WO2023173506A1 - 基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法 - Google Patents
基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法 Download PDFInfo
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- G06F30/00—Computer-aided design [CAD]
- G06F30/10—Geometric CAD
- G06F30/15—Vehicle, aircraft or watercraft design
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
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Definitions
- the invention relates to the field of computer origami fitting of generalized cylindrical surfaces in computer graphics, and specifically relates to a modeling method of fitting generalized cylindrical surfaces based on unfolded mine-derived origami units.
- Origami is an ancient Eastern art of paper folding that has been used to build three-dimensional structures by folding two-dimensional flat sheets of material embedded with designed crease patterns.
- An origami structure that satisfies the expandable constraints required can be fabricated on a flat surface without Cut, which is attractive for manufacturing, the structure can even fold flat, which makes sense for efficient storage and transportation.
- Benefiting from scale-independent features, a preliminary prototype of origami may be suitable for applications in sizes ranging from nanometer to meter to macroscale.
- origami Constructing a given cylinder by origami is a simple case of the inverse origami design problem since its Gaussian curvature is zero.
- This cylindrical origami structure has also received widespread attention from researchers and engineers and has potential applications such as biomedical foldable scaffolds, tunable mechanical memory structures, and spatially deployable arms.
- Origami structures obtained from most design methods are in a partially folded state when unfolded. Even though it is possible to generate a finer approximation corresponding to the target surface by increasing the number of units used to synthesize the crease map, such an origami approximation exhibits a corrugated surface, which is very problematic in exterior design in some specific engineering scenarios. Corrugations, or smooth surfaces, play a vital role, such as aircraft wings. Smooth surfaces allow wind and rain to pass easily through the surface of the object.
- a new method for fitting generalized cylindrical surfaces consisting of expandable rectangular blocks in which a crease pattern inspired by water-elastic tessellation is embedded is proposed.
- the rectangular block is a basic unit, unified by optimizing the width w and height h.
- the resulting crease pattern with such uniform rectangular blocks is expandable and can be fabricated on a flat piece of material without additional cutting.
- To approximate a cylindrical target surface from a 2D crease pattern the collinear creases between adjacent rows are folded simultaneously and appropriately like a hinge, while the internal creases in each row remain unfolded. This approximation can be interpreted as a discrete version of the generalized cylindrical target surface and can still be stored compactly by collapsing all embedded creases.
- the present invention also studies how two other relationships between w and h, namely w ⁇ h and w>h, affect the foldability of the unfolded structure, and introduces mine-derived units to simulate The target generalized cylindrical surface is combined to achieve control of the completely folded shape.
- a generalized cylindrical surface modeling method based on unfolded mine-derived origami units including the following steps:
- S2 The user interactively inputs configuration information to control the generation of the target surface
- step S1 is specifically:
- the target grid model consists of N r ⁇ N c mine units, that is, N r rows and N c columns.
- the width w of the square mine unit is equal to the height h.
- the unit of the mine tiled in odd rows is set as module B O.
- the six creases inside the unit intersect at an internal vertex.
- the crease pattern is shown in Figure 1(a).
- the mine unit module tiled in even-numbered rows is shown in Figure 1(b).
- the origami units are staggered in each row, as shown in Figure 1(c).
- step S2 specifically includes the following content:
- contour curve The user specifies the curve control point, and the NURBS curve (Non-uniform rational basis spline) is generated from the control point, which is the 2D contour curve ⁇ of the cylindrical surface (note: the coordinate system is the space rectangular coordinate system, consisting of x, y and z respectively. axis. At this time, the contour curve is located in the x-z plane);
- the contour curve ⁇ is known, and the user inputs the surface width W, and scans the curve along the y-axis for a distance W, then the surface formed by its scanning path is the target surface ⁇ T , which is the final fitted cylindrical surface;
- step S3 specifically includes the following content:
- the height error of the mine unit can be reduced and the difficulty of production can be reduced;
- step S4 specifically includes the following content:
- the shape is a square, and the patterns of the mine units in odd-numbered rows and even-numbered rows are shown in Figure 2(a) and Figure 2(b) respectively.
- the target grid structure constructed by the mine unit structure is shown in Figure 3(c).
- the origami model When the origami model is completely folded, its final state will self-intersect (see Figure 3(d)), which does not meet the effective configuration and needs to Make adjustments to its crease map.
