&p&数学物理的闲书，我倒是有几本。&/p&&p&第一本：《物理之书》。作者：【美】克利福德.皮科夫著，严诚廷译。&/p&&figure&&img data-rawheight=&1080& src=&https://pic2.zhimg.com/50/v2-cbd787cfe232d486fc4a72c_b.jpg& data-size=&normal& data-rawwidth=&1440& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic2.zhimg.com/50/v2-cbd787cfe232d486fc4a72c_r.jpg&&&/figure&&p&以下就只看封面了：&/p&&figure&&img data-rawheight=&1440& src=&https://pic3.zhimg.com/50/v2-6308fdc9b43fccecdfaf_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic3.zhimg.com/50/v2-6308fdc9b43fccecdfaf_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic4.zhimg.com/50/v2-13fe33016_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/50/v2-13fe33016_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic3.zhimg.com/50/v2-2fc0ffa7a2fd41f71d265c02f8d6b5a1_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic3.zhimg.com/50/v2-2fc0ffa7a2fd41f71d265c02f8d6b5a1_r.jpg&&&/figure&&p&这本讲解分形的书很有意思：&/p&&figure&&img data-rawheight=&1440& src=&https://pic4.zhimg.com/50/v2-d6fde56e25aef_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/50/v2-d6fde56e25aef_r.jpg&&&/figure&&p&分形，相信大家不陌生。分形的图以其复杂而著称。我们看下图，注意要从1看到15，它们是逐张放大的，它们可是分形的典型代表哦：&/p&&figure&&img data-rawheight=&910& src=&https://pic4.zhimg.com/50/v2-e7c981b27b3c1aa4aafbc1929bee8bd6_b.jpg& data-size=&normal& data-rawwidth=&718& class=&origin_image zh-lightbox-thumb& width=&718& data-original=&https://pic4.zhimg.com/50/v2-e7c981b27b3c1aa4aafbc1929bee8bd6_r.jpg&&&/figure&&p&用书中的话来说：分形是初等数学中开出来的一朵最美丽的数学之花。&/p&&p&再看其它的书：&/p&&figure&&img data-rawheight=&1440& src=&https://pic4.zhimg.com/50/v2-a16cd15eb8192edeac0c_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/50/v2-a16cd15eb8192edeac0c_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic4.zhimg.com/50/v2-afc707acfaf4_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/50/v2-afc707acfaf4_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic2.zhimg.com/50/v2-2a8c690ea3324857deeb_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic2.zhimg.com/50/v2-2a8c690ea3324857deeb_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic2.zhimg.com/50/v2-0a3ca3a007abe1aad2729_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic2.zhimg.com/50/v2-0a3ca3a007abe1aad2729_r.jpg&&&/figure&&figure&&img data-rawheight=&1575& src=&https://pic2.zhimg.com/50/v2-1dfe8ff8_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic2.zhimg.com/50/v2-1dfe8ff8_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic4.zhimg.com/50/v2-cde4b7fd972b_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/50/v2-cde4b7fd972b_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic4.zhimg.com/50/v2-28c15d8bdc4dbfcb997c8a_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/50/v2-28c15d8bdc4dbfcb997c8a_r.jpg&&&/figure&&figure&&img data-rawheight=&1080& src=&https://pic2.zhimg.com/50/v2-b567f16ad8f7e6db25f95ed_b.jpg& data-size=&normal& data-rawwidth=&1440& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic2.zhimg.com/50/v2-b567f16ad8f7e6db25f95ed_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic3.zhimg.com/50/v2-ec84fb4d3c0e7f1c7ec80_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic3.zhimg.com/50/v2-ec84fb4d3c0e7f1c7ec80_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic2.zhimg.com/50/v2-eda420a9b65daad3d2e1026_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic2.zhimg.com/50/v2-eda420a9b65daad3d2e1026_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic3.zhimg.com/50/v2-b58bb63fb80e98c18791_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic3.zhimg.com/50/v2-b58bb63fb80e98c18791_r.jpg&&&/figure&&p&还有一些，总共大约近百本。其中还有很老的书，例如《π的奥秘》、《ｅ的奥秘》等等。&/p&&p&这些书很有意思，阅读可以很轻松，真正是“闲书”系列。&/p&&p&另外，还有三本出版社送给我的参考书，如下：&/p&&figure&&img data-rawheight=&1440& src=&https://pic1.zhimg.com/50/v2-dc6a274f2f408e995a75d74bd5c7e48d_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic1.zhimg.com/50/v2-dc6a274f2f408e995a75d74bd5c7e48d_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic2.zhimg.com/50/v2-cdd68e3ec0395eef916a1_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic2.zhimg.com/50/v2-cdd68e3ec0395eef916a1_r.jpg&&&/figure&&figure&&img data-rawheight=&1440& src=&https://pic1.zhimg.com/50/v2-d9ed92a28e37a94a03a8be_b.jpg& data-size=&normal& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic1.zhimg.com/50/v2-d9ed92a28e37a94a03a8be_r.jpg&&&/figure&&p&这最后一本，写的类似章回小说，很有意思。&/p&&p&另外，再推荐一本书：&/p&&figure&&img data-rawheight=&1707& src=&https://pic2.zhimg.com/50/v2-641cd21a3e535a0f4ec44a2fa7e23e24_b.jpg& data-size=&normal& data-rawwidth=&1280& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic2.zhimg.com/50/v2-641cd21a3e535a0f4ec44a2fa7e23e24_r.jpg&&&/figure&&p&书中若干内容摘录：&/p&&figure&&img data-rawheight=&1707& src=&https://pic2.zhimg.com/50/v2-4a42bb7f3ca1c07adf22970_b.jpg& data-size=&normal& data-rawwidth=&1280& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic2.zhimg.com/50/v2-4a42bb7f3ca1c07adf22970_r.jpg&&&/figure&&figure&&img data-rawheight=&1707& src=&https://pic3.zhimg.com/50/v2-e51aeea2ffa59ebb12c5a87_b.jpg& data-size=&normal& data-rawwidth=&1280& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic3.zhimg.com/50/v2-e51aeea2ffa59ebb12c5a87_r.jpg&&&/figure&&figure&&img data-rawheight=&1707& src=&https://pic3.zhimg.com/50/v2-2b3a205ab60ed3c867cfe951_b.jpg& data-size=&normal& data-rawwidth=&1280& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic3.zhimg.com/50/v2-2b3a205ab60ed3c867cfe951_r.jpg&&&/figure&&figure&&img data-rawheight=&1707& src=&https://pic1.zhimg.com/50/v2-59cd61ad889edfdfb3ab4afea444adda_b.jpg& data-size=&normal& data-rawwidth=&1280& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic1.zhimg.com/50/v2-59cd61ad889edfdfb3ab4afea444adda_r.jpg&&&/figure&&p&内容还是很不错的。&/p&
数学物理的闲书，我倒是有几本。第一本：《物理之书》。作者：【美】克利福德.皮科夫著，严诚廷译。以下就只看封面了：这本讲解分形的书很有意思：分形，相信大家不陌生。分形的图以其复杂而著称。我们看下图，注意要从1看到15，它们是逐张放大的，它们可是…
