$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}s_x & 0 \\\\0 & s_y\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ffb\u8f6c<\/h4>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & a \\\\0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5207\u53d8<\/h4>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & a \\\\0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n\u65cb\u8f6c\u65f6\u9ed8\u8ba4\u7ed5\u5750\u6807\u539f\u70b9\u65cb\u8f6c<\/p>\n\n\n\n$$\\mathbf{R}_\\theta=\\begin{bmatrix}\\cos \\theta & -\\sin \\theta \\\\\\sin \\theta & \\cos \\theta\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u65cb\u8f6c$-\\theta$ \u65f6$$\\mathbf{R}_{-\\theta}=\\begin{bmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta\\end{bmatrix}= \\mathbf{R}_{\\theta}^T= \\mathbf{R}_{\\theta}^{-1}$$ <\/p>\n\n\n\n\u4e5f\u5c31\u662f\u65cb\u8f6c$\\theta$ \u65f6\u7684\u8f6c\u7f6e\u548c\u9006\u8fd0\u7b97<\/p>\n\n\n\n\u7ebf\u6027\u53d8\u6362<\/h4>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}+\\begin{bmatrix}t_x \\\\t_y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n\u70b9\u548c\u5411\u91cf\u7684\u8868\u793a\u65b9\u6cd5\u548c\u8ba1\u7b97<\/h3>\n\n\n\n2D\u70b9 = $(x, y, 1)^{\\top}$2D\u5411\u91cf = $(x, y, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u5229\u7528\u9f50\u6b21\u5750\u6807\u7684\u8fdb\u884c\u5e73\u79fb\u53d8\u6362<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\w^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}x+t_x \\\\y+t_y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u4e8c\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n\u4eff\u5c04\u53d8\u6362<\/h3>\n\n\n\n\u4eff\u5c04\u53d8\u6362=\u7ebf\u6027\u53d8\u6362+\u5e73\u79fb<\/p>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & t_x \\\\c & d & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y\\right)=\\begin{bmatrix}s_x & 0 & 0 \\\\0 & s_y & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n$$\\mathbf{R}(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y\\right)=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u9006\u53d8\u6362<\/h3>\n\n\n\n$M^{-1}$ \u5728\u77e9\u9635\u548c\u51e0\u4f55\u610f\u4e49\u4e0a\u90fd\u662f\u53d8\u6362 $M$ \u7684\u9006\u53d8\u6362<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u590d\u5408\u53d8\u6362<\/h3>\n\n\n\n\u5b8c\u6210\u4ee5\u4e0b\u53d8\u6362\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & a \\\\0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5207\u53d8<\/h4>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & a \\\\0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n\u65cb\u8f6c\u65f6\u9ed8\u8ba4\u7ed5\u5750\u6807\u539f\u70b9\u65cb\u8f6c<\/p>\n\n\n\n$$\\mathbf{R}_\\theta=\\begin{bmatrix}\\cos \\theta & -\\sin \\theta \\\\\\sin \\theta & \\cos \\theta\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u65cb\u8f6c$-\\theta$ \u65f6$$\\mathbf{R}_{-\\theta}=\\begin{bmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta\\end{bmatrix}= \\mathbf{R}_{\\theta}^T= \\mathbf{R}_{\\theta}^{-1}$$ <\/p>\n\n\n\n\u4e5f\u5c31\u662f\u65cb\u8f6c$\\theta$ \u65f6\u7684\u8f6c\u7f6e\u548c\u9006\u8fd0\u7b97<\/p>\n\n\n\n\u7ebf\u6027\u53d8\u6362<\/h4>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}+\\begin{bmatrix}t_x \\\\t_y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n\u70b9\u548c\u5411\u91cf\u7684\u8868\u793a\u65b9\u6cd5\u548c\u8ba1\u7b97<\/h3>\n\n\n\n2D\u70b9 = $(x, y, 1)^{\\top}$2D\u5411\u91cf = $(x, y, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u5229\u7528\u9f50\u6b21\u5750\u6807\u7684\u8fdb\u884c\u5e73\u79fb\u53d8\u6362<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\w^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}x+t_x \\\\y+t_y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u4e8c\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n\u4eff\u5c04\u53d8\u6362<\/h3>\n\n\n\n\u4eff\u5c04\u53d8\u6362=\u7ebf\u6027\u53d8\u6362+\u5e73\u79fb<\/p>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & t_x \\\\c & d & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y\\right)=\\begin{bmatrix}s_x & 0 & 0 \\\\0 & s_y & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n$$\\mathbf{R}(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y\\right)=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u9006\u53d8\u6362<\/h3>\n\n\n\n$M^{-1}$ \u5728\u77e9\u9635\u548c\u51e0\u4f55\u610f\u4e49\u4e0a\u90fd\u662f\u53d8\u6362 $M$ \u7684\u9006\u53d8\u6362<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u590d\u5408\u53d8\u6362<\/h3>\n\n\n\n\u5b8c\u6210\u4ee5\u4e0b\u53d8\u6362\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & a \\\\0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n\u65cb\u8f6c\u65f6\u9ed8\u8ba4\u7ed5\u5750\u6807\u539f\u70b9\u65cb\u8f6c<\/p>\n\n\n\n$$\\mathbf{R}_\\theta=\\begin{bmatrix}\\cos \\theta & -\\sin \\theta \\\\\\sin \\theta & \\cos \\theta\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u65cb\u8f6c$-\\theta$ \u65f6$$\\mathbf{R}_{-\\theta}=\\begin{bmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta\\end{bmatrix}= \\mathbf{R}_{\\theta}^T= \\mathbf{R}_{\\theta}^{-1}$$ <\/p>\n\n\n\n\u4e5f\u5c31\u662f\u65cb\u8f6c$\\theta$ \u65f6\u7684\u8f6c\u7f6e\u548c\u9006\u8fd0\u7b97<\/p>\n\n\n\n\u7ebf\u6027\u53d8\u6362<\/h4>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}+\\begin{bmatrix}t_x \\\\t_y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n\u70b9\u548c\u5411\u91cf\u7684\u8868\u793a\u65b9\u6cd5\u548c\u8ba1\u7b97<\/h3>\n\n\n\n2D\u70b9 = $(x, y, 1)^{\\top}$2D\u5411\u91cf = $(x, y, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u5229\u7528\u9f50\u6b21\u5750\u6807\u7684\u8fdb\u884c\u5e73\u79fb\u53d8\u6362<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\w^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}x+t_x \\\\y+t_y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u4e8c\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n\u4eff\u5c04\u53d8\u6362<\/h3>\n\n\n\n\u4eff\u5c04\u53d8\u6362=\u7ebf\u6027\u53d8\u6362+\u5e73\u79fb<\/p>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & t_x \\\\c & d & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y\\right)=\\begin{bmatrix}s_x & 0 & 0 \\\\0 & s_y & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n$$\\mathbf{R}(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y\\right)=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u9006\u53d8\u6362<\/h3>\n\n\n\n$M^{-1}$ \u5728\u77e9\u9635\u548c\u51e0\u4f55\u610f\u4e49\u4e0a\u90fd\u662f\u53d8\u6362 $M$ \u7684\u9006\u53d8\u6362<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u590d\u5408\u53d8\u6362<\/h3>\n\n\n\n\u5b8c\u6210\u4ee5\u4e0b\u53d8\u6362\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u65cb\u8f6c\u65f6\u9ed8\u8ba4\u7ed5\u5750\u6807\u539f\u70b9\u65cb\u8f6c<\/p>\n\n\n\n