- the adjusted mine unit derivative structure is used to construct the target grid structure (see Figure 3(g)), which is finally flat and folded into a regular shape without self-intersection (see Figure 3(h)).
- G r is parallel to the x-axis, indicating the direction of increasing the area by increasing the number of mine unit columns. Its border area is:
- the target grid structure constructed from S-shaped mine origami is shown in Figure 4(c).
- the grid model is folded flat (see Figure 4(d))
- the B O module vertices C 2 and C 6 will intersect (where C 2 and C 6 are shown in Figure 4 ( a))
- the B E module the point pair D 1 , D 6 and D 3 , D 8 will intersect (where D 1 , D 6 and D 3 , D 8 are shown in Figure 4(b)), at this time it is necessary to Mine units are adjusted:
- the B E module change the size of the rectangle D 1 D 3 D 8 D 6 from w ⁇ h to 2w 2 ⁇ h, where And add a width of w 1 and a height of w 1 to its left and right sides respectively.
- the target grid structure is constructed using the adjusted mine unit derivative structure (see Figure 4(g)). Although it cannot satisfy completely flat folding, the approximate structure of the origami can be completely folded and can avoid self-intersection (see Figure 4(h) ));
- the bounding volume of the origami structure is:
- V S N c (wh)A S
- a S is the mapping area of the origami structure to the xz plane in the fully folded state:
- w′ 2 ⁇ w 2 , where ⁇ (0,1) is the scaling factor.
- This mode is called ShortII (SII) at this time, and the target grid structure it constructs is shown in Figure 5(e).
- SII ShortII
- the structure belongs to the origami inlay of the adhesive surface, that is, the completely folded structure exists between the two parallel surfaces in contact with the approximate surface of the origami (see Figure 5(f)), where G c and G r respectively represents the longitudinal and transverse growth directions of the mine origami unit (where G r is parallel to the x-axis, and G c is parallel to the y-axis).
- G c and G r respectively represents the longitudinal and transverse growth directions of the mine origami unit (where G r is parallel to the x-axis, and G c is parallel to the y-axis).
- its volume is:
- the present invention mainly uses the unfolded mine origami-derived structure to fit the target surface with generalized cylinder characteristics.
- the water mine is a type of origami pattern.
- the internal vertices of the pattern unit have six adjacent vertices, forming six sides, among which the distribution is four valley folds and two mountain folds.
- the origami pattern used in the present invention is based on a mine-derived structure, which is a derived origami structure invented based on the consideration of building a corrugated surface.
- the unit module of the generalized cylindrical surface is constructed using rectangular blocks, the mine unit is embedded in the rectangular block, and the flat folding constraint is satisfied, and the target grid model is discretized to build.
- origami structures Four different types were constructed based on the different ratios of height to width of mine-derived origami units.
- the invention develops a new form that can construct smooth surfaces, which can play a vital role in specific application scenarios. For example, when using this origami model to construct the surface of an aircraft wing, the smooth surface can greatly reduce air resistance.
- Figure 1 shows the different styles of mine units and the distribution of crease maps
- Figure 6 is a schematic flow chart of the implementation of the present invention.
- the present invention is a generalized cylindrical surface modeling method based on unfolded mine-derived origami units, which includes the following steps:
- S2 The user interactively inputs configuration information to control the generation of the target surface
- step S1 of the present invention includes:
- the target grid model consists of N r ⁇ N c mine units, that is, N r rows and N c columns.
- the width w of the square mine unit is equal to the height h.
- the unit of the mine tiled in odd rows is set as module B O.
- the six creases inside the unit intersect at an internal vertex.
- the crease pattern is shown in Figure 1(a).
- the mine unit module tiled in even-numbered rows is shown in Figure 1(b).
- the origami units are staggered in each row, as shown in Figure 1(c).
- step S2 specifically includes the following content:
- contour curve The user specifies the curve control point, and the NURBS curve (Non-uniform rational basis spline) is generated from the control point, which is the 2D contour curve ⁇ of the cylindrical surface (note: the coordinate system is the space rectangular coordinate system, consisting of x, y and z respectively. axis. At this time, the contour curve is located in the x-z plane);
- the contour curve ⁇ is known, and the user inputs the surface width W, and scans the curve along the y-axis for a distance W, then the surface formed by its scanning path is the target surface ⁇ T , which is the final fitted cylindrical surface;
- step S3 of the present invention includes:
- the height error of the mine unit can be reduced and the difficulty of production can be reduced;
- step S4 specifically includes the following content:
- the target grid structure constructed by the mine unit structure is shown in Figure 3(c).