&p&编辑器神器 &a href=&//link.zhihu.com/?target=http%3A//www.texstudio.org/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&TeXstudio&/a&，谁用谁知道，用过都说好。&/p&&p&经常写TeX建议至少通读一本TeX的书，可以节省很多调试的时间。&/p&&p&引文不应该直接插入正文，考虑用bibtex。引作者名字的格式要统一。&/p&&p&文章体量大的话用input分出来章节，模块化管理还便于调试。&/p&&p&数学专业的学术写作参考 &a href=&//link.zhihu.com/?target=ftp%3A//ftp.dante.de/tex-archive/info/intro-scientific/scidoc.pdf& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Writing Scientific Documents Using LaTeX&/a&&/p&&p&punctuation（不知道中文比较恰当的翻译是什么）很重要 。什么时候该用-，--和---都是学问。这一点&a href=&//link.zhihu.com/?target=https%3A//www.tug.org/twg/mactex/tutorials/ltxprimer-1.0.pdf& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&ltxprimer&/a&这本书写的非常不错。读读Academic writing 之类的书，学术写作是很繁琐的一件事情。举例如下&/p&&ol&&li&在不希望分割的词组之间加~，例如 some words blah blah~\cite{citation} （引文不应该出现在行首），或者 Michael Jackson~et~al. （et al是不应该分开的）&/li&&li&微分是\mathrm{d} (&a href=&//link.zhihu.com/?target=https%3A//tex.stackexchange.com/questions/14821/whats-the-proper-way-to-typeset-a-differential-operator& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&What's the proper way to typeset a differential operator?&/a&)&/li&&li&更多的例子 &a href=&//link.zhihu.com/?target=https%3A//tex.stackexchange.com/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&TeX - LaTeX Stack Exchange&/a&&/li&&/ol&&p&编译太慢的话可以考虑是不是要清字体缓存。&/p&&p&PDF预览炸了的话可以看看是不是synctex出问题了……（此条供有缘人参考）&/p&&p&&br&&/p&&p&texdoc is awesome! :)&/p&
编辑器神器 ，谁用谁知道，用过都说好。经常写TeX建议至少通读一本TeX的书，可以节省很多调试的时间。引文不应该直接插入正文，考虑用bibtex。引作者名字的格式要统一。文章体量大的话用input分出来章节，模块化管理还便于调试。数学专业的学术写…
&p&为何要引入四元数？首先是因为欧拉角有万向节死锁的问题。&/p&&p&3D游戏或者3D电影中，比如黑客帝国中酷炫的旋转是怎么实现的？&/p&&figure&&img src=&https://pic3.zhimg.com/50/v2-979bc20dbd7e9cf3ebdefb0c5912317b_b.jpg& data-rawwidth=&320& data-rawheight=&179& class=&content_image& width=&320&&&/figure&&p&旋转的算法有很多，这里主要介绍其中一种：欧拉角。&/p&&p&&b&1 欧拉角&/b&&/p&&p&&b&1.1 欧拉角的算法思想是什么&/b&&/p&&p&陌生的你来到了成都，站在盐市口茫然四顾，想知道春熙路怎么走？&/p&&p&这个时候你选择了去问路，得到了两种回答：&/p&&ul&&li&往东经104°04′、北纬30°40′走&/li&&li&右转后一直走&/li&&/ul&&p&第一种回答，告诉了你春熙路的绝对坐标，可是很反人类啊！&/p&&p&第二种回答，告诉了你春熙路的相对坐标，很具有操作性。&/p&&p&欧拉角算法的思想就是采用的第二种回答的方式，优点在于很好理解。&/p&&p&&b&1.2 具体实现步骤&/b&&/p&&p&&a href=&//link.zhihu.com/?target=https%3A//en.wikipedia.org/wiki/Euler_angles& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&维基百科&/a& 中，有这么一副动图，清楚的表明了如何通过欧拉角来完成旋转：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-35da80e428ca5750491ffd_b.jpg& data-rawwidth=&368& data-rawheight=&348& class=&content_image& width=&368&&&/figure&&p&具体来拆解下旋转步骤，先看图：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-df645c9b0d0a01504d5f_b.jpg& data-rawwidth=&400& data-rawheight=&451& class=&content_image& width=&400&&&/figure&&p&图中有两组坐标：&/p&&ul&&li&&img src=&//www.zhihu.com/equation?tex=xyz& alt=&xyz& eeimg=&1&& 为全局坐标，保持不动&/li&&li&&img src=&//www.zhihu.com/equation?tex=XYZ& alt=&XYZ& eeimg=&1&& 为局部坐标，随着物体一起运动&/li&&/ul&&p&旋转步骤如下：&/p&&ul&&li&物体绕&b&全局&/b&的 &img src=&//www.zhihu.com/equation?tex=z& alt=&z& eeimg=&1&& 轴旋转 &img src=&//www.zhihu.com/equation?tex=%5Calpha+& alt=&\alpha & eeimg=&1&& 角&/li&&li&继续绕&b&自己&/b&的 &img src=&//www.zhihu.com/equation?tex=X& alt=&X& eeimg=&1&& 轴（也就是图中的 &img src=&//www.zhihu.com/equation?tex=N& alt=&N& eeimg=&1&& 轴）旋转 &img src=&//www.zhihu.com/equation?tex=%5Cbeta+& alt=&\beta & eeimg=&1&& 角&/li&&li&最后绕&b&自己&/b&的 &img src=&//www.zhihu.com/equation?tex=Z& alt=&Z& eeimg=&1&& 轴旋转 &img src=&//www.zhihu.com/equation?tex=%5Cgamma+& alt=&\gamma & eeimg=&1&& 角&/li&&/ul&&p&这里有一副动图很直观的展示了旋转过程（角度标记的有点不一样： &img src=&//www.zhihu.com/equation?tex=%5Cpsi+& alt=&\psi & eeimg=&1&& 对应 &img src=&//www.zhihu.com/equation?tex=%5Calpha+& alt=&\alpha & eeimg=&1&& ， &img src=&//www.zhihu.com/equation?tex=%5Ctheta+& alt=&\theta & eeimg=&1&& 对应 &img src=&//www.zhihu.com/equation?tex=%5Cbeta+& alt=&\beta & eeimg=&1&& ， &img src=&//www.zhihu.com/equation?tex=%5Cphi+& alt=&\phi & eeimg=&1&& 对应 &img src=&//www.zhihu.com/equation?tex=%5Cgamma+& alt=&\gamma & eeimg=&1&& ），图来自&a href=&//link.zhihu.com/?target=https%3A//gfycat.com/fr/gifs/detail/flakyconventionalbarnowl& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&gfycat&/a&：&/p&&figure&&img src=&https://pic2.zhimg.com/50/v2-c74e8dbd170834_b.jpg& data-rawwidth=&240& data-rawheight=&138& class=&content_image& width=&240&&&/figure&&p&可能你感到奇怪，为什么第一步是绕着全局坐标旋转？因为要和世界保持联系，否则就和世界完全没有关系了。&/p&&p&还不理解？没有关系，自己动手试试（有三个可以操作的点，分别对应三个角度）：&/p&&figure&&img src=&https://pic3.zhimg.com/50/v2-549df3a0ac9f82cc63ccb1_b.jpg& data-rawwidth=&2300& data-rawheight=&804& class=&origin_image zh-lightbox-thumb& width=&2300& data-original=&https://pic3.zhimg.com/50/v2-549df3a0ac9f82cc63ccb1_r.jpg&&&/figure&&blockquote&此处有互动内容，&a href=&//link.zhihu.com/?target=http%3A//www.matongxue.com/madocs/442.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&点击此处前往操作。&/a&&/blockquote&&p&很显然，按照不同的旋转步骤，旋转的结果是不一样的。&/p&&p&就好比刚才问路的时候，回答你，“左转再右转”，和“右转再左转”，肯定到达的地点是不一样的。&/p&&p&我们需要把上面的旋转步骤按照顺序标记为 &img src=&//www.zhihu.com/equation?tex=zXZ& alt=&zXZ& eeimg=&1&& ，加上角度就是一个完整的欧拉角：&/p&&p&&img src=&//www.zhihu.com/equation?tex=zXZ-%28%5Calpha+%2C+%5Cbeta+%2C+%5Cgamma+%29& alt=&zXZ-(\alpha , \beta , \gamma )& eeimg=&1&&&/p&&p&&b&2 万向节死锁（Gimbal Lock）&/b&&/p&&p&局部坐标是很直观，但是导致欧拉角有一个重大缺陷，万向节死锁！&/p&&p&本节大部分参考了博文： &a href=&//link.zhihu.com/?target=http%3A//blog.csdn.net/AndrewFan/article/details/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&欧拉角与万向节死锁（图文版）&/a& ，博主： &a href=&//link.zhihu.com/?target=http%3A//blog.csdn.net/andrewfan& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&andrewfan&/a& 。&/p&&p&&b&2.1 什么是万向节（Gimbal）&/b&&/p&&blockquote&&b&平衡环架（英语：Gimbal）为一具有枢纽的装置，使得一物体能以单一轴旋转。由彼此垂直的枢纽轴所组成的一组三只平衡环架，则可使架在最内的环架的物体维持旋转轴不变，而应用在船上的陀螺仪、罗盘、饮料杯架等用途上，而不受船体因波浪上下震动、船身转向的影响。&/b&&br&&br&----维基百科&/blockquote&&p&长这个样子：&/p&&figure&&img src=&https://pic3.zhimg.com/50/v2-2f3d8c9c2de8c7f225abd_b.jpg& data-rawwidth=&250& data-rawheight=&250& class=&content_image& width=&250&&&/figure&&p&中间有一根竖轴，穿过一个金属圆盘。金属圆盘称为转子，竖轴称为旋转轴。转子用金属制成，应该是了增加质量，从而增大惯性。竖轴外侧是三层嵌套的圆环，它们互相交叉，带来了三个方向自由度的旋转。&/p&&p&看着不停转来转去，有点晕，接下来看下静态的。图来自百度百科：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-115ad26c55_b.jpg& data-rawwidth=&269& data-rawheight=&220& class=&content_image& width=&269&&&/figure&&p&&b&2.2 欧拉角与万向节&/b&&/p&&p&其实欧拉角的工作方式与万向节几乎一样。&/p&&p&看几幅动图就知道（图来自 &a href=&//link.zhihu.com/?target=http%3A//www.ogre3d.org/tikiwiki/tiki-index.php%3Fpage%3DEuler%2BAngle%2BClass& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Euler Angle Class&/a& ）。&/p&&p&这个旋转叫pitch，中文是俯仰：&/p&&figure&&img src=&https://pic3.zhimg.com/50/v2-c9097bcf4ad39e39c684a2fdae7ef7d8_b.jpg& data-rawwidth=&256& data-rawheight=&192& class=&content_image& width=&256&&&/figure&&p&这个旋转叫Yaw，中文叫偏航：&/p&&figure&&img src=&https://pic3.zhimg.com/50/v2-61f08ff3a9b8e2749d31_b.jpg& data-rawwidth=&256& data-rawheight=&192& class=&content_image& width=&256&&&/figure&&p&这个旋转叫Roll，中文叫桶滚：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-bba8ed67cb1f4a6f495a157df0a5208d_b.jpg& data-rawwidth=&256& data-rawheight=&192& class=&content_image& width=&256&&&/figure&&p&可以看出，确实工作方式和万向节一样。