&figure&&img src=&https://pic4.zhimg.com/v2-d49c2d65ebab69_b.jpg& data-rawwidth=&943& data-rawheight=&474& class=&origin_image zh-lightbox-thumb& width=&943& data-original=&https://pic4.zhimg.com/v2-d49c2d65ebab69_r.jpg&&&/figure&&p&重做生成城市配景电池的时候，我想在城市周围生成一些山地，但是这样做肯定不如在山地中生成城市合理，所以我需要一个从无到有的生成地形的电池。&/p&&p&之前记得有人做过生成地形的电池的，找了好久没找到，还不如自己做一个。&/p&&figure&&img src=&https://pic4.zhimg.com/v2-b701b19d3fd7ec56a6f393_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&5935& data-rawheight=&863& class=&origin_image zh-lightbox-thumb& width=&5935& data-original=&https://pic4.zhimg.com/v2-b701b19d3fd7ec56a6f393_r.jpg&&&/figure&&p&1.画个矩形曲面，等分它，得到上面的网格点，拍平数据。&/p&&p&拍平的原因是因为行列之间的点并没有什么差别和联系，不用区别运算。&/p&&figure&&img src=&https://pic2.zhimg.com/v2-ce44b1dee0c8411ce40ffdb1807c28ab_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1312& data-rawheight=&768& class=&origin_image zh-lightbox-thumb& width=&1312& data-original=&https://pic2.zhimg.com/v2-ce44b1dee0c8411ce40ffdb1807c28ab_r.jpg&&&/figure&&p&2.做这种地形，我们最先想到的就是点干扰，所以我们直接从这些点之中选几个作为干扰点。&/p&&p&当然使用populate2d也是可以的，方法很多&/p&&p&如果就这样直接进行距离干扰的话，所有的山都会一样高，很假&/p&&figure&&img src=&https://pic3.zhimg.com/v2-5d56fad11bb0b87a63fdb7b5f75da771_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1085& data-rawheight=&516& class=&origin_image zh-lightbox-thumb& width=&1085& data-original=&https://pic3.zhimg.com/v2-5d56fad11bb0b87a63fdb7b5f75da771_r.jpg&&&/figure&&p&所以在进行下一步前先对这些干扰点进行z轴方向上的随机移动，这样可以从全局的角度调整各点的干扰权重&/p&&figure&&img src=&https://pic3.zhimg.com/v2-1ceda336c20c30d76c3fb6_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1062& data-rawheight=&503& class=&origin_image zh-lightbox-thumb& width=&1062& data-original=&https://pic3.zhimg.com/v2-1ceda336c20c30d76c3fb6_r.jpg&&&/figure&&p&虽然效果好了一些，但是感觉各个山头之间都是独立的，没有连绵的感觉&/p&&p&所以我又“随机”连接两个干扰点，将生成的直线也加入干扰物体，这样山脉也有了&/p&&p&使用list length*number slider接list item的方法控制山脉的位置：&/p&&figure&&img src=&https://pic2.zhimg.com/v2-2ebfd9eeca98e_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1312& data-rawheight=&768& class=&origin_image zh-lightbox-thumb& width=&1312& data-original=&https://pic2.zhimg.com/v2-2ebfd9eeca98e_r.jpg&&&/figure&&figure&&img src=&https://pic1.zhimg.com/v2-77d31b4eaa_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1312& data-rawheight=&768& class=&origin_image zh-lightbox-thumb& width=&1312& data-original=&https://pic1.zhimg.com/v2-77d31b4eaa_r.jpg&&&/figure&&p&3.根据距离加函数干扰&/p&&p&因为我想让不是山地的地方完全水平，z轴坐标统统归零，以方便对接之后的电池，所以加了一些控制&/p&&figure&&img src=&https://pic3.zhimg.com/v2-0f27edbff87_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1312& data-rawheight=&768& class=&origin_image zh-lightbox-thumb& width=&1312& data-original=&https://pic3.zhimg.com/v2-0f27edbff87_r.jpg&&&/figure&&p&4.还有一个硬伤，就是地形太光滑了，只好加个随机移动了&/p&&p&但是我不想让水平的地方和山脚处参加移动，所以用dispatch分离出了海拔较低的点&/p&&p&对较高的点一通操作，再用编织还原数据，最后成面&/p&&figure&&img src=&https://pic2.zhimg.com/v2-4d03a394a6bdc3b1994be_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1312& data-rawheight=&768& class=&origin_image zh-lightbox-thumb& width=&1312& data-original=&https://pic2.zhimg.com/v2-4d03a394a6bdc3b1994be_r.jpg&&&/figure&&figure&&img src=&https://pic1.zhimg.com/v2-b0067d2fdf0cac5f4f2c66c59abe929f_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1062& data-rawheight=&497& class=&origin_image zh-lightbox-thumb& width=&1062& data-original=&https://pic1.zhimg.com/v2-b0067d2fdf0cac5f4f2c66c59abe929f_r.jpg&&&/figure&&p&这次依旧是走全参随机无参考物件的浪漫路线，想换方案只有调种子&/p&&figure&&img src=&https://pic2.zhimg.com/v2-6ee75acbebb95_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1098& data-rawheight=&535& class=&origin_image zh-lightbox-thumb& width=&1098& data-original=&https://pic2.zhimg.com/v2-6ee75acbebb95_r.jpg&&&/figure&&p&顺便还能做个好看的窗花（？）&/p&&figure&&img src=&https://pic3.zhimg.com/v2-3f6be051aebfaad254fec_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1312& data-rawheight=&768& class=&origin_image zh-lightbox-thumb& width=&1312& data-original=&https://pic3.zhimg.com/v2-3f6be051aebfaad254fec_r.jpg&&&/figure&&p&链接: &a href=&https://link.zhihu.com/?target=https%3A//pan.baidu.com/s/1nxikE9N& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://&/span&&span class=&visible&&pan.baidu.com/s/1nxikE9&/span&&span class=&invisible&&N&/span&&span class=&ellipsis&&&/span&&/a& 密码: wbea&/p&&p&使用1.0及以下的版本打开 会丢失数个电池，照着电池图补一下就行了┓( ??` )┏&/p&
重做生成城市配景电池的时候，我想在城市周围生成一些山地，但是这样做肯定不如在山地中生成城市合理，所以我需要一个从无到有的生成地形的电池。之前记得有人做过生成地形的电池的，找了好久没找到，还不如自己做一个。1.画个矩形曲面，等分它，得到上面的…
犀牛建模视频教程
&a href=&//link.zhihu.com/?target=http%3A//i.youku.com/u/UMzQwNzI3Njgw& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&优酷网-中国第一视频网,提供视频播放,视频发布,视频搜索&/a&
犀牛建模视频教程
呃，&br&在时间线上看到了这个问题，忍不住上来分享。&br&&b&不知道rhino的建筑全模是个什么样子？&/b&&br&&br&&br&我是否可以理解为，你就想看一下用rhion建出来的全模。&br&那么就得分类说一下。&br&方案阶段的模型可能只是为了推敲方案，体块关系，也可能只是为了一张好看的效果图。建出的模型可能不是你说的全模，也没有必要达到你所想表达的全模吧。&br&那你所说的模型应该是在做施工图的时候，为了验证图纸，或者研究某些细节，所一步一步搭起来的模型。是吧？&br&那么，上周我刚交了一个这样的施工图，正好手中也有自己建的模型，也是对自己有个总结。&br&话不多说，直接上图：&br&&figure&&img src=&https://pic2.zhimg.com/50/7abf1caf2a948c257b01ca_b.jpg& data-rawwidth=&1100& data-rawheight=&595& class=&origin_image zh-lightbox-thumb& width=&1100& data-original=&https://pic2.zhimg.com/50/7abf1caf2a948c257b01ca_r.jpg&&&/figure&&figure&&img src=&https://pic2.zhimg.com/50/ed6c330cf8ec7e4153acbdf1fd5e526e_b.jpg& data-rawwidth=&1146& data-rawheight=&649& class=&origin_image zh-lightbox-thumb& width=&1146& data-original=&https://pic2.zhimg.com/50/ed6c330cf8ec7e4153acbdf1fd5e526e_r.jpg&&&/figure&还是必须说一下禁止一切形式的转载，如果有人看到被转载，还需提醒一下我。谢谢&br&隐去了建筑的外表皮，这就是我参与的一个体育馆的施工图，&br&会有很多这样的模型的，只是商业机密的原因，也懒的发，&br&&br&模型是在画图的过程中，慢慢建起来的，所有东西也是照着图纸来的。当然，最耗时的也是他的外表皮，钢结构及屋顶的搭建，用犀牛建模也就是为了验证那些地方会不会出错。&br&施工图的立面和剖面上的钢结构背景，基本实在模型里make2D的。&br&&br&建模的工程中，自己还是时不时地会去和su比较。去比较做这类有弧线的建筑时，rhion到底有什么优势？su到底有什么局限。&br&&br&所以，在做栏杆的建模时，我想通了。&br&&figure&&img src=&https://pic2.zhimg.com/50/258e00cc5ad418b1da1d5587e6cad0da_b.jpg& data-rawwidth=&1146& data-rawheight=&649& class=&origin_image zh-lightbox-thumb& width=&1146& data-original=&https://pic2.zhimg.com/50/258e00cc5ad418b1da1d5587e6cad0da_r.jpg&&&/figure&&br&&figure&&img src=&https://pic1.zhimg.com/50/347c9f6ae_b.jpg& data-rawwidth=&1146& data-rawheight=&649& class=&origin_image zh-lightbox-thumb& width=&1146& data-original=&https://pic1.zhimg.com/50/347c9f6ae_r.jpg&&&/figure&这种一圈栏杆的定位。su中的圆，只能增加圆边线的精确度，但可能做出来还是不准确。犀牛中的误差，就会比su小很多，在里面建好的栏杆，我可以直接导成dwg，直接放到施工图里&br&&figure&&img src=&https://pic1.zhimg.com/50/7b36bea81f_b.jpg& data-rawwidth=&1145& data-rawheight=&643& class=&origin_image zh-lightbox-thumb& width=&1145& data-original=&https://pic1.zhimg.com/50/7b36bea81f_r.jpg&&&/figure&&figure&&img src=&https://pic3.zhimg.com/50/98c4dc62fec739bae8d3cfc601c84578_b.jpg& data-rawwidth=&670& data-rawheight=&589& class=&origin_image zh-lightbox-thumb& width=&670& data-original=&https://pic3.zhimg.com/50/98c4dc62fec739bae8d3cfc601c84578_r.jpg&&&/figure&&figure&&img src=&https://pic3.zhimg.com/50/0dbba1eebbeb5f747349_b.jpg& data-rawwidth=&1146& data-rawheight=&649& class=&origin_image zh-lightbox-thumb& width=&1146& data-original=&https://pic3.zhimg.com/50/0dbba1eebbeb5f747349_r.jpg&&&/figure&每一根栏杆的位置，当时还都是定的2度左右一个，这样从场地看上来，会感觉他们都冲着，不过后来深化的时候，审图意见说栏杆间距太远，还是给改的面目全非了。&br&&br&利益相关：&a href=&//link.zhihu.com/?target=http%3A//www.sketchupbar.com/forum.php& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&sketchupbar&/a& 版主一枚&p&&a href=&//link.zhihu.com/?target=http%3A//weixin.qq.com/r/3UP749rEJ4dYrZ6A9xbP& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&weixin.qq.com/r/3UP749r&/span&&span class=&invisible&&EJ4dYrZ6A9xbP&/span&&span class=&ellipsis&&&/span&&/a& (二维码自动识别)&/p&&br&二维码是我的公众号，&br&&b&谢绝转载&/b&