- the origami model When the origami model is completely folded, its final state will self-intersect (see Figure 3(d)), which does not meet the effective configuration and needs to Make adjustments to its crease map.
- D 1 , D 3 and D 8 are the vertices in Figure 3(f);
- the adjusted mine unit derivative structure is used to construct the target grid structure (see Figure 3(g)), which is finally flat and folded into a regular shape without self-intersection (see Figure 3(h)).
- G r is parallel to the x-axis, indicating the direction of increasing the area by increasing the number of mine unit columns. Its border area is:
- the target grid structure constructed from S-shaped mine origami is shown in Figure 4(c).
- the grid model is folded flat (see Figure 4(d))
- the B O module vertices C 2 and C 6 will intersect (where C 2 and C 6 are shown in Figure 4 ( a))
- the B E module the point pair D 1 , D 6 and D 3 , D 8 will intersect (where D 1 , D 6 and D 3 , D 8 are shown in Figure 4(b)), at this time it is necessary to Mine units are adjusted:
- the B E module change the size of the rectangle D 1 D 3 D 8 D 6 from w ⁇ h to 2w 2 ⁇ h, where the length value And add a width of w 1 and a height of w 1 to its left and right sides respectively.
- the target grid structure is constructed using the adjusted mine unit derivative structure (see Figure 4(g)). Although it cannot satisfy completely flat folding, the approximate structure of the origami can be completely folded and can avoid self-intersection (see Figure 4(h) ));
- the bounding volume of the origami structure is:
- V S N c (wh)A S
- a S is the mapping area of the origami structure to the xz plane in the fully folded state:
- w′ 2 ⁇ w 2 , where ⁇ (0,1) is the scaling factor.
- This mode is called ShortII (SII) at this time, and the target grid structure it constructs is shown in Figure 5(e).
- SII ShortII