&/p&&p&&b&2.3 死锁的产生&/b&&/p&&p&为了解释清楚问题，画了一个简单的万向节示意图（金属圆盘就省略了，丑点儿也就别管了）：&/p&&figure&&img src=&https://pic2.zhimg.com/50/v2-78acd2fb18e37d77252f6_b.jpg& data-rawwidth=&326& data-rawheight=&325& class=&content_image& width=&326&&&/figure&&p&把三个Gimbal环用不同的颜色做了标记，底部三个轴向，RGB分别对应 &img src=&//www.zhihu.com/equation?tex=XYZ& alt=&XYZ& eeimg=&1&& 。&/p&&p&假设现在这个万向节被放在一艘船上，船头的方向沿着+Z轴，也就是右前方。&/p&&p&&b&2.3.1 桶滚&/b&&/p&&p&现在假设，船体发生了摇晃，是沿着前方进行旋转的摇晃，也就是桶滚。由于转子和旋转轴具有较大的惯性，只要没有直接施加扭矩，就会保持原有的姿态。由于上图中绿色的活动的连接头处是可以灵活转动的，此时将发生相对旋转，从而出现以下的情形：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-c59686dbabcd7c68df8178cf_b.jpg& data-rawwidth=&420& data-rawheight=&340& class=&content_image& width=&420&&&/figure&&p&&b&2.3.2 俯仰&/b&&/p&&p&再次假设，船体发生了pitch摇晃，也就是俯仰。同样，由于存在相应方向的可以相对旋转的连接头（红色连接头），转子和旋转轴将仍然保持平衡，如下图：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-5ecfced6a8a_b.jpg& data-rawwidth=&420& data-rawheight=&340& class=&content_image& width=&420&&&/figure&&p&&b&2.3.3 偏航&/b&&/p&&p&最后假设，船体发生了yaw摇晃，也就是偏航，此时船体在发生水平旋转。相对旋转发生在蓝色连接头。如下图：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-29ea182b7bb1f34662ca_b.jpg& data-rawwidth=&420& data-rawheight=&340& class=&content_image& width=&420&&&/figure&&p&最终，在船体发生Pitch、Yaw、Roll的情况下，万向节都可以通过自身的调节，而让转子和旋转轴保持平衡。&/p&&p&&b&2.3.4 死锁&/b&&/p&&p&现在看起来，这个万向节一切正常，在船体发生任意方向摇晃都可以通过自身调节来应对。然而，真的是这样吗？&/p&&p&假如，船体发生了剧烈的变化，此时船首仰起了90度（这是要翻船的节奏。。。。），此时的陀螺仪调节状态如下图：&/p&&figure&&img src=&https://pic1.zhimg.com/50/v2-fcbc59ef7cd9c346a7ec2373dfca903e_b.jpg& data-rawwidth=&360& data-rawheight=&332& class=&content_image& width=&360&&&/figure&&p&此时，船体再次发生转动，沿着当前世界坐标的+Z轴（蓝色轴，应该正指向船底）进行转动，那么来看看发生了什么情况：&/p&&figure&&img src=&https://pic3.zhimg.com/50/v2-cabe060d9bebb30d976c7035164bcc75_b.jpg& data-rawwidth=&420& data-rawheight=&340& class=&content_image& width=&420&&&/figure&&p&现在，转子不平衡了，万向节的三板斧不起作用了。它失去了自身的调节能力。那么这是为什么呢？&/p&&p&之前万向节之所以能通过自身调节，保持平衡，是因为存在可以相对旋转的连接头。在这种情况下，已经不存在可以相对旋转的连接头了。&/p&&p&那么连接头呢？去了哪里？显然，它还是在那里，只不过从上图中，我们清楚地看到：&/p&&ul&&li&红色连接头：可以给予一个相对俯仰的自由度。&/li&&li&绿色连接头：可以给予一个相对偏航的自由度。&/li&&li&蓝色连接头：可以给予一个相对偏航的自由度。&/li&&/ul&&p&没错，三个连接头，提供的自由度只对应了俯仰和偏航两个自由度，桶滚自由度丢失了。&/p&&p&我们可以回头去试试之前的 &a href=&//link.zhihu.com/?target=http%3A//www.matongxue.com/madocs/442.html& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&操作页面&/a& ，在下面这样子的情况下其实就是死锁了：&/p&&figure&&img src=&https://pic4.zhimg.com/50/v2-8b612cbd78cd8a0ffc5e85_b.jpg& data-rawwidth=&2300& data-rawheight=&804& class=&origin_image zh-lightbox-thumb& width=&2300& data-original=&https://pic4.zhimg.com/50/v2-8b612cbd78cd8a0ffc5e85_r.jpg&&&/figure&&p&&b&3 视频&/b&&/p&&p&还不懂？没有关系，这里还有做得非常好的视频，以供参考：&/p&&a class=&video-box& href=&//link.zhihu.com/?target=https%3A//v.qq.com/x/cover/ew/ew.html& target=&_blank& data-video-id=&611136& data-video-playable=&true& data-name=&欧拉角旋转以及万向节死锁_腾讯视频& data-poster=&https://pic3.zhimg.com/v2-346d49b10eee41a0c38de.jpg& data-lens-id=&&&
&img class=&thumbnail& src=&https://pic3.zhimg.com/v2-346d49b10eee41a0c38de.jpg&&&span class=&content&&
&span class=&title&&欧拉角旋转以及万向节死锁_腾讯视频&span class=&z-ico-extern-gray&&&/span&&span class=&z-ico-extern-blue&&&/span&&/span&
&span class=&url&&&span class=&z-ico-video&&&/span&https://v.qq.com/x/cover/ew/ew.html&/span&
&/a&&p&我对视频主要内容总结如下：&/p&&ul&&li&万向节死锁的根源在于欧拉角的定义方式&/li&&li&万向节死锁的结果，不是说不能旋转了，而是会导致旋转不自然&/li&&li&要规避万向节死锁，需要选择合适的旋转顺序（有12种旋转顺序）&/li&&/ul&&p&&b&4 总结&/b&&/p&&p&在编程中很难规避死锁问题，所以现在很多时候都使用四元数实现旋转，四元数那又是另外的话题了。&/p&
为何要引入四元数？首先是因为欧拉角有万向节死锁的问题。3D游戏或者3D电影中，比如黑客帝国中酷炫的旋转是怎么实现的？旋转的算法有很多，这里主要介绍其中一种：欧拉角。1 欧拉角1.1 欧拉角的算法思想是什么陌生的你来到了成都，站在盐市口茫然四顾，想知…
我来装回b吧。以下代码出自这 &a href=&//link.zhihu.com/?target=https%3A//github.com/PRML/PRMLT& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&PRML/PRMLT&/a&.在老版本Matlab上，基本都比内置函数快个1~2个数量级，新版Matlab没测过了。&br&两组数据欧式距离（&a class=&internal& href=&https://www.zhihu.com/people/yan-xiao-peng&&阎小鹏&/a&上面提了）&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&D&span class=&p&&=&/span&&span class=&nf&&sqdist&/span&&span class=&p&&(&/span&X, Y&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&n&&D&/span&&span class=&p&&=&/span&&span class=&nb&&bsxfun&/span&&span class=&p&&(@&/span&&span class=&n&&plus&/span&&span class=&p&&,&/span&&span class=&nb&&dot&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&&span class=&o&&'&/span&&span class=&p&&,&/span&&span class=&nb&&dot&/span&&span class=&p&&(&/span&&span class=&n&&Y&/span&&span class=&p&&,&/span&&span class=&n&&Y&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&))&/span&&span class=&o&&-&/span&&span class=&mi&&2&/span&&span class=&o&&*&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&o&&'*&/span&&span class=&n&&Y&/span&&span class=&p&&);&/span&
&/code&&/pre&&/div&&br&kmeans聚类算法（这是&a class=&internal& href=&https://www.zhihu.com/people/bobye&&豆豆叶&/a&上面提到的最新改进版）其实这俩都是我写的。&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&[label, energy, m] &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&litekmeans&/span&&span class=&p&&(&/span&X, k&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&c&&% Perform k-means clustering.&/span&
&span class=&c&&% Input:&/span&
&span class=&c&&%
X: d x n data matrix&/span&
&span class=&c&&%
init: k number of clusters or label (1 x n vector)&/span&
&span class=&c&&% Output:&/span&
&span class=&c&&%
label: 1 x n cluster label&/span&
&span class=&c&&%
energy: optimization target value&/span&
&span class=&c&&%
m: d x k center of clusters&/span&
&span class=&c&&% Written by Mo Chen ().&/span&
&span class=&n&&n&/span& &span class=&p&&=&/span& &span class=&nb&&size&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&);&/span&
&span class=&n&&idx&/span& &span class=&p&&=&/span& &span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&n&/span&&span class=&p&&;&/span&
&span class=&n&&last&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&span class=&n&&label&/span& &span class=&p&&=&/span& &span class=&nb&&ceil&/span&&span class=&p&&(&/span&&span class=&n&&k&/span&&span class=&o&&*&/span&&span class=&nb&&rand&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&));&/span&
&span class=&c&&% random initialization&/span&
&span class=&k&&while&/span& &span class=&n&&any&/span&&span class=&p&&(&/span&&span class=&n&&label&/span& &span class=&o&&~=&/span& &span class=&n&&last&/span&&span class=&p&&)&/span&
&span class=&p&&[&/span&&span class=&o&&~&/span&&span class=&p&&,&/span&&span class=&o&&~&/span&&span class=&p&&,&/span&&span class=&n&&last&/span&&span class=&p&&(:)]&/span& &span class=&p&&=&/span& &span class=&n&&unique&/span&&span class=&p&&(&/span&&span class=&n&&label&/span&&span class=&p&&);&/span&
