科普:Planar Quad Mesh

panhao1

其实planar quad mesh也不是什么新鲜玩意 但是最近好像比较火 自从2007的几篇paper之后 最近又出现新的paper 不过无非是补充那几个展平方法或是说明4边形网格是如何省材料 其实大家研究平面4边形网格的真正原因是SQP方法（二次规划法）的运用使得很多建筑师关于paneling曲面的梦想成了现实，再就是4边形更好看，因为离散平面4边形网格（PQmesh）比离散三角面网格更加有规律，加上构件的设计和施工更加合理。总之就是发现最近很多网站都在做PQmesh或是PQmesh trip，就连犀牛也在开发类似的插件，不过要看懂文献还真要点功夫，其中涉及很多计算几何（B-Spline公式）和线性代数的一些算法（如矩阵求和的几种算法） 下面给大家提供1种比较简单的PQmesh算法 其实算法很多 目前最好的算法是求阴影轮廓（属于共轭曲线）的一种算法 希望大家对曲面分割有一个基本认识，至少明白这些分割方法的优劣吧。{:3_55:}

panhao1

1是translate Surface 其实很类似网格放样 很直观的方法 大家可以看看盖里事务所的研究人员的文献（这里的图都是来自2007年发表的一篇文献） 先说基本原理 就是对边平行的空间4边形式平面4边形

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panhao1

这里再补充一些基本常识

1 Triangle Meshes

Triangulation is the most straightforward way to segment a freeform surface into planar panels. Numerous examples exist in architecture. While triangulating a surface might seem a simple task, its application in architecture can be delicate due to numerous requirements:

• visual appearance, meeting design goals
• equally distributed panel sizes and shapes
• alignment of triangulation with features or creases
• minimum and maximum panel sizes
• freeform surfaces consisting of several patches

Usually there are contradictory requirements, leading to the need of carefully balanced solutions. An example can be seen in the project Blob.

A major issue we are concerned with is flexibility: In a design process requirements may be readjusted or even changed in a rigorous way on very short notice. We have developed design tools that allow us to adapt to such changes spontaneously. This also fits nicely into the paradigm of parametric modelling: our design tools and software development skills let us close the gaps in parametric modelling tools.

In the following rendering a triangle mesh featuring almost equilateral triangles is shown. The second rendering reveals how we may achieve this: by letting the incircles of the triangles form a packing.

panhao1

Hybrid Meshes

We have developed tools that allow us to segment freeform surfaces into planar panels using many different appealing patterns. Apart from pure triangle meshes, we offer segmentation into hybrid meshes. These are meshes consisting of patterns of triangles and planar quads, pentagons or hexagons. Panelizations based on hybrid meshes may offer the following advantages for architecture over triangle meshes:

• better translucency,
• less beams and joints,
• less material loss when cutting panels.

At the same time hybrid meshes offer the same design flexibilities as triangle meshes. Hybrid meshes can be used to panelize virtually any freeform surface. The following images show simple examples for generating hybrid mesh patterns consisting of triangles and planar quads. We simply add some diagonals to a given quad mesh. This results in triangles and non-planar quads, which we make planar by minimally moving vertex positions. This technology has been applied to the Museum of Islamic Arts at the Louvre Museum.

panhao1

Single Curved Panels

Single curved surfaces result from bending planar surface sheets without stretching or tearing them. Materials which are frequently used in architecture such as metal sheets, glass and wood can more easily be shaped to resemble single curved surfaces (including special cases like cylinders or cones) than general double curved surfaces. We have developed technologies which allow us to

• design surfaces composed of single curved panels
• cover surfaces with single curve panels, in particular cylindrical or conical panels
• meet restrictions on panel sizes, curvature radii, or other constraints imposed by the manufacturing process
• cover surfaces with panels having a nearly straight planar development (e.g. wooden panels)
• support a single curved skin with a quad-structure of planar beams which exhibits torsion-free nodes

Evolute’s research on these topics is supported by the EU under the IAPP project ARC and performed jointly with the Geometric Modeling and Industrial Geometry research group at Vienna University of Technology and RFR. For details about our research results please have a look at the scientific publications.

panhao1

Ruled Panels

Ruled surfaces contain a continuous set of straight lines; in other words, they are swept by a moving line in space. The presence of straight lines facilitates the realization as an architectural structure. Ruled surface panels are also preferable shapes for certain materials like fibre reinforced concrete, since the cheap and fast process of hot wire cutting can be applied to manufacture moulds for the production of panels. In view of these advantages, we can offer solutions for a wide range of tasks in this area, including the following ones:

• We have developed tools for the design of surfaces which are composed of ruled patches.
• We can cover any input surface by ruled panels, staying as close as possible to the input surface. Depending on the curvature behavior of the design surface we may even achieve an overall smooth surface.
• The underlying optimization algorithms have been developed so that we can satisfy constraints imposed by the manufacturing process.

Evolute’s research on these topics has been supported by the Austrian funding agency FFG under the FIT-IT-Konzept program.

This technology has been applied to the following projects: Cagliari Contemporary Arts Center, Toolpath computation for milling with cylindric cutter, ModuleWorks.

archiglory

感谢分享{:3_64:}

claudemit

被科普了{:3_56:}

麓山小农

建筑的同志，应该多读读数学的教材，虽然可能读的不大懂 但有些可以做尝试性的学习和研究

bryson

潘大可以多给我们说说mesh的东西哦。。。

jackhu88

1# panhao1 楼主，在哪里可以找到文献啊？

panhao1

12# jackhu88 网上遍地是 不过都是e文的 生词较多 常备字典

WENDYJOLIE

很有用~~~~

musofan

6# panhao1 niubility啊 前几天聊天就被科普 当时朋友说gh的精华是tree和mesh我就不懂mesh怎么nb了 现在慢慢发现还是mesh的nb啊 学习学习

claudemit

华南理工好强。。。膜拜下

86220267

华南理工好强。。。膜拜下 claudemit 发表于 2010-10-6 21:45

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潘神是华科的{:3_53:}

harchy

真是科普啊，谢谢啊

panxinzhang

。。。。。厉害。。。

闲池阁

pqmesh v5.

STL2000

。。。。。厉害。。。