- the structure belongs to the origami inlay of the adhesive surface, that is, the completely folded structure exists between the two parallel surfaces in contact with the approximate surface of the origami (see Figure 5(f)), where G c and G r represent the longitudinal and transverse growth directions of the mine origami unit respectively (G r is parallel to the x-axis, and G c is parallel to the y-axis).
- G c and G r represent the longitudinal and transverse growth directions of the mine origami unit respectively (G r is parallel to the x-axis, and G c is parallel to the y-axis).
- its volume is:
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Abstract
本发明公开了基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,首先由用户交互式输入轮廓曲线及曲面宽度,经扫描形成目标广义圆柱曲面。其次对于平铺在该曲面上的水雷单元大小进行优化统一化。最后按水雷单元的高度与宽度大小不同分多种情况,将水雷单元平铺在目标广义圆柱曲面构成目标网格模型。与传统利用水雷折纸折叠拟合目标广义圆柱曲面不同的是,本发明采用的是未折叠的折纸单元,这一方面避免了产生波纹表面,同时也开发了离散化拟合目标广义圆柱曲面的新形式。
Description
本发明涉及计算机图形学中计算机折纸拟合广义圆柱曲面的领域,具体涉及基于未折叠水雷衍生折纸单元拟合广义圆柱曲面建模方法。
折纸是一种古老的东方折纸艺术,已被用来通过折叠嵌入设计的折痕图案的二维平板材料来构建三维结构,一种满足可展开约束所需的折纸结构可以在平面上制造而无需切割,这对制造业具有吸引力,这种结构甚至可以进行平坦折叠,这对于有效的储存和运输来说很有意义。受益于尺度无关的特征,折纸的一种初步原型可能适用于尺寸范围从纳米级、米级到宏观级的应用。
通过折纸构造给定的柱面是反折纸设计问题的一个简单情况,因为它的高斯曲率为零。这种圆柱形折纸结构也受到了研究人员和工程师的广泛关注,并具有潜在的应用前景,如生物医学可折叠支架、可调谐机械记忆结构、空间可展开臂。从大多数设计方法获得的折纸结构在展开时处于部分折叠状态。即使可以通过增加用于合成折痕图的单位的数量来生成一个与目标表面相对应的更精细的近似,这样的折纸近似显示出一个波纹表面,在一些特定工程场景中的外观设计中,非波纹即光滑表面起着至关重要的作用,例如飞机机翼等,光滑的表面可以让风雨轻松通过物体表面。
发明内容
在本发明中,提出了一种拟合广义圆柱曲面的新方法,该圆柱面由可展开的矩形块组成,其中嵌入了受水弹镶嵌启发的折痕图案。矩形块是一个基本单元,通过优化宽w和 高h统一化。生成的具有这种统一矩形块的折痕图案是可展开的,无需额外切割即可在一块平板材料上制造。为了从2D折痕图案逼近圆柱目标曲面,相邻行之间的共线折痕像一个铰链一样同时地并且适当地折叠,而每一行中的内部折痕保持未折叠的状态。这种近似可以解释为广义圆柱目标曲面的离散版本,并且仍然可以通过折叠所有嵌入的折痕来紧凑地存储。除了w=h的块之外,本发明还研究了w和h之间的其他两个关系,即w<h和w>h,如何影响展开结构的可折叠性,并引入水雷衍生单元以拟合目标广义圆柱曲面,实现对完全折叠后形状的控制。
为了实现上述目的,本发明提供的技术方案如下:
一种基于未折叠水雷衍生折纸单元拟合广义圆柱曲面建模方法,包括以下步骤:
S1:引入利用未折叠水雷单元折纸拟合目标曲面,构建网格模型的相关概念;
S2:用户交互式输入配置信息,控制生成目标曲面;
S3:通过优化统一化水雷单元的大小;
S4:平铺水雷单元拟合目标曲面,构建目标网格模型。
进一步地,上述步骤S1具体为:
S11、目标网格单元构成。目标网格模型由N
r×N
c个水雷单元构成,即N
r行、N
c列。其中方形水雷单元宽度w等于高度h,首先设置平铺在奇数行的水雷的单元为模块B
O,单元内部六条折痕相交于一个内部顶点,其折痕样式如图1(a)所示,其次设置平铺在偶数行的水雷单元模块为B
E,其在模块B
O的基础上,交换水雷单元的左边和右边,折痕样式如图1(b)所示。折纸的单元在每一行交错排列,如图1(c)。
进一步地,上述步骤S2具体包括以下内容:
S21、生成轮廓曲线。用户指定曲线控制点,由该控制点生成NURBS曲线(Non-uniform rational basis spline),即为圆柱曲面的2D轮廓曲线Γ(注意:其中坐标系为空间直角坐标系,分别由x,y和z轴组成。此时,该轮廓曲线位于x-z平面);
S22、生成目标曲面。已知轮廓曲线Γ,由用户输入曲面宽度W,将该曲线沿y轴扫 描距离W,则其扫描路径形成的面即为目标曲面Φ
T,即为最终拟合的圆柱曲面;
进一步地,上述步骤S3具体包括以下内容:
S31、轮廓曲线取样。对于轮廓曲线Γ,在其上取N
r+1个采样点,分别设置顶点为s
i(i=1,…,N
r+1),则轮廓曲线被分为N
r段,设置每一段的长度为h
j(j=1,…,N
r),此时h
j可能会有所不同,这会造成制作难度,为了降低制作难度,此时引入迭代优化以减小高度误差,定义长度残差r
j:
由该误差值建立优化目标函数:
通过该迭代优化过程,可减少水雷单元高度误差,降低制作难度;
进一步地,上述步骤S4具体包括以下内容:
在拟合过程中,无法满足每次目标曲面的宽度正好满足:W=N
cw,所以需要对水雷单元的大小进行灵活的调整。首先设置水雷单元宽度为w=W/N
r,此时w和h有三种可能的比例关系,这三种比例关系造成不同的折叠问题,分别就这三种情况进行模型构建的构建研究:
S41、当w=h,则构成Square(E)型水雷折纸结构:
此时,对于每个未折叠的水雷单元,形状为一个正方形,且奇数行及偶数行的水雷单元样式分别见图2(a)和图2(b)。
对于整个折纸结构平坦折叠状态下,见图2(c)。将该折纸结构放置x-z平面,其 中G
r表示与x轴平行且指向x轴正方向的一个方向向量,当增加水雷单元的列数时,会在G
r方向增加水雷模块,且该折纸结构不会发生自相交,其边框面积为:
S42、当w<h,则构成Tall(T)型水雷折纸结构:
此时,水雷单元结构构建的目标网格结构见图3(c),当折纸模型进行完全折叠时,其最终状态会发生自相交(见图3(d)),则不符合有效配置,需要对其折痕图进行调整。
此时,运用调整后的水雷单元衍生结构构建目标网格结构(见图3(g)),其最终平坦折叠为规则的形状,且不会发生自相交(见图3(h)),此时将其放置在x-z平面,其中G
r与x轴平行,表示增加水雷单元列数而增加面积的方向,其边框面积为:
S43、当w>h时,则构成Short(S)型折纸结构:
由S型水雷折纸构建目标网格结构见图4(c)。根据折纸模型的折叠特性,当网格模型进行平坦折叠时(见图4(d)),在B
O模块中,顶点C
2和C
6会发生相交(其中C
2、C
6见图4(a)),在B
E模块中点对D
1、D
6和D
3、D
8会发生相交(其中D
1、D
6和D
3、D
8见图4(b)),此时需要对水雷单元进行调整:
首先对于奇数行,即B
O模块,将矩形块F
1C
1C
5F
3和C
3F
2F
4C
7分别添加在该水雷单元两侧,其大小为w
1×h,其中(见图4(e)):
其次对于偶数行,即B
E模块,将矩形D
1D
3D
8D
6的大小由w×h变为2w
2×h,其中
并在其左右两侧分别增添宽为w
1,高为
的矩形块G
1D
1D
4G
3、G
3D
4D
6G
5、D
3G
2G
4D
5以及D
5G
4G
6D
8,(见图4(f));
运用调整后的水雷单元衍生结构构建目标网格结构(见图4(g)),尽管无法满足完全平坦折叠,但是该折纸的近似结构能够进行完全折叠并且能够避免自相交(见图4(h));
当该折纸结构进行平坦折叠时,该折纸结构的边界体积为:
V
S=N
c(w-h)A
S
其中A
S是该折纸结构在完全折叠状态下向x-z平面的映射面积:
代入面积得体积为:
S44、由Short型折纸结构,发现其在完全折叠状态下,其与下表面的接触仅由零面积的点及边组成(见图4(h)),这样形成的尖锐端面可能会对下接触面造成损伤,为了解决这个问题,在Short型的基础上进行折痕图的修改:
引入变量:
w′
2=λw
2,其中λ∈(0,1)是缩放因子。
由类型Short的正方形C
1C
3C
7C
5(见图1(a))和D
1D
3D
8D
6(见图1(b))被转换为由缩放参数λ确定的“Tall”型情况,然后通过参照类型“Tall”来分割顶点,以此来修改矩形C
1C
3C
7C
5和D
1D
3D
8D
6中的折痕。
对于奇数行单元模块B
O,参照类型“Tall”(见图3(e)),此时样式如见图5(a)。
引入参数d∈(0,w′
1),将中间部分的左侧移动到左侧,并镜像对称地将中间部分右侧移动到右侧,因此奇数行以及偶数行的水雷单元分别被分割为图5(c)、(d),则水雷单元的宽度被分割为w″
1、w′
2、2d、w′
2和w″
15个部分,其中w″
1为:
w″
1=w′
1-d