&span class=&c&&% remove empty clusters&/span&
&span class=&n&&E&/span& &span class=&p&&=&/span& &span class=&n&&sparse&/span&&span class=&p&&(&/span&&span class=&n&&idx&/span&&span class=&p&&,&/span&&span class=&n&&last&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&c&&% transform label into indicator matrix&/span&
&span class=&n&&m&/span& &span class=&p&&=&/span& &span class=&n&&X&/span&&span class=&o&&*&/span&&span class=&p&&(&/span&&span class=&n&&E&/span&&span class=&o&&./&/span&&span class=&n&&sum&/span&&span class=&p&&(&/span&&span class=&n&&E&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&));&/span&
&span class=&c&&% compute centers &/span&
&span class=&p&&[&/span&&span class=&n&&val&/span&&span class=&p&&,&/span&&span class=&n&&label&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&n&&min&/span&&span class=&p&&(&/span&&span class=&nb&&dot&/span&&span class=&p&&(&/span&&span class=&n&&m&/span&&span class=&p&&,&/span&&span class=&n&&m&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&&span class=&o&&'/&/span&&span class=&mi&&2&/span&&span class=&o&&-&/span&&span class=&n&&m&/span&&span class=&o&&'*&/span&&span class=&n&&X&/span&&span class=&p&&,[],&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span& &span class=&c&&% assign labels&/span&
&span class=&k&&end&/span&
&span class=&n&&energy&/span& &span class=&p&&=&/span& &span class=&nb&&dot&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&(:),&/span&&span class=&n&&X&/span&&span class=&p&&(:),&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&&span class=&o&&+&/span&&span class=&mi&&2&/span&&span class=&o&&*&/span&&span class=&n&&sum&/span&&span class=&p&&(&/span&&span class=&n&&val&/span&&span class=&p&&);&/span&
&/code&&/pre&&/div&&br&kernel k-means&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&[label, energy, model] &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&knKmeans&/span&&span class=&p&&(&/span&X, k, kn&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&n&&n&/span& &span class=&p&&=&/span& &span class=&nb&&size&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&);&/span&
&span class=&n&&K&/span& &span class=&p&&=&/span& &span class=&n&&kn&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&n&&X&/span&&span class=&p&&);&/span&
&span class=&n&&last&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&span class=&n&&label&/span& &span class=&p&&=&/span& &span class=&nb&&ceil&/span&&span class=&p&&(&/span&&span class=&n&&k&/span&&span class=&o&&*&/span&&span class=&nb&&rand&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&));&/span&
&span class=&k&&while&/span& &span class=&n&&any&/span&&span class=&p&&(&/span&&span class=&n&&label&/span& &span class=&o&&~=&/span& &span class=&n&&last&/span&&span class=&p&&)&/span&
&span class=&p&&[&/span&&span class=&o&&~&/span&&span class=&p&&,&/span&&span class=&o&&~&/span&&span class=&p&&,&/span&&span class=&n&&last&/span&&span class=&p&&(:)]&/span& &span class=&p&&=&/span& &span class=&n&&unique&/span&&span class=&p&&(&/span&&span class=&n&&label&/span&&span class=&p&&);&/span&
&span class=&c&&% remove empty clusters&/span&
&span class=&n&&E&/span& &span class=&p&&=&/span& &span class=&n&&sparse&/span&&span class=&p&&(&/span&&span class=&n&&last&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&n&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&n&&E&/span& &span class=&p&&=&/span& &span class=&n&&E&/span&&span class=&o&&./&/span&&span class=&n&&sum&/span&&span class=&p&&(&/span&&span class=&n&&E&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&);&/span&
&span class=&n&&T&/span& &span class=&p&&=&/span& &span class=&n&&E&/span&&span class=&o&&*&/span&&span class=&n&&K&/span&&span class=&p&&;&/span&
&span class=&p&&[&/span&&span class=&n&&val&/span&&span class=&p&&,&/span& &span class=&n&&label&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&n&&max&/span&&span class=&p&&(&/span&&span class=&n&&T&/span&&span class=&o&&-&/span&&span class=&nb&&dot&/span&&span class=&p&&(&/span&&span class=&n&&T&/span&&span class=&p&&,&/span&&span class=&n&&E&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span&&span class=&o&&/&/span&&span class=&mi&&2&/span&&span class=&p&&,[],&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&k&&end&/span&
&span class=&n&&energy&/span& &span class=&p&&=&/span& &span class=&n&&trace&/span&&span class=&p&&(&/span&&span class=&n&&K&/span&&span class=&p&&)&/span&&span class=&o&&-&/span&&span class=&mi&&2&/span&&span class=&o&&*&/span&&span class=&n&&sum&/span&&span class=&p&&(&/span&&span class=&n&&val&/span&&span class=&p&&);&/span&
&/code&&/pre&&/div&&br&kmedoids聚类&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&[label, energy，index] &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&kmedoids&/span&&span class=&p&&(&/span&X, init&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&n&&X&/span& &span class=&p&&=&/span& &span class=&n&&X&/span&&span class=&o&&-&/span&&span class=&n&&mean&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&);&/span&
&span class=&c&&% reduce chance of numerical problems&/span&
&span class=&n&&v&/span& &span class=&p&&=&/span& &span class=&nb&&dot&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&n&&D&/span& &span class=&p&&=&/span& &span class=&n&&v&/span&&span class=&o&&+&/span&&span class=&n&&v&/span&&span class=&o&&'-&/span&&span class=&mi&&2&/span&&span class=&o&&*&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&o&&'*&/span&&span class=&n&&X&/span&&span class=&p&&);&/span&
&span class=&c&&% Euclidean distance matrix&/span&
&span class=&n&&D&/span&&span class=&p&&(&/span&&span class=&nb&&sub2ind&/span&&span class=&p&&([&/span&&span class=&n&&d&/span&&span class=&p&&,&/span&&span class=&n&&d&/span&&span class=&p&&],&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&d&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&d&/span&&span class=&p&&))&/span& &span class=&p&&=&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span&
&span class=&c&&% reduce chance of numerical problems&/span&
&span class=&n&&last&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&span class=&n&&label&/span& &span class=&p&&=&/span& &span class=&nb&&ceil&/span&&span class=&p&&(&/span&&span class=&n&&k&/span&&span class=&o&&*&/span&&span class=&nb&&rand&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&));&/span&