呃， 在时间线上看到了这个问题，忍不住上来分享。 不知道rhino的建筑全模是个什么样子？ 我是否可以理解为，你就想看一下用rhion建出来的全模。 那么就得分类说一下。 方案阶段的模型可能只是为了推敲方案，体块关系，也可能只是为了一张好看的效果图。建…
我来我来，这个问题强答一发好了，刚刚结束的一学期才用rhino结合revit出了一次全模。&br&&br&答主作为学生用rhino也有一年多两年了，但是以前一直是用的rhino来辅助su进行曲面建模,或者就是做个看起来炫酷实际上没什么太大用处的表皮，捏个看起来爽一点的形态，表皮空间形态对位也常常不经过深究，感觉就只是注重表现却不是认真做建筑的啊！！（摔！）。没有利用rhino 这个cad软件的核心去辅助深化。总有一种暴殄天物的负罪感。。&br&&br&所以这次终于就决定不能再这么low&br&原本是想尽力用这套流程做个接近扩初程度的图纸的。。但是无奈revit并没有很熟练。而且好多工程工艺，细部构造，实际情况并不了解。所以虽然说是全模，但也就停留在方案阶段。&br&&br&说说这次做设计的流程，希望能给新手一点启示，也希望老司机给提出一些意见，欢迎各位拍砖。&br&先给各流程用到的软件：&br&前期（调研， 建立地形，前期分析）---rhino,grasshopper &br&中期（方案构思，确定，生形，排平面等）---rhino，grasshopper&br&后期（梁柱楼板门窗梯，剖面，全模型, 出图）---rhino，revit&br&表现（排版，表现图，文本）---rhino, su,adobe 系列&br&所以，每一步都有rhino，（以前一直没觉得rhino这么强大来着）&br&&br&脑补了一下按照整个流程讲下来的工程量。。。。太可怕&br&所以这里就简单果断的提出一些核心的思想来配图说明。&br&&br&&b&要诀之一----- rhino就是加强版的cad!!!&/b&&br&前期的地形cad导入rhino进行清理场地 图层排布 效率非常高 因为rhino有强大的图层管理能力和选取能力&br&&figure&&img src=&https://pic4.zhimg.com/50/dd9e595d1b8a04d312db3fb_b.jpg& data-rawwidth=&1328& data-rawheight=&74& class=&origin_image zh-lightbox-thumb& width=&1328& data-original=&https://pic4.zhimg.com/50/dd9e595d1b8a04d312db3fb_r.jpg&&&/figure&&br&而且还可以从google上利用插件把地形给搞下来（楼主没下插件，直接用的su的添加位置导出来再进的rhino）&br& 用指令: _Contour&br&&figure&&img src=&https://pic2.zhimg.com/50/cbc2f39aee26bd7d474ce7d_b.jpg& data-rawwidth=&52& data-rawheight=&40& class=&content_image& width=&52&&&/figure& 来生成等高线之类的东西,真的是非常方便。&br&&br&下面就是基本清理好的场地&br&&br&&figure&&img src=&https://pic4.zhimg.com/50/83e2f25ebfa902ae52c3ec_b.jpg& data-rawwidth=&1662& data-rawheight=&787& class=&origin_image zh-lightbox-thumb& width=&1662& data-original=&https://pic4.zhimg.com/50/83e2f25ebfa902ae52c3ec_r.jpg&&&/figure&&br&&br&&br&&br&像总平平面这些也是都可以用rhino手工撸，跟cad一样一样的。而且更快更优。&br&&figure&&img src=&https://pic3.zhimg.com/50/b09f3c93ed13cbe5efb8e4c_b.jpg& data-rawwidth=&878& data-rawheight=&750& class=&origin_image zh-lightbox-thumb& width=&878& data-original=&https://pic3.zhimg.com/50/b09f3c93ed13cbe5efb8e4c_r.jpg&&&/figure&&br&&br&&b&要诀之二----- 结合gh 分析好用 逼格也不低&/b&&br&做完基本的场地，如果你要用rhino来做一做前期分析，那简直是再方便不过了。因为cad的数据，地形的数据都在rhino里了。如果熟悉grasshopper可以分分钟（说起来是分分钟，真要认真做思路和想算法得想很久）出一个比较理性的分析结果。如果你会代码，会爬数据，那恭喜你，分析结果又可以更上一层楼。&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/d2f5a7ac4f84b066e9da02d17a669b92_b.jpg& data-rawwidth=&1403& data-rawheight=&776& class=&origin_image zh-lightbox-thumb& width=&1403& data-original=&https://pic1.zhimg.com/50/d2f5a7ac4f84b066e9da02d17a669b92_r.jpg&&&/figure&&br&&br&上面就是答主结合地形和人流（人流量是结合区位YY+少量照片）&br& 利用gh kangaroo和遗传算法，装**算的，使人分布最广的 商业点安排。&br&&br&而且分析出结果以后，如果懒癌发作，不想过多的后期处理，想直接上板，print就好了，简直不能更方便！&br&&figure&&img src=&https://pic4.zhimg.com/50/98f4e8a3d4e2fefecef5_b.jpg& data-rawwidth=&1556& data-rawheight=&772& class=&origin_image zh-lightbox-thumb& width=&1556& data-original=&https://pic4.zhimg.com/50/98f4e8a3d4e2fefecef5_r.jpg&&&/figure&&br&&br&这是导出图片后大致的效果，答主觉得直接打印的效果不必su差&br&&figure&&img src=&https://pic4.zhimg.com/50/a084f766743cfffebddd3_b.jpg& data-rawwidth=&1919& data-rawheight=&375& class=&origin_image zh-lightbox-thumb& width=&1919& data-original=&https://pic4.zhimg.com/50/a084f766743cfffebddd3_r.jpg&&&/figure&&br&&br&&br&&b&要诀三---rhino结合gh可以做方案的探讨 ，快速准确调整得到结果&/b&&br&&br&&br&有很多人觉得rhino的体块推敲速度慢，其实只是没有习惯它的操作，这个回答可以在很大程度上解决这个问题&a href=&https://www.zhihu.com/question/& class=&internal&&Rhino 有什么奇技淫巧？ - 建筑设计&/a&&br&下图是答主对于基本形体的一些平面和形体的探讨&br&&figure&&img src=&https://pic4.zhimg.com/50/9bfe14c6dd_b.jpg& data-rawwidth=&1396& data-rawheight=&553& class=&origin_image zh-lightbox-thumb& width=&1396& data-original=&https://pic4.zhimg.com/50/9bfe14c6dd_r.jpg&&&/figure&&br&最终 为了结构的合理性（至少在答主本人和老师的忍受范围之内）因为毕竟答主希望自己的东西的可实施性得保证选择了一个形体做结构研究&br&&br&结构研究这一步就是gh大显身手的时候了，因为只要你以你的逻辑将结构模型和体块模型做出来就可以得到直观的反馈，这比cad的“感觉成立“要准确，比其它方式做模型研究要有效率，因为你可以对结构进行实时监测和修改并与老师讨论，从这个方面来讲是绝对必要且有效率的，这很大程度呈现了方案的合理性，答主觉得我很难相信二维的图纸。&br&这种东西必须完全对上，空间合理，才能有底气。&br&&figure&&img src=&https://pic4.zhimg.com/50/25cc0eda5a73eaf990c020a7c50bbe6c_b.jpg& data-rawwidth=&1337& data-rawheight=&677& class=&origin_image zh-lightbox-thumb& width=&1337& data-original=&https://pic4.zhimg.com/50/25cc0eda5a73eaf990c020a7c50bbe6c_r.jpg&&&/figure&&br&&br&&b&要诀四---和bim软件对接，再深化处理&/b&&br&对于小型一点的作业或者项目，答主觉得rhino完全有能力从头到尾处理好，包括到门窗楼梯等等的绘制出图。&br&强推蒯爷的回答&br&&a href=&https://www.zhihu.com/question//answer/& class=&internal&&习惯了 su 的建模思路，用 rhino 做方方正正的建筑如何提高效率？ - 蒯鼎的回答&/a&&br&这里同时推荐下原厂大大jessen的课程，讲的非常详细且实用。&br&&a href=&//link.zhihu.com/?target=http%3A//ke.qq.com/course/124710& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&ke.qq.com/course/124710&/span&&span class=&invisible&&&/span&&/a&&br&然而这次答主的综合体并不是完全规则的，办公楼每层都有微小偏转，而且门窗墙体等等构件数量巨大，完全交给rhino这个cad软件是效率是比较低的。所以，我利用grasshopper的 humingbird和gerevit和各种文件格式完成了痛苦的rhino和revit对接（不得不说这一步对于rhino gh revit的熟练度均有一定要求而且有很多意想不到的问题）&br&然后revit的事情了 坑坑得做吧 把你觉得有用的地方全做出来。&br&&br&&br&&br&下面是基于gh的数据和rhino模型在revit中重建的模型（表皮是最后导入rhino做的）&br&&figure&&img src=&https://pic2.zhimg.com/50/63cd1d9f0c_b.jpg& data-rawwidth=&1024& data-rawheight=&804& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic2.zhimg.com/50/63cd1d9f0c_r.jpg&&&/figure&&br&有了这东西 平立剖就可以比较方便的导出了&br&&figure&&img src=&https://pic4.zhimg.com/50/12f1a049cf760b5f6a90e_b.jpg& data-rawwidth=&297& data-rawheight=&345& class=&content_image& width=&297&&&/figure& 然而答主的revit并不是非常精通，它自动导出的图很难达到建筑学生所需求的标准（line weight之类的）所以答主又把这些图纸重新在rhino里进行了加工。（再说一次，rhino处理技术图纸真的很爽）&br&&figure&&img src=&https://pic1.zhimg.com/50/3f03a002624eba26ece8a10_b.jpg& data-rawwidth=&1294& data-rawheight=&655& class=&origin_image zh-lightbox-thumb& width=&1294& data-original=&https://pic1.zhimg.com/50/3f03a002624eba26ece8a10_r.jpg&&&/figure&&br&而且这个时候你就可以把rvt的模型重新导回rhino，方便调整全模型并添加表现元素&br&而且一些rhino容易做的工作就交给rhino，比如幕墙表皮。&br&&br&&b&这样，你就能得到一个梁柱板楼梯要什么有什么的全模型了&/b&（至少基本建筑构件是全的）&br&&figure&&img src=&https://pic3.zhimg.com/50/56ac24a403aa34bbc6cc24b098062aed_b.jpg& data-rawwidth=&1042& data-rawheight=&780& class=&origin_image zh-lightbox-thumb& width=&1042& data-original=&https://pic3.zhimg.com/50/56ac24a403aa34bbc6cc24b098062aed_r.jpg&&&/figure&剩下的什么小人小桌子 就随便你加啦！怎么好看怎么漂亮怎么来！