MVP\u53d8\u6362\u7684\u76ee\u7684\u662f\u5c06\u4e09\u7ef4\u7684\u573a\u666f\uff0c\u4e5f\u5c31\u662f\u76f8\u673a\u770b\u5230\u7684\u573a\u666f\uff0c\u8f6c\u5316\u5230\u88c1\u526a\u7a7a\u95f4\u4e2d\uff0c\u4e3a\u5149\u6805\u5316\u5c06\u573a\u666f\u663e\u793a\u5230\u5c4f\u5e55\u4e0a\u505a\u51c6\u5907\u3002<\/p>\n\n\n\n
MVP\u5206\u522b\u7531Model\u77e9\u9635\uff0cView\u77e9\u9635\uff0cProjection\u77e9\u9635\u4e09\u4e2a\u9f50\u6b21\u5750\u6807\u77e9\u9635\u6784\u6210\u3002\u8ba1\u7b97\u51faMVP\u77e9\u9635\u540e\u5e76\u6309\u987a\u5e8f\u76f8\u4e58\uff0c\u518d\u5c06\u6bcf\u4e2a\u5750\u6807\u4f9d\u6b21\u4e58MVP\u5750\u6807\uff0c\u5c31\u80fd\u5f97\u5230\u8be5\u5750\u6807\u5728\u88c1\u526a\u7a7a\u95f4\u4e2d\u7684\u5750\u6807\u3002<\/p>\n\n\n\n
[latex]MVP=Projection\\times View\\times Model[\/latex]<\/p>\n\n\n\n
Model\u77e9\u9635\u7528\u4e8e\u628a\u6a21\u578b\u653e\u7f6e\u5230\u573a\u666f\u4e2d
View\u77e9\u9635\u7528\u4e8e\u5c06\u76f8\u673a\u653e\u5230\u5750\u6807\u7cfb\u539f\u70b9
Projection\u77e9\u9635\u7528\u4e8e\u6839\u636e\u76f8\u673a\u7684\u89c6\u9525\u5f62\u6210\u88c1\u526a\u7a7a\u95f4<\/p>\n\n\n\n
\u5728\u5b8c\u6210\u4e0a\u8ff0\u6b65\u9aa4\u540e\uff0c\u5373MVP\u53d8\u6362\u540e\uff0c\u518d\u5c06\u88c1\u526a\u7a7a\u95f4\u8fdb\u884c\u5149\u6805\u5316\uff0c\u5c31\u80fd\u5f97\u5230\u56fe\u50cf\u4e86\u3002<\/p>\n\n\n\n
Model\u4e3a\u6a21\u578b\u77e9\u9635\uff0c\u7528\u6765\u5c06\u7269\u4f53\u9876\u70b9\u5728\u6a21\u578b\u7a7a\u95f4\u4e0b\u7684\u5750\u6807\u8f6c\u6362\u4e3a\u5728\u4e16\u754c\u7a7a\u95f4\u4e0b\u7684\u5750\u6807\u3002
\u6a21\u578b\u77e9\u9635\u7531\u4e09\u90e8\u5206\u6784\u6210\uff1a\u5e73\u79fb\u77e9\u9635\u3001\u65cb\u8f6c\u77e9\u9635\u3001\u7f29\u653e\u77e9\u9635\u3002<\/p>\n\n\n\n
\u6a21\u578b\u7a7a\u95f4\u4e2d\u662f\u4ee5\u6a21\u578b\u7684\u4e00\u4e2a\u9876\u70b9\u4e3a\u539f\u70b9\u5efa\u7acb\u5750\u6807\u7cfb\u7684\u7a7a\u95f4\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u6a21\u578b\u7684\u81ea\u5df1\u7684\u7a7a\u95f4\uff0c\u6bcf\u4e2a\u6a21\u578b\u90fd\u6709\u81ea\u5df1\u6a21\u578b\u7a7a\u95f4\u3002\u6b64\u65f6\u6a21\u578b\u8fd8\u672a\u88ab\u653e\u5165\u573a\u666f\u4e2d\u3002<\/p>\n\n\n\n