$$\\mathbf{R}_\\theta=\\begin{bmatrix}\\cos \\theta & -\\sin \\theta \\\\\\sin \\theta & \\cos \\theta\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u65cb\u8f6c$-\\theta$ \u65f6$$\\mathbf{R}_{-\\theta}=\\begin{bmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta\\end{bmatrix}= \\mathbf{R}_{\\theta}^T= \\mathbf{R}_{\\theta}^{-1}$$ <\/p>\n\n\n\n\u4e5f\u5c31\u662f\u65cb\u8f6c$\\theta$ \u65f6\u7684\u8f6c\u7f6e\u548c\u9006\u8fd0\u7b97<\/p>\n\n\n\n\u7ebf\u6027\u53d8\u6362<\/h4>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}+\\begin{bmatrix}t_x \\\\t_y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n\u70b9\u548c\u5411\u91cf\u7684\u8868\u793a\u65b9\u6cd5\u548c\u8ba1\u7b97<\/h3>\n\n\n\n2D\u70b9 = $(x, y, 1)^{\\top}$2D\u5411\u91cf = $(x, y, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u5229\u7528\u9f50\u6b21\u5750\u6807\u7684\u8fdb\u884c\u5e73\u79fb\u53d8\u6362<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\w^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}x+t_x \\\\y+t_y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u4e8c\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n\u4eff\u5c04\u53d8\u6362<\/h3>\n\n\n\n\u4eff\u5c04\u53d8\u6362=\u7ebf\u6027\u53d8\u6362+\u5e73\u79fb<\/p>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & t_x \\\\c & d & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y\\right)=\\begin{bmatrix}s_x & 0 & 0 \\\\0 & s_y & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n$$\\mathbf{R}(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y\\right)=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u9006\u53d8\u6362<\/h3>\n\n\n\n$M^{-1}$ \u5728\u77e9\u9635\u548c\u51e0\u4f55\u610f\u4e49\u4e0a\u90fd\u662f\u53d8\u6362 $M$ \u7684\u9006\u53d8\u6362<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u590d\u5408\u53d8\u6362<\/h3>\n\n\n\n\u5b8c\u6210\u4ee5\u4e0b\u53d8\u6362\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u65cb\u8f6c$-\\theta$ \u65f6$$\\mathbf{R}_{-\\theta}=\\begin{bmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta\\end{bmatrix}= \\mathbf{R}_{\\theta}^T= \\mathbf{R}_{\\theta}^{-1}$$ <\/p>\n\n\n\n
\u4e5f\u5c31\u662f\u65cb\u8f6c$\\theta$ \u65f6\u7684\u8f6c\u7f6e\u548c\u9006\u8fd0\u7b97<\/p>\n\n\n\n
$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}$$<\/p>\n\n\n\n
$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix}=\\begin{bmatrix}a & b \\\\c & d\\end{bmatrix}\\begin{bmatrix}x \\\\y\\end{bmatrix}+\\begin{bmatrix}t_x \\\\t_y\\end{bmatrix}$$<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n\u70b9\u548c\u5411\u91cf\u7684\u8868\u793a\u65b9\u6cd5\u548c\u8ba1\u7b97<\/h3>\n\n\n\n2D\u70b9 = $(x, y, 1)^{\\top}$2D\u5411\u91cf = $(x, y, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u5229\u7528\u9f50\u6b21\u5750\u6807\u7684\u8fdb\u884c\u5e73\u79fb\u53d8\u6362<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\w^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}x+t_x \\\\y+t_y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u4e8c\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n\u4eff\u5c04\u53d8\u6362<\/h3>\n\n\n\n\u4eff\u5c04\u53d8\u6362=\u7ebf\u6027\u53d8\u6362+\u5e73\u79fb<\/p>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & t_x \\\\c & d & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y\\right)=\\begin{bmatrix}s_x & 0 & 0 \\\\0 & s_y & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n$$\\mathbf{R}(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y\\right)=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u9006\u53d8\u6362<\/h3>\n\n\n\n$M^{-1}$ \u5728\u77e9\u9635\u548c\u51e0\u4f55\u610f\u4e49\u4e0a\u90fd\u662f\u53d8\u6362 $M$ \u7684\u9006\u53d8\u6362<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u590d\u5408\u53d8\u6362<\/h3>\n\n\n\n\u5b8c\u6210\u4ee5\u4e0b\u53d8\u6362\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
2D\u70b9 = $(x, y, 1)^{\\top}$2D\u5411\u91cf = $(x, y, 0)^{\\top}$<\/p>\n\n\n\n
