated until the change rate Dr in the area ratio becomes a predetermined value or less.

When the rate of change Dr of the area ratio is less than the predetermined value, the trial plane at that time is determined as the reference template in step 48.

In the next step 50, the physical properties of the material such as Young's modulus E, Poisson's ratio ν, shear rigidity G, etc. are input from the keyboard 12 to the personal computer 10.
The physical properties of the material are taken in, and the pattern is evaluated in step 52 by shape comparison based on the free-form surface development method.
That is, in order to evaluate whether or not the reference template determined in step 48 is appropriate for the material to be used, it is assumed that the reference template is attached to the curved surface, and the physical properties of the material to be used are given to give the stress distribution and strain distribution in the template. Etc. are displayed inside the pattern,
If it is more than the standard value, it is judged that there is a defect in the material used and the pattern is corrected. In this modification of the template, the template is reduced in size in step 54, and an external force is applied to the template boundary line using the physical property values of the material in step 56 to perform tensile calculation and tensile deformation using the finite element method. Step 58
Then, the reference pattern and the deformed pattern are compared, and if the reduced and stretched pattern becomes the same as the reference pattern determined in step 48, the reduced pattern becomes a modified template for the material used. The template deformed at 60 is determined as the final template. In step 58, if the shape of the reduced and tensile deformed pattern does not become the same as the shape of the reference pattern, the reduced and tensile deformed pattern is reduced again in a similar manner and tensile deformed, and the strain energy is calculated by the above method. The deformation is calculated and corrected until it becomes the evaluation value or less or the area ratio change rate becomes 1% or less. As a result, it is possible to design a pattern free of defects that takes material characteristics into consideration. In addition, the pattern can be automatically designed and evaluated from the design drawing, and it can be modified according to the physical properties of the material.
Higher quality can be promoted. It should be noted that steps 52 to 60 for modifying the pattern for the material used.
The details are shown in FIG.

Next, the dummy triangular mesh will be described in detail. In the conventional technique, when the expansion calculation is performed without giving special consideration to the method of creating the trial plane, the perimeter of the trial plane after deformation becomes several percent longer than the perimeter of the curved surface to be developed, and the trial plane should be expanded. It was difficult to design a pattern that fits well on a curved surface. For example, in the case of automobile seats, when the skin material cut and sewn based on the pattern is covered by the inner cushion material of the seat, wrinkles do not occur in the skin material and the cushion shape itself is not deformed. Needs to fit well, which was difficult with the prior art. In order to solve this problem, it is necessary that the perimeter of the trial plane after deformation does not change with respect to the perimeter of the trial plane before deformation. Specifically, a highly rigid mesh such as a triangular mesh may be arranged along the periphery of the trial plane so that the perimeter of the trial plane is not deformed.

That is, the curved surface to be developed is divided into meshes, and a trial plane having meshes corresponding to each of the meshes is assumed, and the shape of the mesh on the trial plane and the mesh on the curved surface corresponding to the mesh are assumed. The trial plane is deformed so as to have the same shape as, the trial plane after deformation is evaluated whether the trial conditions satisfy the setting conditions, and the pattern design method for designing the template based on the trial plane satisfying the setting conditions is used. A trial plane whose perimeter is within a predetermined range with respect to the perimeter of the curved surface to be developed may be assumed, and a dummy mesh having high rigidity may be arranged along the perimeter of this trial plane.

The predetermined range is preferably within -2% to + 0.1% with respect to the peripheral length of the curved surface to be developed. If this is assumed to be less than 2% with respect to the perimeter of the curved surface to be developed, the shape of the trial plane after deformation will be affected, and this effect will cause the cloth to be cut and covered on the curved surface (bonded together). This is because when the appearance is wrinkled and it is assumed to be longer than 0.1% with respect to the perimeter of the curved surface to be developed, when the cloth is cut and the curved surface is covered, the appearance of the cloth becomes slack. Moreover, the position where the dummy mesh is arranged may be inside (FIG. 7), outside (FIG. 10), or both inside and outside (FIG. 11) of the trial plane. The rigidity of the dummy mesh is Young's modulus, which is 1 for meshes other than the dummy mesh.
A value of 00 to 10000 is suitable. Further, the shape of the mesh including the outer peripheral line of the trial plane may be the same as the shape of the corresponding mesh of the curved surface to be developed, and the rigidity of the mesh including the outer peripheral line may be increased as described above.

As described above, assuming a trial plane having a perimeter within a predetermined range and arranging a dummy mesh having high rigidity along the periphery of this trial plane, a dummy mesh having high rigidity when the trial plane is deformed is formed. This makes it possible to design a pattern that fits the curved surface to be developed without deforming the periphery of the trial plane. When the curved surface is divided into a plurality of patterns and developed in a plane, the peripheral lengths of the adjacent patterns can be made equal to each other. Therefore, similarly, it is possible to design a pattern group that fits the curved surface well.