&span class=&k&&while&/span& &span class=&n&&any&/span&&span class=&p&&(&/span&&span class=&n&&label&/span& &span class=&o&&~=&/span& &span class=&n&&last&/span&&span class=&p&&)&/span&
&span class=&p&&[&/span&&span class=&o&&~&/span&&span class=&p&&,&/span&&span class=&o&&~&/span&&span class=&p&&,&/span&&span class=&n&&last&/span&&span class=&p&&(:)]&/span& &span class=&p&&=&/span& &span class=&n&&unique&/span&&span class=&p&&(&/span&&span class=&n&&label&/span&&span class=&p&&);&/span&
&span class=&c&&% remove empty clusters&/span&
&span class=&p&&[&/span&&span class=&o&&~&/span&&span class=&p&&,&/span& &span class=&n&&index&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&n&&min&/span&&span class=&p&&(&/span&&span class=&n&&D&/span&&span class=&o&&*&/span&&span class=&n&&sparse&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&n&/span&&span class=&p&&,&/span&&span class=&n&&last&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&),[],&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&c&&% find k medoids&/span&
&span class=&p&&[&/span&&span class=&n&&val&/span&&span class=&p&&,&/span& &span class=&n&&label&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&n&&min&/span&&span class=&p&&(&/span&&span class=&n&&D&/span&&span class=&p&&(&/span&&span class=&n&&index&/span&&span class=&p&&,:),[],&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&c&&% assign labels&/span&
&span class=&k&&end&/span&
&span class=&n&&energy&/span& &span class=&p&&=&/span& &span class=&n&&sum&/span&&span class=&p&&(&/span&&span class=&n&&val&/span&&span class=&p&&);&/span&
&/code&&/pre&&/div&&br&Gram-Schmidt正交化&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&[Q, R] &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&gson&/span&&span class=&p&&(&/span&X&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&p&&[&/span&&span class=&n&&d&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&nb&&size&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&);&/span&
&span class=&n&&m&/span& &span class=&p&&=&/span& &span class=&n&&min&/span&&span class=&p&&(&/span&&span class=&n&&d&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&span class=&n&&R&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&n&&m&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&span class=&n&&Q&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&n&&d&/span&&span class=&p&&,&/span&&span class=&n&&m&/span&&span class=&p&&);&/span&
&span class=&k&&for&/span& &span class=&nb&&i&/span& &span class=&p&&=&/span& &span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&m&/span&
&span class=&n&&R&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&nb&&i&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&nb&&i&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&n&&Q&/span&&span class=&p&&(:,&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&nb&&i&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&&span class=&o&&'*&/span&&span class=&n&&X&/span&&span class=&p&&(:,&/span&&span class=&nb&&i&/span&&span class=&p&&);&/span&
&span class=&n&&v&/span& &span class=&p&&=&/span& &span class=&n&&X&/span&&span class=&p&&(:,&/span&&span class=&nb&&i&/span&&span class=&p&&)&/span&&span class=&o&&-&/span&&span class=&n&&Q&/span&&span class=&p&&(:,&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&nb&&i&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&&span class=&o&&*&/span&&span class=&n&&R&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&nb&&i&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&nb&&i&/span&&span class=&p&&);&/span&
&span class=&n&&R&/span&&span class=&p&&(&/span&&span class=&nb&&i&/span&&span class=&p&&,&/span&&span class=&nb&&i&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&n&&norm&/span&&span class=&p&&(&/span&&span class=&n&&v&/span&&span class=&p&&);&/span&
&span class=&n&&Q&/span&&span class=&p&&(:,&/span&&span class=&nb&&i&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&n&&v&/span&&span class=&o&&/&/span&&span class=&n&&R&/span&&span class=&p&&(&/span&&span class=&nb&&i&/span&&span class=&p&&,&/span&&span class=&nb&&i&/span&&span class=&p&&);&/span&
&span class=&k&&end&/span&
&span class=&n&&R&/span&&span class=&p&&(:,&/span&&span class=&n&&m&/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=&p&&=&/span& &span class=&n&&Q&/span&&span class=&o&&'*&/span&&span class=&n&&X&/span&&span class=&p&&(:,&/span&&span class=&n&&m&/span&&span class=&o&&+&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&/code&&/pre&&/div&&br&Gauss分布pdf&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&y &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&loggausspdf&/span&&span class=&p&&(&/span&X, mu, Sigma&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&n&&d&/span& &span class=&p&&=&/span& &span class=&nb&&size&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&n&&X&/span& &span class=&p&&=&/span& &span class=&nb&&bsxfun&/span&&span class=&p&&(@&/span&&span class=&n&&minus&/span&&span class=&p&&,&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&n&&mu&/span&&span class=&p&&);&/span&
&span class=&p&&[&/span&&span class=&n&&U&/span&&span class=&p&&,&/span&&span class=&n&&p&/span&&span class=&p&&]=&/span& &span class=&n&&chol&/span&&span class=&p&&(&/span&&span class=&n&&Sigma&/span&&span class=&p&&);&/span&
&span class=&k&&if&/span& &span class=&n&&p&/span& &span class=&o&&~=&/span& &span class=&mi&&0&/span&
&span class=&n&&error&/span&&span class=&p&&(&/span&&span class=&s&&'ERROR: Sigma is not PD.'&/span&&span class=&p&&);&/span&
&span class=&k&&end&/span&
&span class=&n&&Q&/span& &span class=&p&&=&/span& &span class=&n&&U&/span&&span class=&o&&'\&/span&&span class=&n&&X&/span&&span class=&p&&;&/span&
&span class=&n&&q&/span& &span class=&p&&=&/span& &span class=&nb&&dot&/span&&span class=&p&&(&/span&&span class=&n&&Q&/span&&span class=&p&&,&/span&&span class=&n&&Q&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&c&&% quadratic term (M distance)&/span&
&span class=&n&&c&/span& &span class=&p&&=&/span& &span class=&n&&d&/span&&span class=&o&&*&/span&&span class=&nb&&log&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&o&&*&/span&&span class=&nb&&pi&/span&&span class=&p&&)&/span&&span class=&o&&+&/span&&span class=&mi&&2&/span&&span class=&o&&*&/span&&span class=&n&&sum&/span&&span class=&p&&(&/span&&span class=&nb&&log&/span&&span class=&p&&(&/span&&span class=&nb&&diag&/span&&span class=&p&&(&/span&&span class=&n&&U&/span&&span class=&p&&)));&/span&