&br&&br&再啰嗦一句：rhino的显示模式如果经过调整一点也不必su差，&br&&figure&&img src=&https://pic1.zhimg.com/50/92a4ffd48d3d377fc1b2_b.jpg& data-rawwidth=&258& data-rawheight=&619& class=&content_image& width=&258&&&/figure&&figure&&img src=&https://pic4.zhimg.com/50/749b04baf08afe35ecf924_b.jpg& data-rawwidth=&554& data-rawheight=&486& class=&origin_image zh-lightbox-thumb& width=&554& data-original=&https://pic4.zhimg.com/50/749b04baf08afe35ecf924_r.jpg&&&/figure&另外 附材质的时候如果你有兴趣可以好好研究下rhino的材质贴图&br&好多东西加上有奇效&figure&&img src=&https://pic3.zhimg.com/50/0b24b402ab6c_b.jpg& data-rawwidth=&845& data-rawheight=&442& class=&origin_image zh-lightbox-thumb& width=&845& data-original=&https://pic3.zhimg.com/50/0b24b402ab6c_r.jpg&&&/figure&&br&&br&这次出图时间太紧张了，因为答主发誓这学期绝对不通宵，到最后一张图都没渲染，全是裸导，再随便加点配景就出图了，效果凑合能看吧。&br&&br&&figure&&img src=&https://pic1.zhimg.com/50/e768cab4a75ce462b4dbdbf2_b.jpg& data-rawwidth=&886& data-rawheight=&988& class=&origin_image zh-lightbox-thumb& width=&886& data-original=&https://pic1.zhimg.com/50/e768cab4a75ce462b4dbdbf2_r.jpg&&&/figure&&br&嗯嗯，其实如果要深入的说，每一个部分都可以扯好多东西，但是毕竟答主能力有限，又比较懒，所以就给出大致的思路和一些核心思维了，各位看官见笑了~&br&另外，十分感谢蒯鼎大神，他100个运算器和100个犀牛命令 他的这本小手册在设计过程中帮了很大的忙&br&这里再自来水一波公众号&br&&p&&a href=&//link.zhihu.com/?target=http%3A//qm.qq.com/cgi-bin/qm/qr%3Fk%3DyBstZqso57N-dpohsMqWAB__F_X7KM0X& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&qm.qq.com/cgi-bin/qm/qr?&/span&&span class=&invisible&&k=yBstZqso57N-dpohsMqWAB__F_X7KM0X&/span&&span class=&ellipsis&&&/span&&/a& (二维码自动识别)&/p&&br&好啦，就这样吧~
我来我来，这个问题强答一发好了，刚刚结束的一学期才用rhino结合revit出了一次全模。 答主作为学生用rhino也有一年多两年了，但是以前一直是用的rhino来辅助su进行曲面建模,或者就是做个看起来炫酷实际上没什么太大用处的表皮，捏个看起来爽一点的形态，表…
&p&我回来答题了。&/p&&p&首先是最老的入门书两本。&/p&&figure&&img src=&https://pic3.zhimg.com/50/c7b288425dedfb_b.jpg& data-caption=&& data-rawwidth=&213& data-rawheight=&232& class=&content_image& width=&213&&&/figure&&p&&br&&/p&&p&这是我看的第一本书~~~差强人意，但是可以看看。&/p&&figure&&img src=&https://pic4.zhimg.com/50/a880bfaa6ec2_b.jpg& data-caption=&& data-rawwidth=&232& data-rawheight=&234& class=&content_image& width=&232&&&/figure&&p&王老师的教材适合入门，新手可以学到挺多东西的。但是这本书确实老了。&/p&&figure&&img src=&https://pic3.zhimg.com/50/ce5a40b841a8ba5ac9792_b.jpg& data-caption=&& data-rawwidth=&134& data-rawheight=&173& class=&content_image& width=&134&&&/figure&&p&&br&&/p&&p&这本书很炫酷，可以看看，但是没啥内容。&/p&&figure&&img src=&https://pic3.zhimg.com/50/016fa5c2bb127_b.jpg& data-caption=&& data-rawwidth=&237& data-rawheight=&275& class=&content_image& width=&237&&&/figure&&p&这个讲了Netlogo，可以学学，书比较老了&/p&&figure&&img src=&https://pic4.zhimg.com/50/303ad6f30c2bbed3e8afab7cd071c05c_b.jpg& data-caption=&& data-rawwidth=&220& data-rawheight=&260& class=&content_image& width=&220&&&/figure&&p&&br&&/p&&p&这本书教你怎么玩折纸（折板）可以学学。。。&/p&&figure&&img src=&https://pic2.zhimg.com/50/a2785fce05ff2b39544faccc0240bf04_b.jpg& data-caption=&& data-rawwidth=&276& data-rawheight=&350& class=&content_image& width=&276&&&/figure&&p&&br&&/p&&p&这本书也是挺好的教程！~~&/p&&p&&br&&/p&&figure&&img src=&https://pic1.zhimg.com/50/675dabae0d_b.jpg& data-caption=&& data-rawwidth=&262& data-rawheight=&360& class=&content_image& width=&262&&&/figure&&p&&br&&/p&&p&这本书非常不错~~~强力推荐~~~~里面有很多思路，但是都是尝试，比如其中的智能体自动规划程序就没解决复杂形体碰撞问题。&/p&&figure&&img src=&https://pic3.zhimg.com/50/c4ff43e9a9a042cde472d4d7_b.jpg& data-caption=&& data-rawwidth=&253& data-rawheight=&328& class=&content_image& width=&253&&&/figure&&p&&br&&/p&&p&这本书也挺好的~~~一定要学&/p&&p&&br&&/p&&figure&&img src=&https://pic3.zhimg.com/50/24393eca9c5ee935ea5156_b.jpg& data-caption=&& data-rawwidth=&298& data-rawheight=&346& class=&content_image& width=&298&&&/figure&&p&&br&&/p&&p&补充一本新书。。。适合要开始学习的新人。&/p&&p&&br&&/p&&p&&br&&/p&&p&再加一本有力度的书~~~远神的新书！！！！&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-3baf4bffbbd980c4cbc0ff1c0d28997a_b.jpg& data-caption=&& data-rawwidth=&600& data-rawheight=&742& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic4.zhimg.com/50/v2-3baf4bffbbd980c4cbc0ff1c0d28997a_r.jpg&&&/figure&&p&毕竟是NCF的创始人~~~大家一定要支持哇~~~~&/p&&p&&a href=&//link.zhihu.com/?target=http%3A//www.ncf-china.com/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&NCF参数化建筑论坛&/a&&/p&&p&这本书手感像毛爷爷&/p&&p&&br&&/p&&p&一时也想不起来太多~~&br&然后再来几本PDF发个网盘，大家不要举报啊！&/p&&p&链接：&a href=&//link.zhihu.com/?target=http%3A//pan.baidu.com/s/1kVmU3R5& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&pan.baidu.com/s/1kVmU3R&/span&&span class=&invisible&&5&/span&&span class=&ellipsis&&&/span&&/a& 密码：o06g&/p&&figure&&img src=&https://pic3.zhimg.com/50/c2edda1bd93dbaf0a4c1e385b1e783ec_b.jpg& data-caption=&& data-rawwidth=&429& data-rawheight=&249& class=&origin_image zh-lightbox-thumb& width=&429& data-original=&https://pic3.zhimg.com/50/c2edda1bd93dbaf0a4c1e385b1e783ec_r.jpg&&&/figure&&p&&br&&/p&&p&这里面好几本书都是很不错的~~~~~看完觉得好记得买纸质书~~~&/p&&p&&br&&/p&&p&我把视频删了。。。各位需要的话请灵活求助马云爸爸&/p&&p&就这样吧。。。以后想起啥好的再补，加班去了~~~~~~~&/p&
我回来答题了。首先是最老的入门书两本。 这是我看的第一本书~~~差强人意，但是可以看看。王老师的教材适合入门，新手可以学到挺多东西的。但是这本书确实老了。 这本书很炫酷，可以看看，但是没啥内容。这个讲了Netlogo，可以学学，书比较老了 这本书教你…
&figure&&img src=&https://pic3.zhimg.com/v2-db5d9eba8a2f5_b.jpg& data-rawwidth=&998& data-rawheight=&160& class=&origin_image zh-lightbox-thumb& width=&998& data-original=&https://pic3.zhimg.com/v2-db5d9eba8a2f5_r.jpg&&&/figure&&h2&前言&/h2&&p&这是我们十分钟paper的第一篇，「十分钟paper」系列是模仿Youtube上的Two Minute Papers 的一个读书报告栏目，GraphiCon的小伙伴们会用简洁有趣的文字，生动形象的图片讲解一些大家感兴趣的paper。本栏目长期接受投稿，如果你想参与进来，请按文末的信息联系我们。&/p&&p&&i&注：本篇论文地址：&a href=&https://link.zhihu.com/?target=http%3A//irc.cs.sdu.edu.cn/CFS/& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&irc.cs.sdu.edu.cn/CFS/&/span&&span class=&invisible&&&/span&&/a&&/i&&/p&&br&&h1&你的模型也许还可以再抢救一下&/h1&&p&作者：GraphiCon - 孙钊，王士玮&/p&&h2&3D打印&/h2&&p&3D打印是近些年兴起的增材制造工艺。相较于传统的模具与减材制造，从想法到产品雏形，它成本低，效率高，非常适合人们去发挥自己的创意。而在网上往往可以看到优秀的3D打印作品：&/p&&br&&br&&p&&em&&figure&&img src=&https://pic4.zhimg.com/v2-647f9a7c1bc958e1e6080_b.jpg& data-rawwidth=&780& data-rawheight=&518& class=&origin_image zh-lightbox-thumb& width=&780& data-original=&https://pic4.zhimg.com/v2-647f9a7c1bc958e1e6080_r.jpg&&&/figure&source: Reuters&/em&&br&&em&&figure&&img src=&https://pic3.zhimg.com/v2-4fd97a299feabb277955dba304ad655b_b.jpg& data-rawwidth=&775& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&775& data-original=&https://pic3.zhimg.com/v2-4fd97a299feabb277955dba304ad655b_r.jpg&&&/figure&source: avax.news&/em&&br&&em&&figure&&img src=&https://pic2.zhimg.com/v2-d733ff004be29663ba55c_b.jpg& data-rawwidth=&900& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&900& data-original=&https://pic2.zhimg.com/v2-d733ff004be29663ba55c_r.jpg&&&/figure&source: &a href=&https://link.zhihu.com/?target=http%3A//howstufworks.com/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&howstufworks.com&/a&&/em&&br&&em&&figure&&img src=&https://pic2.zhimg.com/v2-01d17ed1bfc0ae6b9a3dc346f659a41a_b.jpg& data-rawwidth=&900& data-rawheight=&595& class=&origin_image zh-lightbox-thumb& width=&900& data-original=&https://pic2.zhimg.com/v2-01d17ed1bfc0ae6b9a3dc346f659a41a_r.jpg&&&/figure&source: &a href=&https://link.zhihu.com/?target=http%3A//nydailynews.com/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&nydailynews.com&/a&&/em&&/p&&br&&br&&h2&3D打印机&/h2&&br&&p&尽管3D打印有熔融沉积，激光烧结，熔化成型等不同工艺类型。但最常见的，（我们也用得起的）是熔融沉积打印机（FDM打印机）。这种打印机的通过喷出塑料丝快速熔融，沉积成模型实体, 打印一个10cm高的模型, 平均起来也要7、8个小时左右。