\u4e16\u754c\u7a7a\u95f4\u53ef\u4ee5\u7406\u89e3\u4e3a\u573a\u666f\uff0c\u4e16\u754c\u7a7a\u95f4\u4e2d\u7684\u5750\u6807\u539f\u70b9\u4e5f\u5c31\u662f\u573a\u666f\u7684\u5750\u6807\u539f\u70b9\u3002\u6a21\u578b\u77e9\u9635\u7684\u4f5c\u7528\u5c31\u662f\u628a\u6a21\u578b\u7a7a\u95f4\u4e2d\u7684\u5750\u6807\u8f6c\u6362\u4e3a\u4e16\u754c\u7a7a\u95f4\u4e2d\u7684\u5750\u6807\uff0c\u4e5f\u5c31\u662f\u5c06\u6a21\u578b\u653e\u5165\u573a\u666f\u4e2d\u3002
\u6a21\u578b\u653e\u5165\u573a\u666f\u65f6\u662f\u8981\u786e\u5b9a\u653e\u5165\u7684\u4f4d\u7f6e\u3001\u6a21\u578b\u5927\u5c0f\u548c\u6a21\u578b\u7684\u65cb\u8f6c\u89d2\u5ea6\uff0c\u8fd9\u4e5f\u5c31\u5bf9\u5e94\u7740\u6a21\u578b\u77e9\u9635\u4e2d\u7684\u5e73\u79fb\u77e9\u9635\u3001\u65cb\u8f6c\u77e9\u9635\u3001\u7f29\u653e\u77e9\u9635\u3002
\u8fdb\u884c\u5b8c\u5e73\u79fb\u3001\u65cb\u8f6c\u3001\u7f29\u653e\u7684\u53d8\u6362\u540e\uff0c\u5c31\u80fd\u786e\u5b9a\u6a21\u578b\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u5728\u573a\u666f\u4e2d\u7684\u5177\u4f53\u4f4d\u7f6e\uff0c\u5373\u5750\u6807\u3002
\u5982\u679c\u4ee5\u4e0a\u4e09\u79cd\u53d8\u6362\u6709\u7f3a\u7701\u7684\u8bdd\uff0c\u6bd4\u5982\u8bf4\u4e0d\u8fdb\u884c\u7f29\u653e\u53d8\u6362\uff0c\u5373\u8868\u793a\u6a21\u578b\u6309\u71671:1\u52a0\u5165\u5230\u573a\u666f\u3002<\/p>\n\n\n\n
\u6211\u4eec\u9700\u8981\u77e5\u9053\u7684\u662f\u6a21\u578b\u7a7a\u95f4\u539f\u70b9\u5728\u4e16\u754c\u7a7a\u95f4\u4e2d\u7684\u4f4d\u7f6e\u3001\u6a21\u578b\u7a7a\u95f4\u5750\u6807\u7cfb\u7cfb\u5728\u4e16\u754c\u7a7a\u95f4\u4e2d\u65cb\u8f6c\u4fe1\u606f\uff0c\u6a21\u578b\u7a7a\u95f4\u5750\u6807\u7cfb\u5728\u4e16\u754c\u7a7a\u95f4\u4e2d\u7684\u7f29\u653e\u4fe1\u606f\u3002\u5176\u5b9e\u5c31\u662f\u5e73\u79fb\u3001\u65cb\u8f6c\u3001\u7f29\u653e\u3002<\/p>\n\n\n\n