To apply the above principle to the present invention, the curved surface to be expanded is divided into meshes, and a trial plane having meshes corresponding to each of the meshes is assumed, and the shape of the mesh on the trial plane is assumed. And deform the trial plane so that the mesh shape on the curved surface corresponding to the mesh becomes equal, evaluate whether the trial plane after deformation satisfies the setting condition, and based on the trial plane satisfying the setting condition, the pattern paper In the design method of the pattern for designing, a trial plane whose perimeter is within a predetermined range with respect to the perimeter of the curved surface to be developed is assumed, and a highly rigid dummy mesh is arranged along the perimeter of this trial plane. It is only necessary to evaluate whether the deformed trial plane satisfies the setting condition based on the rate of change in the area ratio between the trial plane and the curved surface or the ratio of the area ratio.

By doing so, it is possible to reduce the number of calculation iterations of the pattern design and design a pattern that fits well to the curved surface to be developed.

Next, in the case where plane expansion is performed under the same conditions as the perimeter of the curved surface and the perimeter of the trial plane, as shown in FIG. 7, a dummy triangle having a high rigidity is provided around the perimeter of the trial plane 22. A method in which the plane on which the mesh 26 is arranged is used as a trial plane (internal dummy triangle mesh method), and a plane in which a highly rigid dummy triangle mesh 26 is arranged on the outer periphery of the boundary of the trial plane 22 as shown in FIG. A method of using a trial plane (external dummy triangle mesh method), and a dummy triangle mesh of the external dummy triangle mesh method is arranged only at the corners of each side as shown in FIG. A description will be given in comparison with a combined method in which a dummy triangular mesh of the rectangular mesh method is arranged. In the external dummy triangular mesh method, the number of elements is increased by the external dummy triangular mesh method as compared with the internal dummy triangular mesh method. On the other hand, in the internal dummy triangle mesh method, as shown in FIG. 7, a triangle mesh located around the boundary of the trial plane 22 is used as a dummy triangle mesh, and the triangle around the boundary of the trial plane 22 is surrounded. A highly rigid dummy triangular mesh 26 is arranged. Comparing the number of elements of the external dummy triangle mesh method and the number of elements of the internal dummy triangle mesh method, the number of elements in FIG. 10 is 2 (MN-1) in the external dummy triangle mesh method, and the number of elements of the internal dummy triangle mesh method is 2 (MN-1). The number of elements is 2 (M-1). (N-1) as described above. Comparing the number of elements of both, it becomes 2 (MN-1)> 2 (M-1). (N-1), and the number of elements of the internal dummy triangle mesh method is smaller than the number of elements of the external dummy triangle mesh method, It is advantageous for the calculation processing time. However, M and N> 3. Further, Table 4 below shows the results of examination on the calculation time of the external dummy triangle mesh method and the internal dummy triangle mesh method. The three-dimensional curved surface shapes * 1 and * 2 in Table 4 are omitted. As can be understood from Table 4, the calculation time of the internal dummy triangle mesh method is
It can be understood that the calculation time is shorter than that of the rectangular mesh method, and that the smaller the number of elements, the shorter the calculation processing time.

[0035]

[Table 4]

If an external dummy triangular mesh is provided for each of the two triangular meshes at the corners of the trial plane as in the combined method shown in FIG. 11, the corner portion is more easily deformed than the internal dummy triangular mesh method. Therefore, the corner portion can be formed in a highly accurate shape. The number of elements in this combined system is 2 (N-1). (M-1)
This is +8, which is eight more than the number of elements in the internal dummy triangular mesh method described above, but less than the number of elements in the external dummy triangular mesh method. However, comparing the number of elements of the above external dummy triangular mesh method, the above internal dummy triangular mesh method and the above combined method, 2 (MN-1)> 2 (M-1). (N-1) +8
> 2 (M-1) · (N-1), and it is clear that the calculation processing time is also in this order, and the calculation time can be shortened as compared with the external dummy triangular mesh method.
In addition, the combination method uses the external dummy 3 only in the above corners.
Of course, it can be arranged at an appropriate position in addition to the arrangement of the rectangular mesh.

Next, the effect of this embodiment will be described in comparison with the prior art. In the prior art, since the plane length is not expanded under the same condition as the circumference length of the curved surface and the circumference length of the trial plane, when the cloth is cut by the obtained pattern and the curved surface to be expanded is covered, the cloth has a lot of slack. The appearance was not good. On the other hand, in this embodiment, since a trial plane whose perimeter is within a predetermined range with respect to the perimeter of the curved surface to be developed is assumed, it is possible to design a template that fits well to the curved surface to be developed.

Further, since the convergence judgment of the repeated calculation of the plane expansion calculation is performed depending on whether or not the area ratio between the curved surface and the plane becomes 1 or a value close to 1, the area and the curved surface which are almost equal to the curved surface are obtained. It is possible to efficiently obtain a template having a perimeter length substantially equal to the perimeter length, that is, a template that is well fitted to the shape of the curved surface, in a short calculation time.