虽然感觉上很慢（相比起模具制造,不知道快了多少）, 但想法满天飞的图形控们, 已经感觉打通了数字世界和原子世界的壁垒好吗！（嗯，就差二次元壁垒等待我们去征服了。）&/p&&br&&p&&em&&figure&&img src=&https://pic4.zhimg.com/v2-5d53debe2c57fe43ab86_b.jpg& data-rawwidth=&1080& data-rawheight=&720& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/v2-5d53debe2c57fe43ab86_r.jpg&&&/figure&source: Makerbot&/em&&/p&&br&&br&&h2&FDM 打印机的缺陷&/h2&&br&&p&但梦想的丰满照进现实就显得如此骨感, 比如: 当你兴冲冲地拿着辛苦了一周的模型，找到了一台最常见的FDM 3d打印机，要它打印这个：&/p&&br&&p&&em&&figure&&img src=&https://pic4.zhimg.com/v2-b7b59ecaeded_b.jpg& data-rawwidth=&795& data-rawheight=&966& class=&origin_image zh-lightbox-thumb& width=&795& data-original=&https://pic4.zhimg.com/v2-b7b59ecaeded_r.jpg&&&/figure&source: &a href=&https://link.zhihu.com/?target=http%3A//nipic.com/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&nipic.com&/a&&/em&&br&FDM打印机：&/p&&br&&p&&em&&figure&&img src=&https://pic3.zhimg.com/v2-314be9a1964aab4ae2f5481b_b.jpg& data-rawwidth=&730& data-rawheight=&440& class=&origin_image zh-lightbox-thumb& width=&730& data-original=&https://pic3.zhimg.com/v2-314be9a1964aab4ae2f5481b_r.jpg&&&/figure&source: &a href=&https://link.zhihu.com/?target=http%3A//weibo.com/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&weibo.com&/a&&/em&&br&打印机在打印模型时，遵循切片分层再分层打印的原理。而每一层的打印，喷头都要遵循工具路径来移动，最终填满整个单层区域。问题就出在这里。&br&大部分FDM打印机的分层打印基于Zigzag曲线（锯齿形或之字形）：&/p&&br&&figure&&img src=&https://pic3.zhimg.com/v2-1d3fff311d4a362a6f43a3b084179fce_b.jpg& data-rawwidth=&395& data-rawheight=&408& class=&content_image& width=&395&&&/figure&&figure&&img src=&https://pic1.zhimg.com/v2-d32d2c60bc4d3b4ec4fd52_b.jpg& data-rawwidth=&627& data-rawheight=&684& class=&origin_image zh-lightbox-thumb& width=&627& data-original=&https://pic1.zhimg.com/v2-d32d2c60bc4d3b4ec4fd52_r.jpg&&&/figure&&br&&br&&p&这种“不撞南墙不回头”的曲线碰到洞，尤其是边缘还不规则的众多洞，折返次数会大大增加，其效果也变得不堪入目:&/p&&br&&figure&&img src=&https://pic4.zhimg.com/v2-bddccf988fcddb7a32d5deb15c298bdd_b.jpg& data-rawwidth=&1024& data-rawheight=&576& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic4.zhimg.com/v2-bddccf988fcddb7a32d5deb15c298bdd_r.jpg&&&/figure&&br&&p&你能忍受兴冲冲地去打印, 满心欢喜地等待7、8个小时, 最后只能拿到一个废渣一样的东西吗？&/p&&br&&p&而面对激增的急转弯个数，FDM打印机表示：&/p&&br&&figure&&img src=&https://pic1.zhimg.com/v2-f36190f41dafaf960a467b5dc0966dc9_b.jpg& data-rawwidth=&245& data-rawheight=&300& class=&content_image& width=&245&&&/figure&&br&&p&FDM打印机在处理高亏格的模型总是少了根筋。对于我们大开的脑洞而言，FDM打印机的表现简直就是呆萌。如果你碰到了这种情况，先别急着删文件，或许你的模型还可以抢救一下。最新的Siggraph 2016文章&a href=&https://link.zhihu.com/?target=http%3A//irc.cs.sdu.edu.cn/CFS/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&[1]&/a&提供了新的打印技巧：连续费马螺旋线（Connected Fermat Spirals）。&/p&&br&&br&&h2&皮埃尔·德·费马&/h2&&br&&p&这里就不得不说一下费马，传说中的业（zhuan）余(ye)数(wa)学(keng)家之王。&/p&&br&&figure&&img src=&https://pic4.zhimg.com/v2-ddc1fab7ed_b.jpg& data-rawwidth=&220& data-rawheight=&294& class=&content_image& width=&220&&&/figure&&br&&p&如果你打开某某百科，搜索皮埃尔o德o费马，你会发现一个奇怪的生平：&/p&&br&&blockquote&&ul&&li&1601年出生于法国，父亲是皮货商（大土豪），母亲出身穿袍贵族，是名副其实的富二代。&/li&&li&高中毕业后，富二代听从了父亲的推荐，欣然去大学学习法律（喵喵喵？不应该抱着柱子大喊我要学数学吗？）。&/li&&li&大学还没毕业，富二代就通过买官得到了律师和参议员的职位。&/li&&li&1631年，费马返回家乡，当上了图卢兹议会议员，从此逐年提升，于1642年进入最高刑事法庭，1646年升任首席发言人。&/li&&li&费马是个廉洁的好官，获得了人们的信任和称赞。&/li&&/ul&&/blockquote&&p&没啦？这就没啦？数学嘞？？？你让那些专职数学家情何以堪？&/p&&br&&p&然而费马作为24k纯业余数学家却是数学家多如繁星的17世纪最多产的数学家。&/p&&br&&blockquote&&ul&&li&在解析几何上，费马做了开创性的基础工作，从方程出发决定点线面曲线曲面。&/li&&li&微积分上，费马建立了求切线、极值的方法以及定积分方法。&/li&&li&概率论上引入了随机变量与数学期望，这是最基础的部分。&/li&&li&光学上提出了最小作用原理，这引出了变分法求函数极值的思想，直接导致了拉格朗日的成就。&/li&&li&数论上，费马小定理、费马大定理为众人所熟知，此外还有众多数论工作为人称道。&/li&&/ul&&/blockquote&&br&&h2&费马与猜想&/h2&&br&&p&费马作为数学家真可以说是前无古人的妙人：喜欢猜想。这些猜想前前后后分别被后人所证明，而费马大定理，也即费马最后的猜想，更是一段佳话。&/p&&br&&blockquote&&ul&&li&费马在阅读《算术》时，在书中空白处写下了这一猜想，并表明已想出一美妙的证法，但是书页空白处太小，写不下。无论费马是否想出了证明，这个猜想给数学带来的推动是毋庸置疑的。&/li&&li&自1753年起，众多数学家接力证明一些特殊情况。&/li&&li&1844年，理想素数理论证明了100以内的情况。&/li&&li&1922年，莫德尔猜想给出了代数曲线角度的新方法。&/li&&li&1983年莫德尔猜想得到证明，打开了新局面。&/li&&li&1986年，安德鲁o怀尔斯依赖于谷山丰猜想、肯o里贝特在弗雷命题上的工作、岩泽理论—弗莱切方法最终给出了费马大定理的证明。&/li&&/ul&&/blockquote&&p&历时三个世纪，证明的接力终于结束，而理论数学的发展，已然沧海桑田。&/p&&br&&br&&h2&费马螺旋线&/h2&&br&&p&&em&以下图片均来自&a href=&https://link.zhihu.com/?target=http%3A//irc.cs.sdu.edu.cn/CFS/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&[1]&/a&&/em&&br&扯了那么多, 只是为了表达我们对费马大神滔滔不绝的景仰之情, 现在终于可以回归正题讲讲费马螺线了。费马螺线，是费马在解析几何研究中提出的以代数方程定义的新曲线。表达式为&img src=&https://www.zhihu.com/equation?tex=r%5En%3Da%CE%B8& alt=&r^n=aθ& eeimg=&1&&。连续费马螺旋线（Connected Fermat Spirals）,下文简称CFS，它是这样的曲线：&/p&&br&&figure&&img src=&https://pic3.zhimg.com/v2-f3dec2cbf321d64846f0_b.jpg& data-rawwidth=&300& data-rawheight=&301& class=&content_image& width=&300&&&/figure&&br&&p&CFS具有两个交错的子螺旋，一个向内和一个向外。CFS有着非常利于3d打印的性质：&/p&&br&&blockquote&&ul&&li&1.符合区域边界；&/li&&li&2.多个CFS是可以连续的连接的；&/li&&li&3.CFS的起点与终点可以在边界上任意选择；&/li&&/ul&&/blockquote&&p&CFS在填充单连通含亏格或多亏格的区域时表现十分出色：&/p&&br&&br&&figure&&img src=&https://pic2.zhimg.com/v2-283f7afe1b1c0d94a660c_b.jpg& data-rawwidth=&341& data-rawheight=&481& class=&content_image& width=&341&&&/figure&&figure&&img src=&https://pic1.zhimg.com/v2-671adf7c7e_b.jpg& data-rawwidth=&666& data-rawheight=&295& class=&origin_image zh-lightbox-thumb& width=&666& data-original=&https://pic1.zhimg.com/v2-671adf7c7e_r.jpg&&&/figure&&figure&&img src=&https://pic3.zhimg.com/v2-3bc26cb10970bef242d8fab6acba973e_b.jpg& data-rawwidth=&459& data-rawheight=&290& class=&origin_image zh-lightbox-thumb& width=&459& data-original=&https://pic3.zhimg.com/v2-3bc26cb10970bef242d8fab6acba973e_r.jpg&&&/figure&&br&&br&&p&实际打印效果也是更加的优秀：&/p&&br&&p&Zigzag：欠填充与边界问题&/p&&br&&figure&&img src=&https://pic4.zhimg.com/v2-4d615efda3c0_b.jpg& data-rawwidth=&135& data-rawheight=&180& class=&content_image& width=&135&&&/figure&&figure&&img src=&https://pic2.zhimg.com/v2-bb88e1de07ceabbb98e815bf_b.jpg& data-rawwidth=&135& data-rawheight=&180& class=&content_image& width=&135&&&/figure&&br&&p&Countour-Parallel：严重的过填充与令人满意的边界&/p&&br&&figure&&img src=&https://pic1.zhimg.com/v2-12f7af736ff501dab0a1cf_b.jpg& data-rawwidth=&136& data-rawheight=&180& class=&content_image& width=&136&&&/figure&&figure&&img src=&https://pic3.zhimg.com/v2-808feee57_b.jpg& data-rawwidth=&129& data-rawheight=&121& class=&content_image& width=&129&&&/figure&&figure&&img src=&https://pic4.zhimg.com/v2-5f2c3020fac5851f49ccab_b.jpg& data-rawwidth=&135& data-rawheight=&180& class=&content_image& width=&135&&&/figure&&br&&p&CFS：很少的过填充与令人满意的边界&br&&/p&&figure&&img src=&https://pic3.zhimg.com/v2-af2ebed4d124fd621c2aecae5aa39db4_b.jpg& data-rawwidth=&135& data-rawheight=&180& class=&content_image& width=&135&&&/figure&&figure&&img src=&https://pic4.zhimg.com/v2-8a83db53a9c651ecebb215f0bc8f3022_b.jpg& data-rawwidth=&129& data-rawheight=&122& class=&content_image& width=&129&&&/figure&&br&&br&&p&&b&那么到底如何将一个单连通区域用CFS填满呢？&/b&&/p&&br&&p&我们将这个区域进行恰当的域分解&a href=&https://link.zhihu.com/?target=http%3A//www.sciencedirect.com/science/article/pii/S0402& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&[2]&/a&，对于每个可螺旋的域，我们采用固定的算法填充。