&span class=&c&&% normalization constant&/span&
&span class=&n&&y&/span& &span class=&p&&=&/span& &span class=&o&&-&/span&&span class=&p&&(&/span&&span class=&n&&c&/span&&span class=&o&&+&/span&&span class=&n&&q&/span&&span class=&p&&)&/span&&span class=&o&&/&/span&&span class=&mi&&2&/span&&span class=&p&&;&/span&
&/code&&/pre&&/div&&br&Naive Bayes分类&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&[u, w] &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&nbBern&/span&&span class=&p&&(&/span&X, t&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&n&&n&/span& &span class=&p&&=&/span& &span class=&nb&&size&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&);&/span&
&span class=&n&&E&/span& &span class=&p&&=&/span& &span class=&n&&sparse&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&n&/span&&span class=&p&&,&/span&&span class=&n&&t&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&n&&nk&/span& &span class=&p&&=&/span& &span class=&n&&sum&/span&&span class=&p&&(&/span&&span class=&n&&E&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&n&&w&/span& &span class=&p&&=&/span& &span class=&n&&full&/span&&span class=&p&&(&/span&&span class=&n&&nk&/span&&span class=&o&&/&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&span class=&n&&mu&/span& &span class=&p&&=&/span& &span class=&n&&X&/span&&span class=&o&&*&/span&&span class=&p&&(&/span&&span class=&n&&E&/span&&span class=&o&&./&/span&&span class=&n&&nk&/span&&span class=&p&&);&/span&
&/code&&/pre&&/div&&br&Viterbi滤波&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&[z, llh] &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&hmmViterbi&/span&&span class=&p&&(&/span&x, A, E, s&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&n&&k&/span& &span class=&p&&=&/span& &span class=&nb&&numel&/span&&span class=&p&&(&/span&&span class=&n&&s&/span&&span class=&p&&);&/span&
&span class=&n&&Z&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&n&&k&/span&&span class=&p&&,&/span&&span class=&n&&n&/span&&span class=&p&&);&/span&
&span class=&n&&Z&/span&&span class=&p&&(:,&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&k&/span&&span class=&p&&;&/span&
&span class=&n&&v&/span& &span class=&p&&=&/span& &span class=&n&&s&/span&&span class=&p&&(:)&/span&&span class=&o&&+&/span&&span class=&n&&M&/span&&span class=&p&&(:,&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&k&&for&/span& &span class=&n&&t&/span& &span class=&p&&=&/span& &span class=&mi&&2&/span&&span class=&p&&:&/span&&span class=&n&&n&/span&
&span class=&p&&[&/span&&span class=&n&&v&/span&&span class=&p&&,&/span&&span class=&n&&idx&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&n&&max&/span&&span class=&p&&(&/span&&span class=&nb&&bsxfun&/span&&span class=&p&&(@&/span&&span class=&n&&plus&/span&&span class=&p&&,&/span&&span class=&n&&A&/span&&span class=&p&&,&/span&&span class=&n&&v&/span&&span class=&p&&),[],&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&c&&% 13.68&/span&
&span class=&n&&v&/span& &span class=&p&&=&/span& &span class=&n&&v&/span&&span class=&p&&(:)&/span&&span class=&o&&+&/span&&span class=&n&&M&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&p&&);&/span&
&span class=&n&&Z&/span& &span class=&p&&=&/span& &span class=&n&&Z&/span&&span class=&p&&(&/span&&span class=&n&&idx&/span&&span class=&p&&,:);&/span&
&span class=&n&&Z&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&k&/span&&span class=&p&&;&/span&
&span class=&k&&end&/span&
&span class=&p&&[&/span&&span class=&n&&llh&/span&&span class=&p&&,&/span&&span class=&n&&idx&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&n&&max&/span&&span class=&p&&(&/span&&span class=&n&&v&/span&&span class=&p&&);&/span&
&span class=&n&&z&/span& &span class=&p&&=&/span& &span class=&n&&Z&/span&&span class=&p&&(&/span&&span class=&n&&idx&/span&&span class=&p&&,:);&/span&
&/code&&/pre&&/div&&br&HMM forward-backward&br&&div class=&highlight&&&pre&&code class=&language-matlab&&&span class=&k&&function&/span&&span class=&w&& &/span&[gamma, alpha, beta, c] &span class=&p&&=&/span&&span class=&w&& &/span&&span class=&nf&&hmmSmoother&/span&&span class=&p&&(&/span&M, A, s&span class=&p&&)&/span&&span class=&w&&&/span&
&span class=&p&&[&/span&&span class=&n&&K&/span&&span class=&p&&,&/span&&span class=&n&&T&/span&&span class=&p&&]&/span& &span class=&p&&=&/span& &span class=&nb&&size&/span&&span class=&p&&(&/span&&span class=&n&&M&/span&&span class=&p&&);&/span&
&span class=&n&&At&/span& &span class=&p&&=&/span& &span class=&n&&A&/span&&span class=&o&&'&/span&&span class=&p&&;&/span&
&span class=&n&&c&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&n&&T&/span&&span class=&p&&);&/span&
&span class=&n&&alpha&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&n&&K&/span&&span class=&p&&,&/span&&span class=&n&&T&/span&&span class=&p&&);&/span&
&span class=&p&&[&/span&&span class=&n&&alpha&/span&&span class=&p&&(:,&/span&&span class=&mi&&1&/span&&span class=&p&&),&/span&&span class=&n&&c&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&)]&/span& &span class=&p&&=&/span& &span class=&n&&normalize&/span&&span class=&p&&(&/span&&span class=&n&&s&/span&&span class=&o&&.*&/span&&span class=&n&&M&/span&&span class=&p&&(:,&/span&&span class=&mi&&1&/span&&span class=&p&&),&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&k&&for&/span& &span class=&n&&t&/span& &span class=&p&&=&/span& &span class=&mi&&2&/span&&span class=&p&&:&/span&&span class=&n&&T&/span&
&span class=&p&&[&/span&&span class=&n&&alpha&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&p&&),&/span&&span class=&n&&c&/span&&span class=&p&&(&/span&&span class=&n&&t&/span&&span class=&p&&)]&/span& &span class=&p&&=&/span& &span class=&n&&normalize&/span&&span class=&p&&((&/span&&span class=&n&&At&/span&&span class=&o&&*&/span&&span class=&n&&alpha&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&))&/span&&span class=&o&&.*&/span&&span class=&n&&M&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&p&&),&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&c&&% 13.59&/span&
&span class=&k&&end&/span&