Further, in the prior art, there is a problem that when the plane is expanded with the perimeter of the curved surface being the same as the perimeter of the quadrangle as the trial plane, the quadrangle as the trial plane may not be obtained. However, in the above-described embodiment, since it is determined whether a tetragon or a pentagon can be formed as a trial plane and a trial plane suitable for each is created,
A trial plane for designing the pattern can be automatically created with any value of the four sides forming the periphery of the curved surface. Therefore, an efficient and versatile pattern design can be performed.

In the prior art, since the same template is prepared regardless of the physical properties of the material, there is a problem that it cannot be evaluated whether it is appropriate for the characteristics of the cloth using the template. In the above embodiment, the relationship between the triangular mesh which is each element inside the pattern paper and the triangular mesh which is an element on the curved surface corresponding to this triangular mesh is given finite by giving material characteristic values such as Young's modulus and Poisson's ratio. The stress applied to each triangular mesh is calculated by the element method, and the stress distribution for the material properties is calculated from the stress,
The distortion distribution etc. are displayed inside the pattern paper. Therefore, it is possible to judge and evaluate whether or not the pattern is appropriate for the material to be used, and it is possible to design an effective and versatile pattern. Further, if it is evaluated that defects (wrinkles, breakage, etc.) occur, the pattern is corrected. In this case, the evaluated value is a value preset according to the material used. In other words, if it is determined that the designed template has defects in the material used, the reference template is reduced in size, and the physical properties of the material such as Young's modulus, Poisson's ratio, shear rigidity, etc. are reduced. The tensile deformation calculation is performed using the finite element method by applying an external force to each value and the pattern boundary line, and if the result is the same as the reference pattern, the calculation ends, and the reduced pattern is used as the pattern for the material to be used. .. On the other hand, if the pattern after the tensile deformation calculation is not the same as the shape of the reference pattern, the reduced pattern is reduced again in a similar manner, and the tension calculation by the above method is repeated until it becomes the same shape as the reference pattern. Therefore, it is possible to design a pattern free of defects in consideration of the physical properties of the material. Further, by incorporating this pattern design method into the pattern design apparatus, even the template evaluation and correction can be performed, and the versatility can be further improved. Therefore, if the designed template is inappropriate for the material to be used, the template can be modified.

[0041]

As described above, according to the present invention, the trial plane is evaluated based on the rate of change of the area ratio between the trial plane and the curved surface or the ratio of the area ratio. Are directly associated with each other, so that it is possible to reduce the number of calculation iterations of the pattern design, and thereby to shorten the calculation time.

Further, in the conventional case, even if the expanded shape is the same, if there is a local residual strain, the calculation is repeated to reduce this strain, so it takes time to converge and the calculation time increases, whereas the mesh In the present invention for comparing the shapes, the residual strain as described above can be ignored if the shapes are converged, so that the values can be converged faster and the calculation time can be shortened.

[Brief description of drawings]

FIG. 1 is a flow chart showing software of an embodiment of the present invention.

FIG. 2 is a diagram comparing developed shapes of the related art and the present invention.

FIG. 3 is a block diagram showing a hardware configuration of an embodiment of the present invention.

FIG. 4 is a schematic view of a curved surface divided into a triangular mesh.

FIG. 5 is a schematic view of a curved surface divided into meshes.

FIG. 6 is a diagram for explaining that a quadrangle is not formed as a trial plane.

FIG. 7 is a schematic view showing a trial plane of a quadrangle.

FIG. 8 is a schematic view showing a pentagonal trial plane.

FIG. 9 is a flow chart for explaining pattern evaluation by material.

FIG. 10 is a schematic view of a trial plane in which a dummy triangular mesh is arranged on the outer circumference.

FIG. 11 is a schematic view of a trial plane in which an external dummy triangular mesh is arranged only in a corner portion.

[Explanation of symbols]

 10 personal computer 12 keyboard 14 display 16 printer 18 curved surface 20 triangular mesh 22 trial plane 24 triangular mesh 26 dummy triangular mesh

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Tetsuzo Inoue Nagakute-machi, Aichi-gun, Aichi Prefecture, Nagachote 1 41st Yokomichi, Toyota Central Research Institute Co., Ltd. (72) Inventor Isao Watanabe Nagakute-cho, Aichi-gun 1st in the 41st place in the Yokomichi Toyota Central Research Institute Co., Ltd. (72) Inventor Kazuyoshi Fujizaka 1 Toyota Town, Toyota City, Aichi Prefecture Toyota Motor Corporation (72) Inventor Yuji Okamoto 1 Toyota Town, Toyota City, Aichi Prefecture Toyota Automobile Co., Ltd.

Claims (1)

[Claims] 1. A curved surface to be expanded is divided into meshes, and a trial plane having meshes corresponding to each of the meshes is assumed, and the shape of the mesh on the trial plane and the mesh on the curved surface corresponding to the mesh are assumed. By deforming the trial plane so that it becomes equal to the shape of, evaluate whether the trial plane after deformation satisfies the setting conditions, and in the pattern design method of designing the pattern based on the trial plane that satisfies the setting conditions,
A method for designing a paper pattern, comprising evaluating whether or not the trial plane after deformation satisfies a setting condition based on the rate of change in the area ratio between the trial plane and the curved surface or the ratio of the area ratio.