这一算法的几何直观如下：&/p&&br&&figure&&img src=&https://pic2.zhimg.com/v2-0bd2f0f62cb3c14fdb1b8a_b.jpg& data-rawwidth=&136& data-rawheight=&180& class=&content_image& width=&136&&&/figure&&figure&&img src=&https://pic1.zhimg.com/v2-c7bae89afe7a4_b.jpg& data-rawwidth=&398& data-rawheight=&90& class=&content_image& width=&398&&&/figure&&br&&blockquote&&ul&&li&从轮廓平行路径（a），&/li&&li&通过断开和重新路由相邻的等值线（b），&/li&&li&多次重复得到螺旋（c）。&/li&&/ul&&/blockquote&&figure&&img src=&https://pic2.zhimg.com/v2-3cf029c10f6d210b9bd5f5eb6a5df90a_b.jpg& data-rawwidth=&398& data-rawheight=&140& class=&content_image& width=&398&&&/figure&&br&&blockquote&&ul&&li&（a）点p及其对应的内点&img src=&https://www.zhihu.com/equation?tex=I%28p%29& alt=&I(p)& eeimg=&1&&和外点&img src=&https://www.zhihu.com/equation?tex=O%28p%29& alt=&O(p)& eeimg=&1&&，以及前点&img src=&https://www.zhihu.com/equation?tex=B%28p%29& alt=&B(p)& eeimg=&1&&和后点&img src=&https://www.zhihu.com/equation?tex=N%28p%29& alt=&N(p)& eeimg=&1&&。&/li&&li&（b）从&img src=&https://www.zhihu.com/equation?tex=p_%7Bin%7D& alt=&p_{in}& eeimg=&1&&开始，在&img src=&https://www.zhihu.com/equation?tex=p_1+%3D+B%28p_%7Bout%7D%29& alt=&p_1 = B(p_{out})& eeimg=&1&&处向内到达&img src=&https://www.zhihu.com/equation?tex=p_2%3DI%28p_1%29& alt=&p_2=I(p_1)& eeimg=&1&&，并继续至&img src=&https://www.zhihu.com/equation?tex=p_3%3DB%28I%28B%28p_2%29%29%29& alt=&p_3=B(I(B(p_2)))& eeimg=&1&&。&/li&&li&（c）不断重复（b）所得的CFS。&/li&&/ul&&/blockquote&&p&而实际应用的算法中，则采取了树的结构用以生成CFS：&/p&&br&&h3&STEP 1&/h3&&p&给定轮廓之间的间隔w，我们使用Clipper算法&a href=&https://link.zhihu.com/?target=http%3A//www.angusj.com/delphi/clipper.php& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&[3]&/a&构造等轮廓的集合L。&/p&&br&&p&通过&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2Cj%7D& alt=&c_{i,j}& eeimg=&1&&索引等轮廓，&/p&&br&&blockquote&&ul&&li&其中：i表示其到区域边界?R?R的距离，&br&&img src=&https://www.zhihu.com/equation?tex=d%28%5Cpartial+R%2Cc_%7Bi%2Cj%7D%29%3D%28i-0.5%29w& alt=&d(\partial R,c_{i,j})=(i-0.5)w& eeimg=&1&&。&br&** j是具有相同距离指数i的等值线的索引号。&/li&&li&例如，若j≠j′,则&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2Cj%7D& alt=&c_{i,j}& eeimg=&1&&与&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2Cj%27%7D& alt=&c_{i,j'}& eeimg=&1&&属于两个分离的口袋。&/li&&li&不失一般性，我们总是假设&img src=&https://www.zhihu.com/equation?tex=c_%7B1%2C1%7D& alt=&c_{1,1}& eeimg=&1&&为外部区域边界&img src=&https://www.zhihu.com/equation?tex=%5Cpartial+R& alt=&\partial R& eeimg=&1&&。&/li&&/ul&&/blockquote&&h3&STEP 2&/h3&&p&我们首先将具有相邻i值的等值线连接到初始图中，例如&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2Cj%7D& alt=&c_{i,j}& eeimg=&1&&与&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2B1%2Cj%7D& alt=&c_{i+1,j}& eeimg=&1&&。&br&为此，我们在&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2Cj%7D& alt=&c_{i,j}& eeimg=&1&&上朝向&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2B1%2Cj%7D& alt=&c_{i+1,j}& eeimg=&1&&定义连接段&img src=&https://www.zhihu.com/equation?tex=O_%7Bi%2Cj%2Cj%27%7D& alt=&O_{i,j,j'}& eeimg=&1&&：&br&&/p&&br&&blockquote&&ul&&li&其中，d(p, c)表示点p沿着一条轮廓到点c的距离。&/li&&li&段&img src=&https://www.zhihu.com/equation?tex=O_%7Bi%2Cj%2Cj%27%7D& alt=&O_{i,j,j'}& eeimg=&1&&通过在两个等值线之间的可能的重新连接点形成。&/li&&li&如果&img src=&https://www.zhihu.com/equation?tex=O_%7Bi%2Cj%2Cj%27%7D%5Cne+%5Coslash+& alt=&O_{i,j,j'}\ne \oslash & eeimg=&1&&，我们在&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2Cj%7D& alt=&c_{i,j}& eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=c_%7Bi%2B1%2Cj%27%7D& alt=&c_{i+1,j'}& eeimg=&1&&之间添加一条边。&/li&&li&分配给边的权重是&img src=&https://www.zhihu.com/equation?tex=O_%7Bi%2Cj%2Cj%27%7D& alt=&O_{i,j,j'}& eeimg=&1&&的长度。&/li&&li&在构建等值线上的初始图之后，我们用&img src=&https://www.zhihu.com/equation?tex=c_%7B1%2C1%7D& alt=&c_{1,1}& eeimg=&1&&作为根 ，计算最小权重生成树，螺旋轮廓树。&/li&&/ul&&/blockquote&&figure&&img src=&https://pic1.zhimg.com/v2-679b741ddb8e5f35423e_b.jpg& data-rawwidth=&858& data-rawheight=&581& class=&origin_image zh-lightbox-thumb& width=&858& data-original=&https://pic1.zhimg.com/v2-679b741ddb8e5f35423e_r.jpg&&&/figure&&br&&h3&STEP 3&/h3&&p&树节点分为两种类型：&/p&&br&&blockquote&&ul&&li&类型I节点具有小于或等于2的度，并且它们对应于形成可螺旋区域的等轮廓线。具体地，每个这样的区域，例如图（a）中的&img src=&https://www.zhihu.com/equation?tex=R_0%2CR_1%2C...%2CR_4& alt=&R_0,R_1,...,R_4& eeimg=&1&&由I型节点的路径形成。&/li&&li&类型II节点具有大于2的度，例如，在图（b）中以浅蓝色着色的节点，并且它们对应于分支等轮廓。这种等轮廓线提供了可螺旋区域和可能的其它类型II节点之间的接口。&/li&&/ul&&/blockquote&&p&为了获得一个全局连续的路径，我们以自下而上的方式重新连接等轮廓线，从叶节点开始到根结束。有两种类型的重新连接操作：&/p&&br&&blockquote&&ul&&li&第一种将可螺旋区域上的等角圆心连接成具有彼此相邻的起点和出口点的单个费马螺旋线。&/li&&/ul&&/blockquote&&figure&&img src=&https://pic2.zhimg.com/v2-3cf029c10f6d210b9bd5f5eb6a5df90a_b.jpg& data-rawwidth=&398& data-rawheight=&140& class=&content_image& width=&398&&&/figure&&br&&blockquote&&ul&&li&第二种操作在最近的点（灰色点）将费马螺旋的起点和出点连接到类型II等轮廓线。&/li&&/ul&&/blockquote&&figure&&img src=&https://pic2.zhimg.com/v2-ebe6_b.jpg& data-rawwidth=&181& data-rawheight=&88& class=&content_image& width=&181&&&/figure&&br&&p&到目前为止获得的工具路径是全局连续的并且覆盖输入区域R，但是仅&img src=&https://www.zhihu.com/equation?tex=C%5E0& alt=&C^0& eeimg=&1&& 连续并且可能遭受高度不均匀的间距。为此要进行曲线优化。&/p&&p&曲线优化&/p&&p&路径优化采取了两种方式：均匀间距优化和平滑优化。下图是效果对比：&/p&&figure&&img src=&https://pic3.zhimg.com/v2-50d6eaeddaa4_b.jpg& data-rawwidth=&503& data-rawheight=&193& class=&origin_image zh-lightbox-thumb& width=&503& data-original=&https://pic3.zhimg.com/v2-50d6eaeddaa4_r.jpg&&&/figure&&br&&br&&p&左：路径优化前。 中间：仅优化均匀间距的路径。 右：使用平滑和均匀间距的优化。&/p&&br&&p&优化可以充分解决过填充的问题，但两种优化同时使用会造成更多的欠填充。因此到底优化到何种地步需要具体问题具体分析。&/p&&br&&p&至此，我们得到了CFS路径。&/p&&br&&br&&h2&效果展示&/h2&&p&CFS路径填充众多复杂不规则图案可以看看下边的范例：&/p&&br&&figure&&img src=&https://pic4.zhimg.com/v2-a2d37218ddd22d7ecad15d7_b.jpg& data-rawwidth=&832& data-rawheight=&568& class=&origin_image zh-lightbox-thumb& width=&832& data-original=&https://pic4.zhimg.com/v2-a2d37218ddd22d7ecad15d7_r.jpg&&&/figure&&figure&&img src=&https://pic3.zhimg.com/v2-db5d9eba8a2f5_b.jpg& data-rawwidth=&998& data-rawheight=&160& class=&origin_image zh-lightbox-thumb& width=&998& data-original=&https://pic3.zhimg.com/v2-db5d9eba8a2f5_r.jpg&&&/figure&&figure&&img src=&https://pic1.zhimg.com/v2-fbe88ae5c14_b.jpg& data-rawwidth=&162& data-rawheight=&216& class=&content_image& width=&162&&&/figure&&figure&&img src=&https://pic3.zhimg.com/v2-77e9be3b3dc2efadd5456a31_b.jpg& data-rawwidth=&162& data-rawheight=&216& class=&content_image& width=&162&&&/figure&&figure&&img src=&https://pic2.zhimg.com/v2-108fddb3d92b670ee4014_b.jpg& data-rawwidth=&163& data-rawheight=&216& class=&content_image& width=&163&&&/figure&&figure&&img src=&https://pic2.zhimg.com/v2-38e258acede94fa2a5c914c143c82c65_b.jpg& data-rawwidth=&163& data-rawheight=&216& class=&content_image& width=&163&&&/figure&&p&不知道你的模型有没有被拯救呢？&/p&&br&&h2&References&/h2&&p&[1] &a href=&https://link.zhihu.com/?target=http%3A//irc.cs.sdu.edu.cn/CFS/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Connected Fermat Spirals for Layered Fabrication&/a&; Haisen Zhao, Fanglin Gu, Qixing Huang, Jorge Garcia, Yong Chen, Changhe Tu, Bedrich Benes, Hao Zhang, Daniel Cohen-Or, Baoquan C SIGGRAPH 2016.