&span class=&nb&&beta&/span& &span class=&p&&=&/span& &span class=&nb&&ones&/span&&span class=&p&&(&/span&&span class=&n&&K&/span&&span class=&p&&,&/span&&span class=&n&&T&/span&&span class=&p&&);&/span&
&span class=&k&&for&/span& &span class=&n&&t&/span& &span class=&p&&=&/span& &span class=&n&&T&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&mi&&1&/span&
&span class=&nb&&beta&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&n&&A&/span&&span class=&o&&*&/span&&span class=&p&&(&/span&&span class=&nb&&beta&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&o&&+&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&&span class=&o&&.*&/span&&span class=&n&&M&/span&&span class=&p&&(:,&/span&&span class=&n&&t&/span&&span class=&o&&+&/span&&span class=&mi&&1&/span&&span class=&p&&))&/span&&span class=&o&&/&/span&&span class=&n&&c&/span&&span class=&p&&(&/span&&span class=&n&&t&/span&&span class=&o&&+&/span&&span class=&mi&&1&/span&&span class=&p&&);&/span&
&span class=&c&&% 13.62&/span&
&span class=&k&&end&/span&
&span class=&nb&&gamma&/span& &span class=&p&&=&/span& &span class=&n&&alpha&/span&&span class=&o&&.*&/span&&span class=&nb&&beta&/span&&span class=&p&&;&/span&
&span class=&c&&% 13.64&/span&
&/code&&/pre&&/div&&br&还有很多不都列了。
我来装回b吧。以下代码出自这 .在老版本Matlab上，基本都比内置函数快个1~2个数量级，新版Matlab没测过了。 两组数据欧式距离（上面提了） function D=sqdist(X, Y)
D=bsxfun(@plus,dot(X,X,1)',dot(Y,Y,1))-2*(X'*Y); kmeans聚类算法（这是
&p&第二天的更新：果然已经被研究过了。。。请移步&a href=&https://www.zhihu.com/question//answer/& class=&internal&& Leng Yq 的回答&/a& 。&/p&&p&以下是民科的原回答。&/p&&br&&p&太！好！玩！啦！&/p&&p&首先，这个错觉可能最好被描述为中间的黑色地毯部分（跟狗没什么关系可以忽略）在&b&运动时比起周围延迟了&/b&。这里，图像的运动包括平移、缩放（即图片到眼睛的距离远近变化）、旋转。&/p&&p&原理暂时只能&b&猜测&/b&：视觉系统处理自然纹理，特别是类似于动物毛发纹理的运动时，不同于简单形状纹理；特别地，在运动时会强调毛发纹理，并减缓其运动速度以增加处理时间（亦可能恰是更长处理时间的结果）。这一差异通常不可察觉，但在毛发纹理被简单纹理包围时由于对比而明显。需要强调，&b&这纯粹是猜想性质的&/b&。尤其是毛发这一类别，可能也包括其他重要的自然纹理。&/p&&p&以下提供简单的实验，建议手机点开图片后实际测试。电脑的话，上下移动答案也有类似的效果。&/p&&p&0. 原图&/p&&figure&&img src=&https://pic2.zhimg.com/50/v2-7affdf8a55db_b.jpg& data-rawwidth=&575& data-rawheight=&767& class=&origin_image zh-lightbox-thumb& width=&575& data-original=&https://pic2.zhimg.com/50/v2-7affdf8a55db_r.jpg&&&/figure&&br&&br&&p&1. 如 &a href=&https://www.zhihu.com/question//answer/& class=&internal&&Lin Heiline 的回答&/a& 所说，依赖于周围的格子。没有格子时，没有这个现象。&/p&&figure&&img src=&https://pic1.zhimg.com/50/v2-2a1b3bac8f612b0ea722_b.jpg& data-rawwidth=&310& data-rawheight=&497& class=&content_image& width=&310&&&/figure&&br&&p&2. 当格子中间是黑色色块时，没有这个现象：&/p&&figure&&img src=&https://pic1.zhimg.com/50/v2-f356aa1db757c15c689d5bb948cde221_b.jpg& data-rawwidth=&575& data-rawheight=&767& class=&origin_image zh-lightbox-thumb& width=&575& data-original=&https://pic1.zhimg.com/50/v2-f356aa1db757c15c689d5bb948cde221_r.jpg&&&/figure&&br&&p&3.当格子中间是自然纹理（石头）时，没有这个现象&/p&&figure&&img src=&https://pic1.zhimg.com/50/v2-3d143ad9f34dbbadb55f5b2_b.jpg& data-rawwidth=&599& data-rawheight=&800& class=&origin_image zh-lightbox-thumb& width=&599& data-original=&https://pic1.zhimg.com/50/v2-3d143ad9f34dbbadb55f5b2_r.jpg&&&/figure&&br&&br&&br&&p&4. 当中间是另一种毛发纹理时，有延迟现象：&/p&&figure&&img src=&https://pic3.zhimg.com/50/v2-35ead0dde9d7d0409f80_b.jpg& data-rawwidth=&599& data-rawheight=&800& class=&origin_image zh-lightbox-thumb& width=&599& data-original=&https://pic3.zhimg.com/50/v2-35ead0dde9d7d0409f80_r.jpg&&&/figure&&br&&br&&br&&p&5. 去掉格子，改为白色背景+水平线，则左右运动无晃动感，垂直运动晃动明显。（建议把手机横过来上下晃，只要和线段平行，晃动的错觉可能就会减弱。对我来说是消失了，但也有人说没影响…）&/p&&figure&&img src=&https://pic1.zhimg.com/50/v2-84bd3a40b012b4e3f0f067_b.jpg& data-rawwidth=&437& data-rawheight=&556& class=&origin_image zh-lightbox-thumb& width=&437& data-original=&https://pic1.zhimg.com/50/v2-84bd3a40b012b4e3f0f067_r.jpg&&&/figure&&p&第五种似乎支持前面说的，这效应来自于&b&运动感知的对比。&/b&&/p&&br&&p&补充：第五幅图的效应不是很可靠，对我来说只在低速下才有明显的区别。也可能只是我的幻觉。。。&/p&&p&再补充：看不到原图效应的话试试下图，原理是类似的。&/p&&figure&&img data-rawwidth=&1024& data-rawheight=&1024& src=&https://pic4.zhimg.com/50/v2-1c6170fca94d5080afe7ddcb7c29efd0_b.jpg& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic4.zhimg.com/50/v2-1c6170fca94d5080afe7ddcb7c29efd0_r.jpg&&&/figure&
第二天的更新：果然已经被研究过了。。。请移步 。以下是民科的原回答。 太！好！玩！啦！首先，这个错觉可能最好被描述为中间的黑色地毯部分（跟狗没什么关系可以忽略）在运动时比起周围延迟了。这里，图像的运动包括平移、缩放（即图片到…
&p&本来是数学专业本硕博毕业，后来没有继续做数学研究，跨界做金融风险管理...&/p&&p&在建模分析的时候，大多数要用到数学知识....&/p&&p&生活中多了解一些数学、物理类知识，真的会改变对事物的认知、对生活的态度!!!&/p&&br&&p&看过的不多，但是真心喜欢这类科普书...&/p&&br&&p&&b&推荐三本数学类的，《什么是数学》《魔鬼数学》《无言的宇宙》；&/b&&/p&&p&&b&
三本物理类的，《上帝掷骰子吗》《超越时空》《果壳中的宇宙》。&/b&&/p&&br&&p&都是特别经典的科普书，有空读一读受益匪浅。&br&&/p&&p&1. &b&什么是数学（&/b&&b& What is
Mathematics&/b&&b&）&/b&&/p&&p&
作者: [美] R·柯朗 H·罗宾 著 / I·斯图尔特 修订 &/p&&p&
译者: 左平 / 张饴慈&/p&&figure&&img src=&https://pic4.zhimg.com/50/fac601d679_b.jpg& data-rawwidth=&297& data-rawheight=&447& class=&content_image& width=&297&&&/figure&&br&&p& 《什么是数学》既是为初学者也是为专家，既是为学生也是为教师，既是为哲学家也是为工程师而写的。&/p&&p&
作为一本世界著名的数学科普读物。书中搜集了许多经典的数学珍品，给出了数学世界的一组有趣的深入浅出的图画，对整个数学领域中的基本概念与方法，做了精深而生动的阐述。&/p&&p&&br&&b&2.&/b&&b&魔鬼数学（&/b&&b&how Not to Be Wrong: The Power of
Mathematical Thinking&/b&&b&）&/b&&br&
作者: &a href=&//link.zhihu.com/?target=https%3A//book.douban.com/search/& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&[美] 乔丹o艾伦伯格（Jordan Ellenberg）&/a&&br&
译者: &a href=&//link.zhihu.com/?target=https%3A//book.douban.com/search/%25E8%%25E5%25B0%258F%25E9%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&胡小锐&/a&&br&&figure&&img src=&https://pic1.zhimg.com/50/93aefe8fbdc6_b.jpg& data-rawwidth=&312& data-rawheight=&426& class=&content_image& width=&312&&&/figure&&br&
如果你是一个有“数学焦虑症”的人，你可能不会相信有一天你会爱上数学。原因在于，我们在学校所学的数学知识看上去不过是一堆沉闷的规则、定律和公理，都是前人传下来的，而且是不容置疑的。在《魔鬼数学》中，世界知名数学家乔丹o艾伦伯格告诉我们这样的认识是错误的。数学与我们所做的每一件事都息息相关，可以帮助我们洞见在混沌和嘈杂的表象之下日常生活的隐性结构和秩序。&/p&&p&
数学是一门告诉我们“如何做才不会犯错”的科学，是经年累月的努力、争论所锤炼出来的。你应该提前多长时间到达机场？民意调查的结果真的能代表人们的意愿吗？为什么父母都是高个子，孩子的身高却比较矮？用什么策略买彩票才能中大奖？《魔鬼数学》运用数学方法分析和解决了很多的日常生活问题，帮助数学门外汉习得用数学思维思考问题的技能。&/p&&p&3. &b&无言的宇宙（&/b&&b&The Universe in
Zero Words:the Story of Mathematics As Told Through Equations&/b&&b&）&/b&&/p&&p&
作者: &a href=&//link.zhihu.com/?target=https%3A//book.douban.com/search/%25E8%25BE%25BE%25E7%25BA%25B3%25C2%25B7%25E9%25BA%25A6%25E8%2582%25AF%25E9%25BD%2590& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&[美]达纳·麦肯齐&/a&&/p&&p&
译者: &a href=&//link.zhihu.com/?target=https%3A//book.douban.com/search/%25E6%259D%258E%25E6%25B0%25B8%25E5%25AD%25A6& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&李永学&/a&&/p&&figure&&img src=&https://pic4.zhimg.com/50/b52abdd1f260_b.jpg& data-rawwidth=&309& data-rawheight=&432& class=&content_image& width=&309&&&/figure&&br&&p& 《无言的宇宙》向你讲述的是人类历史上24个美丽而伟大的公式背后的故事，从基本的1 + 1 = 2到揭示电磁现象的“麦克斯韦方程”，从著名的E = mc?到神秘的“汉密尔顿的四元数方程”，清晰地解释了每一个方程的含义、谁（如何）发现了它，他们在人类发展史上和现实生活中发挥的巨大作用。&/p&&p&