\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n
$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\w^{\\prime}\\end{bmatrix}=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}x+t_x \\\\y+t_y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n
\u4eff\u5c04\u53d8\u6362=\u7ebf\u6027\u53d8\u6362+\u5e73\u79fb<\/p>\n\n\n\n
$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & t_x \\\\c & d & t_y \\\\0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n
$$\\mathbf{S}\\left(s_x, s_y\\right)=\\begin{bmatrix}s_x & 0 & 0 \\\\0 & s_y & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
$$\\mathbf{R}(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
$$\\mathbf{T}\\left(t_x, t_y\\right)=\\begin{bmatrix}1 & 0 & t_x \\\\0 & 1 & t_y \\\\0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
$M^{-1}$ \u5728\u77e9\u9635\u548c\u51e0\u4f55\u610f\u4e49\u4e0a\u90fd\u662f\u53d8\u6362 $M$ \u7684\u9006\u53d8\u6362<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u590d\u5408\u53d8\u6362<\/h3>\n\n\n\n\u5b8c\u6210\u4ee5\u4e0b\u53d8\u6362\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u5b8c\u6210\u4ee5\u4e0b\u53d8\u6362\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u9700\u8981\u5148\u65cb\u8f6c\u3001\u518d\u5e73\u79fb\uff0c\u56e0\u4e3a\u65cb\u8f6c\u65f6\u4ee5\u5750\u6807\u539f\u70b9\u4e3a\u4e2d\u5fc3\u7684\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u5982\u679c\u5148\u5e73\u79fb\u3001\u518d\u65cb\u8f6c\uff0c\u5219\u4f1a\u5f97\u5230\u5df2\u4e0b\u7ed3\u679c\uff1a<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u8fde\u7eed\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n\u5206\u89e3\u590d\u6742\u53d8\u6362<\/h3>\n\n\n\n\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u6ca1\u6709\u4ea4\u6362\u5f8b\u7684\uff0c\u6240\u4ee5\uff1a$$R_{45} \\cdot T_{(1,0)} \\neq T_{(1,0)} \\cdot R_{45}$$<\/p>\n\n\n\n
\u77e9\u9635\u7684\u8ba1\u7b97\u662f\u4ece\u53f3\u5f80\u5de6\u7684\uff1a<\/p>\n\n\n\n
$$T_{(1,0)} \\cdot R_{45}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}=\\begin{bmatrix}1 & 0 & 1 \\\\0 & 1 & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}\\cos 45^{\\circ} & -\\sin 45^{\\circ} & 0 \\\\\\sin 45^{\\circ} & \\cos 45^{\\circ} & 0 \\\\0 & 0 & 1\\end{bmatrix}\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n
\u5982\u679c\u6709\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u5e8f\u5217A1\uff0cA2\uff0cA3\uff0c\u2026<\/p>\n\n\n\n
$$\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot\\begin{bmatrix}x \\\\y \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n
\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u7387\u6765\u6539\u53d8\u8fd0\u7b97\u987a\u5e8f\uff0c\u5148\u7b97\u524d\u9762\u7684 $\\mathbf{A}_n \\cdots \\mathbf{A}_2 \\cdot \\mathbf{A}_1 \\cdot $\uff0c\u7136\u540e\u518d\u548c\u5750\u6807\u5411\u91cf\u76f8\u4e58<\/p>\n\n\n\n
\u5982\u679c\u9700\u8981\u65cb\u8f6c\u4e00\u4e2a\u9876\u70b9\u4e0d\u5728\u5750\u6807\u539f\u70b9\u7684\u56fe\u50cf\uff0c\u9700\u8981\u5148\u5c06\u56fe\u50cf\u5e73\u79fb\u5230\u5750\u6807\u539f\u70b9\uff0c\u518d\u65cb\u8f6c\uff0c\u6700\u540e\u518d\u5e73\u79fb\u56de\u53bb<\/p>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n\u4e09\u7ef4\u9f50\u6b21\u5750\u6807<\/h2>\n\n\n\n3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n\u4e09\u7ef4\u4eff\u5c04\u53d8\u6362<\/h2>\n\n\n\n \u77e9\u9635\u8868\u793a<\/h3>\n\n\n\n$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n \u4eff\u5c04\u53d8\u6362 <\/h3>\n\n\n\n\u7f29\u653e<\/h4>\n\n\n\n$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u5e73\u79fb<\/h4>\n\n\n\n$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u65cb\u8f6c<\/h4>\n\n\n\n<\/figure >\n<\/figure><\/noscript>\n\n\n\n\u7ed5x\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nx\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5y\u8f74\u65cb\u8f6c<\/h5>\n\n\n\ny\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7ed5z\u8f74\u65cb\u8f6c<\/h5>\n\n\n\nz\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f<\/h5>\n\n\n\n\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n<\/p>\n\n\n\n<\/h2>\n","protected":false},"excerpt":{"rendered":"\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
$$\\mathbf{T}(\\mathbf{c}) \\cdot \\mathbf{R}(\\alpha) \\cdot \\mathbf{T}(-\\mathbf{c})$$<\/p>\n\n\n\n
3D\u70b9 = $(x, y, z, 1)^{\\top}$3D\u5411\u91cf = $(x, y, z, 0)^{\\top}$<\/p>\n\n\n\n
\u5411\u91cf+\u5411\u91cf=\u5411\u91cf\u70b9-\u70b9=\u5411\u91cf\u70b9+\u5411\u91cf=\u5750\u6807\u70b9+\u70b9=$\\begin{bmatrix}x \/ w \\\\y \/ w \\\\z \/ w \\\\1\\end{bmatrix}, w \\neq 0$<\/p>\n\n\n\n
$$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime} \\\\z^{\\prime} \\\\1\\end{bmatrix}=\\begin{bmatrix}a & b & c & t_x \\\\d & e & f & t_y \\\\g & h & i & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}\\cdot\\begin{bmatrix}x \\\\y \\\\z \\\\1\\end{bmatrix}$$<\/p>\n\n\n\n
$$\\mathbf{S}\\left(s_x, s_y, s_z\\right)=\\begin{bmatrix}s_x & 0 & 0 & 0 \\\\0 & s_y & 0 & 0 \\\\0 & 0 & s_z & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
$$\\mathbf{T}\\left(t_x, t_y, t_z\\right)=\\begin{bmatrix}1 & 0 & 0 & t_x \\\\0 & 1 & 0 & t_y \\\\0 & 0 & 1 & t_z \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
x\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_x(\\alpha)=\\begin{bmatrix}1 & 0 & 0 & 0 \\\\0 & \\cos \\alpha & -\\sin \\alpha & 0 \\\\0 & \\sin \\alpha & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
y\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_y(\\alpha)=\\begin{bmatrix}\\cos \\alpha & 0 & \\sin \\alpha & 0 \\\\0 & 1 & 0 & 0 \\\\-\\sin \\alpha & 0 & \\cos \\alpha & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
z\u5750\u6807\u4e0d\u53d8$$\\mathbf{R}_z(\\alpha)=\\begin{bmatrix}\\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\\\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{bmatrix}$$<\/p>\n\n\n\n
\u7ed5\u8f74n\u65cb\u8f6c\u89d2\u5ea6\u03b1\uff0c\u9ed8\u8ba4\u65cb\u8f6c\u8f74n\u8fc7\u539f\u70b9$$\\mathbf{R}(\\mathbf{n}, \\alpha)=\\cos (\\alpha) \\mathbf{I}+(1-\\cos (\\alpha)) \\mathbf{n} \\mathbf{n}^T+\\sin (\\alpha) \\underbrace{\\begin{bmatrix}0 & -n_z & n_y \\\\n_z & 0 & -n_x \\\\-n_y & n_x & 0\\end{bmatrix}}_{\\mathbf{N}}$$<\/p>\n\n\n\n
<\/p>\n\n\n\n
\u4e8c\u7ef4\u53d8\u6362 \u7ebf\u6027\u53d8\u6362 \u7f29\u653e $$\\begin{bmatrix}x^{\\prime} \\\\y^{\\prime}\\end{bmatrix …<\/p>","protected":false},"author":1,"featured_media":754,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,14,30,3],"tags":[17,15,16,12,22],"class_list":["post-703","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-games","category-games101","category-30","category-3","tag-games","tag-games101","tag-16","tag-12","tag-22"],"_links":{"self":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/comments?post=703"}],"version-history":[{"count":6,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":1969,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/703\/revisions\/1969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/754"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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