此时称该模式为ShortII(SII),其构建的目标网格结构见图5(e)。当该结构处于完全折叠状态时,则属于可粘面的折纸镶嵌即完全折叠的结构存在于折纸近似面所接触的两个平行面之间(见图5(f)),其中G
c和G
r分别表示水雷折纸单元纵向及横向的生长方向(其中G
r与x轴平行,G
c与y轴平行)。此时,对于该类型的完全折叠结构,其体积为:
本发明的有益效果为:
本发明主要利用未折叠水雷折纸衍生结构来拟合具有广义圆柱体特性的目标曲面。水雷是折纸纹样的一种,其纹样单元内部顶点有六个相邻顶点,构成六条边,其中分布情况为四个谷折和两个山折。本发明使用的折纸纹样是基于水雷衍生结构,该结构是基于对构建无波纹表面的考虑而发明的一种衍生折纸结构。利用矩形块构建广义圆柱曲面的单元模块,在矩形块中嵌入水雷单元,且满足平坦折叠约束,离散化构建目标网格模型。根据水雷衍生折纸单元的高度与宽度的比例不同,构建了四种不同类型的折纸结构。本发明开发了一种能够构造光滑表面的新形式,能够在特定应用场景中起至关重要的作用,例如利用该折纸模型构建飞机机翼表面时,光滑的表面能够大大减少空气阻力。
图1为水雷单元的不同样式及折痕图的分布情况;
图2目标表面Φ
T以及Square类型网格模型;
图3 Tall类型衍生结构网格模型;
图4 Short类型衍生结构网格模型;
图5 ShortII类型衍生结构网格模型。
图6本发明实施的流程示意图。
以下将结合附图所示的各实施方式对本发明进行详细描述。但这些实施方式并不限制本发明,本领域的普通技术人员根据这些实施方式所做出的结构、方法、或功能上的变换均包含在本发明的保护范围内。
如图6所示,本发明是基于未折叠水雷衍生折纸单元拟合广义圆柱曲面建模方法,包括以下步骤:
S1:引入利用未折叠水雷单元折纸拟合目标曲面,构建网格模型的相关概念;
S2:用户交互式输入配置信息,控制生成目标曲面;
S3:通过优化统一化水雷单元的大小;
S4:平铺水雷单元拟合目标曲面,构建目标网格模型。
作为本发明的优选实施例,本发明步骤S1具体内容包括:
S11、目标网格单元构成。目标网格模型由N
r×N
c个水雷单元构成,即N
r行、N
c列。其中方形水雷单元宽度w等于高度h,首先设置平铺在奇数行的水雷的单元为模块B
O,单元内部六条折痕相交于一个内部顶点,其折痕样式如图1(a)所示,其次设置平铺在偶数行的水雷单元模块为B
E,其在模块B
O的基础上,交换水雷单元的左边和右边,折痕样式如图1(b)所示。折纸的单元在每一行交错排列,如图1(c)。
作为本发明的优选实施例,上述步骤S2具体包括以下内容:
S21、生成轮廓曲线。用户指定曲线控制点,由该控制点生成NURBS曲线(Non-uniform rational basis spline),即为圆柱曲面的2D轮廓曲线Γ(注意:其中坐标系为空间直角坐标系,分别由x,y和z轴组成。此时,该轮廓曲线位于x-z平面);
S22、生成目标曲面。已知轮廓曲线Γ,由用户输入曲面宽度W,将该曲线沿y轴扫 描距离W,则其扫描路径形成的面即为目标曲面Φ
T,即为最终拟合的圆柱曲面;
作为本发明的优选实施例,本发明步骤S3具体内容包括:
S31、轮廓曲线取样。对于轮廓曲线Γ,在其上取N
r+1个采样点,分别设置顶点为s
i(i=1,…,N
r+1),则轮廓曲线被分为N
r段,设置每一段的长度为h
j(j=1,…,N
r),此时h
j可能会有所不同,这会造成制作难度,为了降低制作难度,此时引入迭代优化以减小高度误差,定义长度残差r
j:
由该误差值建立优化目标函数:
通过该迭代优化过程,可减少水雷单元高度误差,降低制作难度;
作为本发明的优选实施例,所述步骤S4具体包括以下内容:
在拟合过程中,无法满足每次目标曲面的宽度正好满足:W=N
cw,所以需要对水雷单元的大小进行灵活的调整。首先设置水雷单元宽度为w=W/N
r,此时w和h有三种可能的比例关系,这三种比例关系造成不同的折叠问题,分别就这三种情况进行模型构建的构建研究:
S41、当w=h,则构成Square(E)型水雷折纸结构:
此时,对于每个未折叠的水雷单元,其形状为一个正方形,且奇数行及偶数行的水雷单元样式分别见图2(a)和图2(b)。
对于整个折纸结构平坦折叠状态下,见图2(c)。将该折纸结构放置x-z平面,其 中G
r表示与x轴平行且指向x轴正方向的一个方向向量,当增加水雷单元的列数时,会在G
r方向增加水雷模块,且该折纸结构不会发生自相交,其边框面积为:
S42、当w<h,则构成Tall(T)型水雷折纸结构:
此时,水雷单元结构构建的目标网格结构见图3(c),当折纸模型进行完全折叠时,其最终状态会发生自相交(见图3(d)),则不符合有效配置,需要对其折痕图进行调整。
D
1、D
3、D
8为图3(f)中的顶点;