&br&[2] &a href=&https://link.zhihu.com/?target=http%3A//www.sciencedirect.com/science/article/pii/S0402& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Automated torch path planning using polygon subdivision for solid freeform fabrication based on welding&/a&; Rajeev Dwivedi, Radovan K Journal of Manufacturing Systems.&br&[3] &a href=&https://link.zhihu.com/?target=http%3A//www.angusj.com/delphi/clipper.php& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Clipper - an open source freeware library for clipping and offsetting lines and polygons&/a&;&a href=&https://link.zhihu.com/?target=http%3A//www.angusj.com/delphi/clipper.php& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://www.&/span&&span class=&visible&&angusj.com/delphi/clipp&/span&&span class=&invisible&&er.php&/span&&span class=&ellipsis&&&/span&&/a&.&/p&&br&&p&_(:3」∠)_ _(?ω?”∠)_ _(:з)∠)_ ∠( ? 」∠)＿ _(:зゝ∠)_&br&请毫不犹豫地关注我们：&br&我们的网站：&a href=&https://link.zhihu.com/?target=https%3A//graphicon.io/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&GraphiCon&/a&&br&知乎专栏：&a href=&https://zhuanlan.zhihu.com/graphicon& class=&internal&&GraphiCon图形控&/a&&br&公众号：GraphiCon&br&&/p&&figure&&img src=&https://pic1.zhimg.com/v2-301e489cf21bed37f5a2bd_b.jpg& data-rawwidth=&201& data-rawheight=&201& class=&content_image& width=&201&&&/figure&如果你有什么想法，建议，或者想加入我们，你可以：&br&给我们发邮件：&a href=&mailto:hi@graphicon.io&&hi@graphicon.io&/a&&br&加入我们的QQ群：SIQGRAPH（）&br&加入我们的slack群：&a href=&https://link.zhihu.com/?target=https%3A//graphicon.herokuapp.com/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&GraphiCon&/a&&br&&p&&figure&&img src=&https://pic4.zhimg.com/v2-ecab54dee366ab7af7473b6_b.jpg& data-rawwidth=&88& data-rawheight=&31& class=&content_image& width=&88&&&/figure&本作品采用&a href=&https://link.zhihu.com/?target=http%3A//creativecommons.org/licenses/by-nc-nd/4.0/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&知识共享署名-非商业性使用-禁止演绎 4.0 国际许可协议&/a&进行许可。&/p&
前言这是我们十分钟paper的第一篇，「十分钟paper」系列是模仿Youtube上的Two Minute Papers 的一个读书报告栏目，GraphiCon的小伙伴们会用简洁有趣的文字，生动形象的图片讲解一些大家感兴趣的paper。本栏目长期接受投稿，如果你想参与进来，请按文末的信…
&p&考虑了一个稍微有点不同的问题：已知圆的个数及每个圆的半径，找出一组圆心坐标使得这些圆互不重叠且尽量紧贴。&/p&&p&思路：当两个圆有重叠时，给它们一个很大的排斥力；当两个圆不重叠时，给它们一个相对小的吸引力；然后最小化总势能。通过设计引力势能等势线的形状，能大致控制最终结果的轮廓（像圆或者像方之类）。例子：&/p&&p&要最后像个圆形的话——&/p&&figure&&img src=&https://pic2.zhimg.com/50/v2-3dfdf08267_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic2.zhimg.com/50/v2-3dfdf08267_r.jpg&&&/figure&&br&&p&要最后像个方形的话——&/p&&figure&&img src=&https://pic1.zhimg.com/50/v2-8e2e1c058cf9d813728faf_b.jpg& data-rawwidth=&800& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic1.zhimg.com/50/v2-8e2e1c058cf9d813728faf_r.jpg&&&/figure&&p&Python 代码：&/p&&div class=&highlight&&&pre&&code class=&language-python3&&&span class=&kn&&import&/span& &span class=&nn&&numpy&/span& &span class=&k&&as&/span& &span class=&nn&&np&/span&
&span class=&kn&&from&/span& &span class=&nn&&scipy.optimize&/span& &span class=&k&&import&/span& &span class=&n&&minimize&/span&
&span class=&kn&&from&/span& &span class=&nn&&scipy.spatial.distance&/span& &span class=&k&&import&/span& &span class=&n&&pdist&/span&
&span class=&kn&&import&/span& &span class=&nn&&matplotlib.pyplot&/span& &span class=&k&&as&/span& &span class=&nn&&plt&/span&
&span class=&kn&&import&/span& &span class=&nn&&os&/span&
&span class=&k&&def&/span& &span class=&nf&&obj_grad_round&/span&&span class=&p&&(&/span&&span class=&n&&x&/span&&span class=&p&&,&/span& &span class=&n&&radii&/span&&span class=&p&&,&/span& &span class=&n&&pair_index_to_index_pair&/span&&span class=&p&&,&/span& &span class=&n&&repulsion_scale&/span&&span class=&p&&,&/span& &span class=&n&&attraction_scale&/span&&span class=&p&&):&/span&
&span class=&n&&n&/span& &span class=&o&&=&/span& &span class=&nb&&len&/span&&span class=&p&&(&/span&&span class=&n&&radii&/span&&span class=&p&&)&/span&
&span class=&n&&centroids&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&reshape&/span&&span class=&p&&(&/span&&span class=&n&&x&/span&&span class=&p&&,&/span& &span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&,&/span& &span class=&mi&&2&/span&&span class=&p&&))&/span&
&span class=&n&&circles_pdist_sqrd&/span& &span class=&o&&=&/span& &span class=&n&&pdist&/span&&span class=&p&&(&/span&&span class=&n&&centroids&/span&&span class=&p&&,&/span& &span class=&s&&'sqeuclidean'&/span&&span class=&p&&)&/span&
&span class=&n&&obj&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&
&span class=&n&&grad&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&zeros&/span&&span class=&p&&((&/span&&span class=&n&&n&/span&&span class=&p&&,&/span& &span class=&mi&&2&/span&&span class=&p&&))&/span&
&span class=&k&&for&/span& &span class=&n&&pidx&/span&&span class=&p&&,&/span& &span class=&n&&dist_sqrd&/span& &span class=&ow&&in&/span& &span class=&nb&&enumerate&/span&&span class=&p&&(&/span&&span class=&n&&circles_pdist_sqrd&/span&&span class=&p&&):&/span&
&span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&n&&j&/span& &span class=&o&&=&/span& &span class=&n&&pair_index_to_index_pair&/span&&span class=&p&&[&/span&&span class=&n&&pidx&/span&&span class=&p&&]&/span&
&span class=&n&&touching_dist_sqrd&/span& &span class=&o&&=&/span& &span class=&p&&(&/span&&span class=&n&&radii&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&]&/span& &span class=&o&&+&/span& &span class=&n&&radii&/span&&span class=&p&&[&/span&&span class=&n&&j&/span&&span class=&p&&])&/span& &span class=&o&&**&/span& &span class=&mi&&2&/span&
&span class=&n&&grad_inc&/span& &span class=&o&&=&/span& &span class=&mi&&4&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&dist_sqrd&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&centroids&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&-&/span& &span class=&n&&centroids&/span&&span class=&p&&[&/span&&span class=&n&&j&/span&&span class=&p&&,&/span& &span class=&p&&:])&/span&
&span class=&n&&scale&/span& &span class=&o&&=&/span& &span class=&n&&repulsion_scale&/span& &span class=&k&&if&/span& &span class=&n&&dist_sqrd&/span& &span class=&o&&&&/span& &span class=&n&&touching_dist_sqrd&/span& &span class=&k&&else&/span& &span class=&n&&attraction_scale&/span&
&span class=&n&&obj&/span& &span class=&o&&+=&/span& &span class=&n&&scale&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&dist_sqrd&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&**&/span& &span class=&mi&&2&/span&
&span class=&n&&grad&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&+=&/span& &span class=&n&&scale&/span& &span class=&o&&*&/span& &span class=&n&&grad_inc&/span&
&span class=&n&&grad&/span&&span class=&p&&[&/span&&span class=&n&&j&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&-=&/span& &span class=&n&&scale&/span& &span class=&o&&*&/span& &span class=&n&&grad_inc&/span&
&span class=&k&&return&/span& &span class=&n&&obj&/span&&span class=&p&&,&/span& &span class=&n&&grad&/span&&span class=&o&&.&/span&&span class=&n&&flatten&/span&&span class=&p&&()&/span&