世界有多大？时间有多长？我们是什么？当我们追问这样的问题时，总是如此高贵又可爱。而无论我们的答案是“寄蜉蝣于天地，渺沧海之一粟。哀吾生之须臾，羡长江之无穷”。还是“在这个世界里，有一个和我们对应的唯一，也许越过时间这个概念，我们就能遇见它。”&/p&&br&&p&&b&1. 上帝掷骰子吗&/b&&/p&&p&
作者: &a href=&//link.zhihu.com/?target=https%3A//book.douban.com/search/%25E6%259B%25B9%25E5%25A4%25A9%25E5%& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&曹天元&/a&&/p&&figure&&img src=&https://pic4.zhimg.com/50/b33c21c3db8_b.jpg& data-rawwidth=&200& data-rawheight=&266& class=&content_image& width=&200&&&/figure&&p&清华的刘兵对此书的总体评价――“努力言说不可言说之事”。 正如副标题所说，这是一本量子物理的“史话”，它不是一本教科书，讲述的是一门科学的历史，或许并不完全准确，但是可以知道这门科学的发展历程，了解很多基本的量子物理理论，了解了批伟大的科学家的事迹。从这个角度来看，这本书对量子论的介绍，实在是精彩绝伦。&/p&&p&&b&
2. 超越时空（ Hyperspace）&/b&&/p&&p&
作者: [美] 加来道雄（Kaku, M） 译者: 刘玉玺 / 曹志良&/p&&figure&&img src=&https://pic4.zhimg.com/50/cab494abaffb7db17390d31_b.jpg& data-rawwidth=&294& data-rawheight=&453& class=&content_image& width=&294&&&/figure&&br&&p&
读到最好的现代物理学普及书籍之一，想象奇崛而又娓娓道来，相关文化史、文艺史的故事信手拈来，毕加索的绘画、厄普代克的诗，海因莱因的小说……比同类书籍一说故事就离不开那些物理学家八卦老梗的恶趣味，好太多。并且，对于谢耳朵从事的超弦理论有很透彻的解释。&/p&&p&必须附上一张谢耳朵，哈哈.........&/p&&figure&&img src=&https://pic3.zhimg.com/50/3a7e5d6eb9c95c7ad62b6fbd68e7c5f7_b.jpg& data-rawwidth=&600& data-rawheight=&600& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic3.zhimg.com/50/3a7e5d6eb9c95c7ad62b6fbd68e7c5f7_r.jpg&&&/figure&&br&&p&&b&
3. 果壳中的宇宙( The Universe In a Nutshell)&/b&&/p&&p&
作者: [英] 史蒂芬·霍金
译者: 吴忠超&/p&&figure&&img src=&https://pic1.zhimg.com/50/8b5f26d5b9c4b2bb72659_b.jpg& data-rawwidth=&600& data-rawheight=&346& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic1.zhimg.com/50/8b5f26d5b9c4b2bb72659_r.jpg&&&/figure&&br&&p&
试着以一种孩子的心态来阅读，来想象，书上说到的任何一个实验虽然都带有它的客观性和严谨性，但我们很容易就能在字里行间体会到作者那份热爱科学的由衷之情，用许多看似幽默的词句引导着我大脑中任何一丝不确定的猜测和推想。时间并不是什么神圣的东西，它也是宇宙中的一员。完后会对周围的世界有种不一样的感觉，可能是一会会，也可能持续很长一段时间。&/p&&p&个人觉得，读数学、物理类的科普书真的比玄幻小说爽得多....&/p&&p&有空多看看书，找到好书一起分享。&/p&&p&未完待续....&/p&
本来是数学专业本硕博毕业，后来没有继续做数学研究，跨界做金融风险管理...在建模分析的时候，大多数要用到数学知识....生活中多了解一些数学、物理类知识，真的会改变对事物的认知、对生活的态度!!! 看过的不多，但是真心喜欢这类科普书... 推荐三本数学…
&p&看闲书能直接看到大师们的深刻思想的话，何必听普通人的杂烩呢。视野差了好几个量级呢。&/p&&p&1. 通向实在之路——宇宙法则的完全指南
罗杰·彭罗斯&/p&&p&不需怎么介绍。标题很准确，一本完全指南。&/p&&p&看了目录你就懂。&/p&&blockquote&第一章 科学的根源&br&第二章 古代定理和现代问题&br&第三章 物理世界里数的种类&br&第四章 奇幻的复数&br&第五章 对数、幂和根的几何学&br&第六章 实数微积分&br&第七章 复数微积分&br&第八章 黎曼曲面和复映射&br&第九章 傅里叶分解和超函数&br&第十章 曲面&br&第十一章 超复数&br&第十二章 n维流形&br&第十三章 对称群&br&第十四章 流形上的微积分&br&第十五章 纤维丛和规范联络&br&第十六章 无限的阶梯&br&第十七章 时空&br&第十八章 闵可夫斯基几何&br&第十九章 麦克斯韦和爱因斯坦的经典场&br&第二十章 拉格朗日量和哈密顿量&br&第二十一章 量子粒子&br&第二十二章 量子代数、几何和自旋&br&第二十三章 纠缠的量子世界&br&第二十四章 狄拉克电子和反粒子&br&第二十五章 粒子物理学的标准模型&br&第二十六章 量子场论&br&第二十七章 大爆炸及其热力学传奇&br&第二十八章 旱期宇宙的推测性理论&br&第二十九章 测量疑难&br&第三十一章 超对称、超维和弦&br&第三十二章 更为狭窄的爱因斯坦途径；圈变量&br&第三十三章 更彻底的观点；扭量理论&br&第三十四章 实在之路通向何方&/blockquote&&p&&br&&/p&&p&2. Ideas and Opinions
Albert Einstein&br&爱因斯坦文集。直接和站在顶点的男人对话吧。&/p&&p&我觉得似乎没有其他几位大神写的那么好，大概爱神意不在此。这本书只是别人整理的爱因斯坦的文稿。考虑到他的深远的影响力，还是把这本书列出来。&/p&&p&爱因斯坦带的学生最后没有出现大师级人物。而费米不仅自己厉害，带学生也厉害，六位诺奖得主，其中第一位是杨振宁。&/p&&p&爱因斯坦还和一位普林斯顿大学的教授合写过一本科普书《物理学的进化》。很精彩，适合初高中生了解物理学的发展历程。有孩子的可以预备一本..&/p&&p&3. The Character of Physical Law/物理之美
理查德 费曼&/p&&p&这是我读过的最好的物理类“闲书”了。充满了对于物理定律有趣而深刻的思考。&br&这是在康奈尔大学的一个系列讲座。从学生到教授都来聆听大师对物理定律独特的诠释。思维如水般自在流淌。&br&这个系列讲座还有录音，耳机里传出费曼的声音，可以感觉到这位导师近在咫尺。&/p&&p&无论是《费曼物理学讲义》还是这本《物理之美》都可以称得上传世之作。&br&每一名物理系的学生都可以叫您一声老师啊.. &br&（比爱因斯坦的Ideas and Opinions好太多。忽然觉得爱神的学生都没能成长为大师级的科学家也是有原因的。与费曼在教学上的倾心付出相比，爱因斯坦几乎把全部精力都用来独自解决难题。）&/p&&p&&br&&/p&&p&4.玻尔哲学文选
尼尔斯 玻尔&br&哥本哈根学派的开创者。&br&之所以哥本哈根是量子力学的圣地，一度也是物理学术中心，都因为一个人——玻尔。&br&玻尔不仅自己厉害，更重要的是，他会忽悠.. 不对，是教导年轻人。&/p&&p&他和爱因斯坦是一辈人。但是他们下一辈人几乎都更信任玻尔的意见。&br&美剧《曼哈顿计划》里，玻尔一出场，几乎所有人都要去拜见这位物理学的领袖。&br&最后的结果证明玻尔的方向非常准确。&br&他同时也慧眼识英才，把当时最优秀的年轻人带入了自己的研究方向。在海森堡和泡利22岁时就把这两位年轻人揽入哥本哈根。&br&而爱因斯坦晚年的研究则是门可罗雀。&/p&&p&和爱因斯坦那场旷日持久的论战时期写的文章是此书最精华的部分。科学史上很难找到比这还要精彩、重要的论战了。&/p&&p&哥本哈根学派的闲书总是和哲学思想有些关联。&/p&&figure&&img src=&https://pic3.zhimg.com/50/a382c597cbd9cf6c37bd47b7f3f8108e_b.jpg& data-caption=&& data-rawwidth=&248& data-rawheight=&330& class=&content_image& width=&248&&&/figure&&p&&br&&/p&&p&这是玻尔的家徽。他认为那个太极图是最能代表“互补思想”的图案，于是放入了徽章中。&/p&&p&互补思想是从波粒二象性/不确定性原理里延伸出来的一种哲学思想。&/p&&p&主要思想是，人类不可能准确描述自然，每一种自然科学的概念在描述宇宙的一部分信息时，总会失去另一部分信息。我们会用轨道去描述物体的运动，但是实际上，宇宙并没有由粒子模型构成的物体，也没有人类观察到的“轨道”。无论是粒子，还是波，都是自然科学的概念，都只是对自然某一面的描述。&/p&&p&&br&&/p&&p&&br&&/p&&p&5. Physics and Philosophy
维纳.海森堡&br&哥本哈根学派最好的“闲书”。玻尔的思想已然深远，而海森堡这本书比他老师玻尔的书更加深刻。&/p&&p&量子力学不只是对科学的变革，更是对人类思维方式，如何看待宇宙的变革。&br&有一段讨论自然科学里的常数让人印象深刻。&br&讨论经典力学、量子力学和热力学、电动力学等物理理论的相容性时更是让人大开眼界。&/p&&p&比费曼的书差的只是年代有些久远了，没有涉及到上个世纪自然科学的后续发展。但是这本书对物理学思想的探讨更为深入，费曼的书则更关注物理定律。&/p&&p&海森堡是玻尔的学生。&br&费曼是玻尔学生的学生。&/p&&p&6. What is Life/生命是什么
薛定谔&br&物理学思想引入到其他领域的典范。&/p&&p&7.确定性的终结
伊利亚.普利高津&br&热力学/统计物理一直是物理学里很特别的一个领域。这位非平衡热力学大师的视角会是怎样的？&br&1977年诺奖得主普利高津从热力学和量子力学的角度对确定性进行的探讨。&/p&&p&&br&&/p&&p&8.Symmetry/对称
Weyl&br&外尔被称为上世纪上半叶出现的最后一位“全能数学家”。师从希尔伯特。&br&多次再版的一本名作。历史、数学、科学、自然中的对称性。&br&（我还没来得及读这本《对称》就离开物理系了，但是觉得应该很精彩吧，毕竟是另一位的大神推荐的...）&/p&&p&9. 科学发现的逻辑 卡尔·波普尔&br&从爱因斯坦到索罗斯都推崇的一种看待世界的方法。&br&这本书可能需要静下心来才能读进去。&/p&&p&10. Scientific American/中文版：环球科学 （杂志）&br&也可以涉猎一下这个时代各个领域的进展。&br&在国内的时候连续六七年，几乎每期不落地看。很好的休闲读物。&/p&&p&&br&&/p&&p&其他的想到再补充。&/p&&hr&&p&还有一个夏天的世界，留给喜欢Summer的你：summer-clover（个人公众号）。&/p&
看闲书能直接看到大师们的深刻思想的话，何必听普通人的杂烩呢。视野差了好几个量级呢。1. 通向实在之路——宇宙法则的完全指南 罗杰·彭罗斯不需怎么介绍。标题很准确，一本完全指南。看了目录你就懂。第一章 科学的根源 第二章 古代定理和现代问题 第三章…
研究生期间，做的方向是双足机器人的步态模型，投了一篇SCI的论文，评委给回了些修改意见，一种很关键的一个模型的数学基础必须解决，否则不给发表。于是天天想天天想，怎么也没有思路，直到有一天在图书馆发呆，看到边上的大一师妹正在研究微积分的拉格朗日中值定理，突然受到启发，然后问题迎刃而解，最后论文顺利发表。总结一下就是，没事要多去图书馆转转，见到师妹要重点关注。
研究生期间，做的方向是双足机器人的步态模型，投了一篇SCI的论文，评委给回了些修改意见，一种很关键的一个模型的数学基础必须解决，否则不给发表。于是天天想天天想，怎么也没有思路，直到有一天在图书馆发呆，看到边上的大一师妹正在研究微积分的拉格朗…
&a href=&//link.zhihu.com/?target=http%3A//weibo.com//profile%3Frightmod%3D1%26wvr%3D6%26mod%3Dpersoninfo%26is_all%3D1& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&假面塞壬的微博&/a&&br&欢迎制服爱好者来微博一起玩~ 答主我致力于把小众爱好发展成大众爱好，哼哼~&br&&br&谢邀。资深制服控，不过我只是为了满足自己哈哈哈(￣▽￣) 穿上不同的制服体验不同的角色对于我来说非常有趣~ 不过邀请我来答这个问题的童鞋们大概也只是想让我上图吧~ 这个问题有请评论里的男人们给出回答(?o?ωo??)&br&&br&之前一个高票回答比较危险，所以还是转移来这里啦~~以后都在这更新好了~~&br&&br&女仆系列~ 不要幻想我是小女仆了，我是要把女仆穿成女王的S大人(￣▽￣)~*&br&&br&&br&真女王系列( ????? )&br&&br&&br&女警官系列~ 缺个手铐还是差了点啊&br&&br&&br&海军系列。硬朗的军装元素和女性柔美曲线的结合非常诱人(?ω?)&br&&br&&br&空姐系列。比较显气质的制服。不过这套的上衣太小了，勒死了快。。而且这套怎么拍都显得好壮我也是醉了。。啊对了答主曾经很想去考空姐，但父母总担心空难之类的(-ι_- )&br&&br&哎，喷子请免开尊口喷我了，省点口水，您看到我别点开我的回答就当没看到我了。还有那些上来就说自己是揽胜车主什么的别给我私信了，我本人又丑又胖怕吓得你们阳痿真的。&br&&br&好多童鞋说我并没有回答问题。我上自己的制服照，一是因为我自己的爱好我拥有各式各样的制服并且我经常会穿了拍个照什么的，二是我觉得我穿制服的样子还可以，发上来看了的人自然会懂为什么男人会喜欢制服。&br&&br&非要我}