此时,运用调整后的水雷单元衍生结构构建目标网格结构(见图3(g)),其最终平坦折叠为规则的形状,且不会发生自相交(见图3(h)),此时将其放置在x-z平面,其中G
r与x轴平行,表示增加水雷单元列数而增加面积的方向,其边框面积为:
S43、当w>h时,则构成Short(S)型折纸结构:
由S型水雷折纸构建目标网格结构见图4(c)。根据折纸模型的折叠特性,当网格模型进行平坦折叠时(见图4(d)),在B
O模块中,顶点C
2和C
6会发生相交(其中C
2、C
6见图4(a)),在B
E模块中点对D
1、D
6和D
3、D
8会发生相交(其中D
1、D
6和D
3、D
8见图4(b)), 此时需要对水雷单元进行调整:
首先对于奇数行,即B
O模块,将矩形块F
1C
1C
5F
3和C
3F
2F
4C
7分别添加在该水雷单元两侧,其大小为w
1×h,其中(见图4(e)),w
1为宽度数值:
其次对于偶数行,即B
E模块,将矩形D
1D
3D
8D
6的大小由w×h变为2w
2×h,其中长度数值
并在其左右两侧分别增添宽为w
1,高为
的矩形块G
1D
1D
4G
3、G
3D
4D
6G
5、D
3G
2G
4D
5以及D
5G
4G
6D
8,(见图4(f));
运用调整后的水雷单元衍生结构构建目标网格结构(见图4(g)),尽管无法满足完全平坦折叠,但是该折纸的近似结构能够进行完全折叠并且能够避免自相交(见图4(h));
当该折纸结构进行平坦折叠时,该折纸结构的边界体积为:
V
S=N
c(w-h)A
S
其中A
S是该折纸结构在完全折叠状态下向x-z平面的映射面积:
代入面积得体积为:
S44、针对上述Short型折纸结构,发现其在完全折叠状态下,其与下表面的接触仅由零面积的点及边组成(见图4(h)),这样形成的尖锐端面可能会对下接触面造成损伤,为了解决这个问题,本发明在Short型的基础上进行折痕图的修改:
引入变量:
w′
2=λw
2,其中λ∈(0,1)是缩放因子。
由类型Short的正方形C
1C
3C
7C
5(见图1(a))和D
1D
3D
8D
6(见图1(b))被转换为由缩放参数λ确定的“Tall”型情况,然后通过参照类型“Tall”来分割顶点,以此来修改矩形 C
1C
3C
7C
5和D
1D
3D
8D
6中的折痕。
对于奇数行单元模块B
O,参照类型“Tall”(见图3(e)),此时样式如见图5(a)。
引入参数d∈(0,w′
1),将中间部分的左侧移动到左侧,并镜像对称地将中间部分右侧移动到右侧,因此奇数行以及偶数行的水雷单元分别被分割为图5(c)、(d),则水雷单元的宽度被分割为w″
1、w′
2、2d、w′
2和w″
15个部分,其中w″
1为:
w″
1=w′
1-d
此时称该模式为ShortII(SII),其构建的目标网格结构见图5(e)。当该结构处于完全折叠状态时,则该结构属于可粘面的折纸镶嵌即完全折叠的结构存在于折纸近似面所接触的两个平行面之间(见图5(f)),其中G
c和G
r分别表示水雷折纸单元纵向及横向的生长方向(其中G
r与x轴平行,G
c与y轴平行)。此时,对于该类型的完全折叠结构,其体积为:
上文所列出的一系列的详细说明仅仅是针对本发明的可行性实施方式的具体说明,它们并非用以限制本发明的保护范围,凡未脱离本发明技术所创的等效方式或变更均应包含在本发明的保护范围之内。
Claims (10)
- 基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,包括如下步骤:S1:引入利用未折叠水雷单元折纸拟合目标曲面,构建初步的目标网格模型;S2:用户交互式输入配置信息,控制生成目标曲面;S3:通过优化统一化水雷单元的大小;S4:平铺水雷单元拟合目标曲面,构建目标网格模型。
- 根据权利要求1所述的基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,所述S1中构建目标网格模型的具体方法:S11、目标网格模型由N r×N c个水雷单元构成,即N r行、N c列,其中方形水雷单元宽度w等于高度h,首先设置平铺在奇数行的水雷的单元为模块B O,单元内部六条折痕相交于一个内部顶点,其次设置平铺在偶数行的水雷单元模块为B E,其在模块B O的基础上,交换水雷单元的左边和右边,折纸的单元在每一行交错排列。
- 根据权利要求1所述的基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,所述S2的具体方法:S21、生成轮廓曲线:用户指定曲线控制点,由该控制点生成NURBS曲线(Non-uniform rational basis spline),即为圆柱曲面的2D轮廓曲线Γ;S22、生成目标曲面:根据轮廓曲线Γ,由用户输入曲面宽度W,将该曲线沿y轴扫描距离W,则其扫描路径形成的面即为目标曲面Φ T,即为最终拟合的圆柱曲面。