&span class=&k&&def&/span& &span class=&nf&&obj_grad_square&/span&&span class=&p&&(&/span&&span class=&n&&x&/span&&span class=&p&&,&/span& &span class=&n&&radii&/span&&span class=&p&&,&/span& &span class=&n&&pair_index_to_index_pair&/span&&span class=&p&&,&/span& &span class=&n&&repulsion_scale&/span&&span class=&p&&,&/span& &span class=&n&&attraction_scale&/span&&span class=&p&&):&/span&
&span class=&n&&n&/span& &span class=&o&&=&/span& &span class=&nb&&len&/span&&span class=&p&&(&/span&&span class=&n&&radii&/span&&span class=&p&&)&/span&
&span class=&n&&centroids&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&reshape&/span&&span class=&p&&(&/span&&span class=&n&&x&/span&&span class=&p&&,&/span& &span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&,&/span& &span class=&mi&&2&/span&&span class=&p&&))&/span&
&span class=&n&&circles_pdist_sqrd&/span& &span class=&o&&=&/span& &span class=&n&&pdist&/span&&span class=&p&&(&/span&&span class=&n&&centroids&/span&&span class=&p&&,&/span& &span class=&s&&'sqeuclidean'&/span&&span class=&p&&)&/span&
&span class=&n&&obj&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&
&span class=&n&&grad&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&zeros&/span&&span class=&p&&((&/span&&span class=&n&&n&/span&&span class=&p&&,&/span& &span class=&mi&&2&/span&&span class=&p&&))&/span&
&span class=&k&&for&/span& &span class=&n&&pidx&/span&&span class=&p&&,&/span& &span class=&n&&dist_sqrd&/span& &span class=&ow&&in&/span& &span class=&nb&&enumerate&/span&&span class=&p&&(&/span&&span class=&n&&circles_pdist_sqrd&/span&&span class=&p&&):&/span&
&span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&n&&j&/span& &span class=&o&&=&/span& &span class=&n&&pair_index_to_index_pair&/span&&span class=&p&&[&/span&&span class=&n&&pidx&/span&&span class=&p&&]&/span&
&span class=&n&&touching_dist_sqrd&/span& &span class=&o&&=&/span& &span class=&p&&(&/span&&span class=&n&&radii&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&]&/span& &span class=&o&&+&/span& &span class=&n&&radii&/span&&span class=&p&&[&/span&&span class=&n&&j&/span&&span class=&p&&])&/span& &span class=&o&&**&/span& &span class=&mi&&2&/span&
&span class=&n&&diff&/span& &span class=&o&&=&/span& &span class=&n&&centroids&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&-&/span& &span class=&n&&centroids&/span&&span class=&p&&[&/span&&span class=&n&&j&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span&
&span class=&k&&if&/span& &span class=&n&&dist_sqrd&/span& &span class=&o&&&&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&:&/span&
&span class=&c&&# repulsion&/span&
&span class=&n&&obj&/span& &span class=&o&&+=&/span& &span class=&n&&repulsion_scale&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&dist_sqrd&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&**&/span& &span class=&mi&&2&/span&
&span class=&n&&grad_inc&/span& &span class=&o&&=&/span& &span class=&mi&&4&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&dist_sqrd&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&*&/span& &span class=&n&&diff&/span&
&span class=&n&&grad&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&+=&/span& &span class=&n&&repulsion_scale&/span& &span class=&o&&*&/span& &span class=&n&&grad_inc&/span&
&span class=&n&&grad&/span&&span class=&p&&[&/span&&span class=&n&&j&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&-=&/span& &span class=&n&&repulsion_scale&/span& &span class=&o&&*&/span& &span class=&n&&grad_inc&/span&
&span class=&k&&else&/span&&span class=&p&&:&/span&
&span class=&c&&# attraction&/span&
&span class=&n&&diff_sqrd&/span& &span class=&o&&=&/span& &span class=&n&&diff&/span& &span class=&o&&**&/span& &span class=&mi&&2&/span&
&span class=&k&&if&/span& &span class=&nb&&max&/span&&span class=&p&&(&/span&&span class=&n&&diff_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&&=&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&:&/span&
&span class=&c&&# do not overlap after projection, needs attraction&/span&
&span class=&k&&if&/span& &span class=&n&&diff_sqrd&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&&=&/span& &span class=&n&&diff_sqrd&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&]:&/span&
&span class=&n&&obj_inc&/span& &span class=&o&&=&/span& &span class=&p&&(&/span&&span class=&n&&diff_sqrd&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&**&/span& &span class=&mi&&2&/span&
&span class=&n&&grad_inc&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&array&/span&&span class=&p&&([&/span&&span class=&mi&&4&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&diff_sqrd&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&]&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&*&/span& &span class=&n&&diff&/span&&span class=&p&&[&/span&&span class=&mi&&0&/span&&span class=&p&&],&/span& &span class=&mi&&0&/span&&span class=&p&&])&/span&
&span class=&k&&else&/span&&span class=&p&&:&/span&
&span class=&n&&obj_inc&/span& &span class=&o&&=&/span& &span class=&p&&(&/span&&span class=&n&&diff_sqrd&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&**&/span& &span class=&mi&&2&/span&
&span class=&n&&grad_inc&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&array&/span&&span class=&p&&([&/span&&span class=&mi&&0&/span&&span class=&p&&,&/span& &span class=&mi&&4&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&diff_sqrd&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&]&/span& &span class=&o&&-&/span& &span class=&n&&touching_dist_sqrd&/span&&span class=&p&&)&/span& &span class=&o&&*&/span& &span class=&n&&diff&/span&&span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&]])&/span&
&span class=&n&&obj&/span& &span class=&o&&+=&/span& &span class=&n&&attraction_scale&/span& &span class=&o&&*&/span& &span class=&n&&obj_inc&/span&
&span class=&n&&grad&/span&&span class=&p&&[&/span&&span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&+=&/span& &span class=&n&&attraction_scale&/span& &span class=&o&&*&/span& &span class=&n&&grad_inc&/span&
&span class=&n&&grad&/span&&span class=&p&&[&/span&&span class=&n&&j&/span&&span class=&p&&,&/span& &span class=&p&&:]&/span& &span class=&o&&-=&/span& &span class=&n&&attraction_scale&/span& &span class=&o&&*&/span& &span class=&n&&grad_inc&/span&
&span class=&k&&return&/span& &span class=&n&&obj&/span&&span class=&p&&,&/span& &span class=&n&&grad&/span&&span class=&o&&.&/span&&span class=&n&&flatten&/span&&span class=&p&&()&/span&
&span class=&k&&def&/span& &span class=&nf&&find_layout&/span&&span class=&p&&(&/span&&span class=&n&&radii&/span&&span class=&p&&,&/span& &span class=&n&&shape&/span&&span class=&o&&=&/span&&span class=&s&&'square'&/span&&span class=&p&&,&/span& &span class=&n&&repulsion_scale&/span&&span class=&o&&=&/span&&span class=&mi&&1&/span&&span class=&n&&e6&/span&&span class=&p&&,&/span& &span class=&n&&attraction_scale&/span&&span class=&o&&=&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span& &span class=&n&&attraction_decay&/span&&span class=&o&&=&/span&&span class=&mf&&0.3&/span&&span class=&p&&,&/span& &span class=&n&&overlapping_tol&/span&&span class=&o&&=&/span&&span class=&mf&&0.&/span&&span class=&p&&,&/span& &span class=&n&&seed&/span&&span class=&o&&=&/span&&span class=&k&&None&/span&&span class=&p&&,&/span& &span class=&n&&verbose&/span&&span class=&o&&=&/span&&span class=&k&&False&/span&&span class=&p&&):&/span&