- 根据权利要求1所述的基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,所述S3的具体方法:S31、轮廓曲线取样:对于轮廓曲线Γ,在其上取N r+1个采样点,分别设置顶点为s i(i=1,…,N r+1),则轮廓曲线被分为N r段,设置每一段的长度为h j(j=1,…,N r)。
- 根据权利要求1所述的基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,所述S4的具体方法如下:设置水雷单元宽度为w=W/N r,并针对w和h三种比例关系,分别进行模型构建,当w=h,则构成Square(E)型水雷折纸结构;当w<h,则构成Tall(T)型水雷折纸结构;当w>h时,则构成Short(S)型折纸结构。
- 根据权利要求6所述的基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,当w<h,构成Tall(T)型水雷折纸结构的方法如下:此时,当折纸模型进行完全折叠时,其最终状态会发生自相交,则不符合有效配置,需要对其折痕图进行调整:运用调整后的水雷单元衍生结构构建目标网格结构,其最终平坦折叠为规则的形状,且不会发生自相交,此时将其放置在x-z平面,其中G r与x轴平行,表示增加水雷单元列数而增加面积的方向,其边框面积为:
- 根据权利要求6所述的基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,当w>h时,构成Short(S)型折纸结构的方法如下:根据折纸模型的折叠特性,当网格模型进行平坦折叠时,在B O模块中,顶点C 2和C 6会发生相交,在B E模块中点对D 1、D 6和D 3、D 8会发生相交,此时对水雷单元进行调整:首先对于奇数行,即B O模块,将矩形块F 1C 1C 5F 3和C 3F 2F 4C 7分别添加在该水雷单元两侧,其大小为w 1×h,其中:其次对于偶数行,即B E模块,将矩形D 1D 3D 8D 6的大小由w×h变为2w 2×h,其中 并在其左右两侧分别增添宽为w 1,高为 的矩形块G 1D 1D 4G 3、G 3D 4D 6G 5、D 3G 2G 4D 5以及D 5G 4G 6D 8,当该折纸结构进行平坦折叠时,该折纸结构的边界体积为:V S=N c(w-h)A S其中A S是该折纸结构在完全折叠状态下向x-z平面的映射面积:代入面积得体积为:
- 根据权利要求9所述的基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法,其特征在于,还包括:在Short型的基础上进行折痕图的修改:引入变量:w′ 2=λw 2,其中λ∈(0,1)是缩放因子;由类型Short的正方形C 1C 3C 7C 5和D 1D 3D 8D 6被转换为由缩放参数λ确定的“Tall”型情况,然后通过参照类型“Tall”来分割顶点,以此来修改矩形C 1C 3C 7C 5和D 1D 3D 8D 6中的折痕;引入参数d∈(0,w′ 1),将中间部分的左侧移动到左侧,并镜像对称地将中间部分右侧移动到右侧,奇数行以及偶数行的水雷单元分别被分割,则水雷单元的宽度被分割为w″ 1、w′ 2、2d、w′ 2和w″ 15个部分,其中w″ 1为:w″ 1=w′ 1-d此时称该模式为ShortII(SII),该结构属于可粘面的折纸镶嵌即完全折叠的结构存在于折纸近似面所接触的两个平行面之间,其中G c和G r分别表示水雷折纸单元纵向及横向的生长方向(其中G r与x轴平行,G c与y轴平行),此时,对于该类型的完全折叠结构,其体积为:
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CN113947661A (zh) * | 2021-09-28 | 2022-01-18 | 江苏大学 | 基于水弹折纸衍生结构拟合广义圆柱体曲面的建模方法 |
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CN117252993A (zh) * | 2023-11-16 | 2023-12-19 | 中铁大桥局集团有限公司 | 特征点提取算法的验证方法、装置、电子设备及存储介质 |
CN117252993B (zh) * | 2023-11-16 | 2024-03-26 | 中铁大桥局集团有限公司 | 特征点提取算法的验证方法、装置、电子设备及存储介质 |
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