&span class=&k&&assert&/span& &span class=&n&&shape&/span& &span class=&o&&==&/span& &span class=&s&&'round'&/span& &span class=&ow&&or&/span& &span class=&n&&shape&/span& &span class=&o&&==&/span& &span class=&s&&'square'&/span&
&span class=&n&&obj_grad&/span& &span class=&o&&=&/span& &span class=&n&&obj_grad_round&/span& &span class=&k&&if&/span& &span class=&n&&shape&/span& &span class=&o&&==&/span& &span class=&s&&'round'&/span& &span class=&k&&else&/span& &span class=&n&&obj_grad_square&/span&
&span class=&n&&n&/span& &span class=&o&&=&/span& &span class=&nb&&len&/span&&span class=&p&&(&/span&&span class=&n&&radii&/span&&span class=&p&&)&/span&
&span class=&n&&n_pairs&/span& &span class=&o&&=&/span& &span class=&nb&&int&/span&&span class=&p&&(&/span&&span class=&n&&n&/span& &span class=&o&&*&/span& &span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span& &span class=&o&&/&/span& &span class=&mi&&2&/span&&span class=&p&&)&/span&
&span class=&n&&pair_index_to_index_pair&/span& &span class=&o&&=&/span& &span class=&p&&[&/span&&span class=&k&&None&/span&&span class=&p&&]&/span& &span class=&o&&*&/span& &span class=&n&&n_pairs&/span&
&span class=&n&&pidx&/span& &span class=&o&&=&/span& &span class=&mi&&0&/span&
&span class=&k&&for&/span& &span class=&n&&i&/span& &span class=&ow&&in&/span& &span class=&nb&&range&/span&&span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&):&/span&
&span class=&k&&for&/span& &span class=&n&&j&/span& &span class=&ow&&in&/span& &span class=&nb&&range&/span&&span class=&p&&(&/span&&span class=&n&&i&/span&&span class=&o&&+&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span& &span class=&n&&n&/span&&span class=&p&&):&/span&
&span class=&n&&pair_index_to_index_pair&/span&&span class=&p&&[&/span&&span class=&n&&pidx&/span&&span class=&p&&]&/span& &span class=&o&&=&/span& &span class=&n&&i&/span&&span class=&p&&,&/span& &span class=&n&&j&/span&
&span class=&n&&pidx&/span& &span class=&o&&+=&/span& &span class=&mi&&1&/span&
&span class=&k&&if&/span& &span class=&n&&seed&/span&&span class=&p&&:&/span&
&span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&random&/span&&span class=&o&&.&/span&&span class=&n&&seed&/span&&span class=&p&&(&/span&&span class=&n&&seed&/span&&span class=&p&&)&/span&
&span class=&k&&if&/span& &span class=&n&&verbose&/span&&span class=&p&&:&/span&
&span class=&n&&options&/span& &span class=&o&&=&/span& &span class=&p&&{&/span&&span class=&s&&'disp'&/span&&span class=&p&&:&/span& &span class=&k&&True&/span&&span class=&p&&}&/span&
&span class=&n&&history_folder&/span& &span class=&o&&=&/span& &span class=&s&&'history_%s'&/span& &span class=&o&&%&/span& &span class=&n&&shape&/span&
&span class=&k&&if&/span& &span class=&ow&&not&/span& &span class=&n&&os&/span&&span class=&o&&.&/span&&span class=&n&&path&/span&&span class=&o&&.&/span&&span class=&n&&exists&/span&&span class=&p&&(&/span&&span class=&n&&history_folder&/span&&span class=&p&&):&/span&
&span class=&n&&os&/span&&span class=&o&&.&/span&&span class=&n&&mkdir&/span&&span class=&p&&(&/span&&span class=&n&&history_folder&/span&&span class=&p&&)&/span&
&span class=&k&&else&/span&&span class=&p&&:&/span&
&span class=&n&&options&/span& &span class=&o&&=&/span& &span class=&k&&None&/span&
&span class=&n&&overlapped&/span& &span class=&o&&=&/span& &span class=&k&&True&/span&
&span class=&n&&n_trials&/span& &span class=&o&&=&/span& &span class=&mi&&1&/span&
&span class=&n&&x0&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&random&/span&&span class=&o&&.&/span&&span class=&n&&randn&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span& &span class=&o&&*&/span& &span class=&n&&n&/span&&span class=&p&&)&/span&
&span class=&k&&while&/span& &span class=&n&&overlapped&/span&&span class=&p&&:&/span&
&span class=&k&&if&/span& &span class=&n&&verbose&/span&&span class=&p&&:&/span&
&span class=&nb&&print&/span&&span class=&p&&(&/span&&span class=&s&&'repulsion scale = %g&/span&&span class=&se&&\t&/span&&span class=&s&&attraction scale = %g'&/span& &span class=&o&&%&/span& &span class=&p&&(&/span&&span class=&n&&repulsion_scale&/span&&span class=&p&&,&/span& &span class=&n&&attraction_scale&/span&&span class=&p&&))&/span&
&span class=&n&&opt_result&/span& &span class=&o&&=&/span& &span class=&n&&minimize&/span&&span class=&p&&(&/span&&span class=&n&&obj_grad&/span&&span class=&p&&,&/span& &span class=&n&&x0&/span&&span class=&p&&,&/span& &span class=&n&&args&/span&&span class=&o&&=&/span&&span class=&p&&(&/span&&span class=&n&&radii&/span&&span class=&p&&,&/span& &span class=&n&&pair_index_to_index_pair&/span&&span class=&p&&,&/span& &span class=&n&&repulsion_scale&/span&&span class=&p&&,&/span& &span class=&n&&attraction_scale&/span&&span class=&p&&),&/span& &span class=&n&&jac&/span&&span class=&o&&=&/span&&span class=&k&&True&/span&&span class=&p&&,&/span& &span class=&n&&options&/span&&span class=&o&&=&/span&&span class=&n&&options&/span&&span class=&p&&)&/span&
&span class=&n&&centroids&/span& &span class=&o&&=&/span& &span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&reshape&/span&&span class=&p&&(&/span&&span class=&n&&opt_result&/span&&span class=&o&&.&/span&&span class=&n&&x&/span&&span class=&p&&,&/span& &span class=&p&&(&/span&&span class=&n&&n&/span&&span class=&p&&,&/span& &span class=&mi&&2&/span&&span class=&p&&))&/span&
&span class=&k&&if&/span& &span class=&n&&verbose&/span&&span class=&p&&:&/span&
&span class=&k&&if&/span& &span class=&ow&&not&/span& &span class=&n&&opt_result&/span&&span class=&o&&.&/span&&span class=&n&&success&/span&&span class=&p&&:&/span&
&span class=&nb&&print&/span&&span class=&p&&(&/span&&span class=&n&&opt_result&/span&&span class=&o&&.&/span&&span class=&n&&message&/span&&span class=&p&&)&/span&
&span class=&n&&draw_layout&/span&&span class=&p&&(&/span&&span class=&n&&centroids&/span&&span class=&p&&,&/span& &span class=&n&&radii&/span&&span class=&p&&,&/span& &span class=&s&&'%s/%02d_attraction=%g.png'&/span& &span class=&o&&%&/span& &span class=&p&&(&/span&&span class=&n&&history_folder&/span&&span class=&p&&,&/span& &span class=&n&&n_trials&/span&&span class=&p&&,&/span& &span class=&n&&attraction_scale&/span&&span class=&p&&))&/span&
&span class=&n&&np&/span&&span class=&o&&.&/span&&span class=&n&&savetxt&/span&&span class=&p&&(&/span&&span class=&s&&'%s/%02d_attraction=%g.dat'&/span& &span class=&o&&%&/span& &span class=&p&&(&/span&&span class=&n&&history_folder&/span&&span class=&p&&,&/span& &span class=&n&&n_trials&/span&&span class=&p&&,&/span& &span class=&n&&attraction_scale&/span&&span class=&p}