\u6a21\u578b\u77e9\u9635=[\u5e73\u79fb]*[\u65cb\u8f6c]*[\u7f29\u653e]<\/strong><\/p>\n\n\n\n \u5728\u65cb\u8f6c\u65f6\u5206\u4e3a\u7ed5X\u8f74\u3001Y\u8f74\u3001Z\u8f74\u548c\u4efb\u610f\u8f74\u65cb\u8f6c\u3002\u7ed5\u4efb\u610f\u8f74\u65cb\u8f6c\u65f6\u9700\u8981\u5b9a\u4e49\u65cb\u8f6c\u8f74\u5411\u91cf[latex]\\overrightarrow{axis}[\/latex]\u5e76\u4f7f\u7528\u7f57\u5fb7\u91cc\u683c\u65af\u65cb\u8f6c\u516c\u5f0f\uff0c\u8fd9\u91cc\u4f7f\u7528\u8be5\u516c\u5f0f\u4e3a\u4f8b\u3002<\/p>\n\n\n\n \u7f29\u653e\uff1a[latex]\\left(s_x, s_y, s_z\\right)[\/latex] [latex] \u6ce8\u610f\u8fd9\u91cc\u76843*3\u77e9\u9635[latex]\\mathbf{R}(\\mathbf{n}, \\alpha)[\/latex]\u9700\u8981\u62d3\u5c55\u62104*4\u77e9\u9635[latex]\\mathbf{R'}(\\mathbf{n}, \\alpha)[\/latex]<\/p>\n\n\n\n [latex] \u6a21\u578b\u77e9\u9635\u4e3b\u8981\u662f\u5728GAMES101-L03 \u53d8\u6362\uff08\u4e8c\u7ef4\u4e0e\u4e09\u7ef4\uff09<\/a>\u4e2d\u8bf4\u660e<\/p>\n\n\n\n View\u4e3a\u89c6\u56fe\u77e9\u9635\uff0c\u7528\u6765\u5c06\u7269\u4f53\u9876\u70b9\u5728\u4e16\u754c\u7a7a\u95f4\u4e0b\u7684\u5750\u6807\u8f6c\u6362\u4e3a\u89c6\u56fe\u7a7a\u95f4\u4e0b\u7684\u5750\u6807\u3002<\/p>\n\n\n\n \u89c6\u56fe\u7a7a\u95f4\u5c31\u662f\u4ee5\u76f8\u673a\u4f5c\u4e3a\u5750\u6807\u7cfb\u539f\u70b9\u5efa\u7acb\u7684\u7a7a\u95f4\uff0c\u76ee\u7684\u662f\u7b80\u5316\u4e0b\u4e00\u6b65\u6295\u5f71\u8fc7\u7a0b\u4e2d\u7684\u8ba1\u7b97\u3002<\/p>\n\n\n\n \u89c6\u56fe\u77e9\u9635\u5c31\u662f\u5c06\u573a\u666f\u4e2d\u3001\u4e5f\u5c31\u662f\u4e16\u754c\u7a7a\u95f4\u4e2d\u6240\u6709\u7684\u5750\u6807\uff0c\u5305\u62ec\u76f8\u673a\uff0c\u8fdb\u884c\u5e73\u79fb\u548c\u65cb\u8f6c\uff0c\u4f7f\u5f97\u76f8\u673a\u5904\u4e8e\u6807\u51c6\u4f4d\uff1a\u76f8\u673a\u5904\u4e8e\u539f\u70b9\uff0c\u5411-Z\u65b9\u5411\u770b\uff0c\u5411\u4e0a\u65b9\u5411\u4e3aY\u3002<\/p>\n\n\n\n \u89c6\u56fe\u77e9\u9635=[\u65cb\u8f6c]*[\u5e73\u79fb]<\/strong><\/p>\n\n\n\n \u76f8\u673a\u4f4d\u7f6e\uff1a[latex]\\overrightarrow{e}=\\left(x_e,y_e,z_e\\right)[\/latex] \u5f85\u5b8c\u5584 [latex] \u6a21\u578b\u77e9\u9635\u4e3b\u8981\u662f\u5728GAMES101-L04 \u53d8\u6362\uff08\u6a21\u578b\u3001\u89c6\u56fe\u3001\u6295\u5f71\uff09<\/a>\u4e2d\u8bf4\u660e\u3002<\/p>\n\n\n\n Projection\u4e3a\u6295\u5f71\u77e9\u9635\uff0c\u7528\u6765\u5c06\u7269\u4f53\u9876\u70b9\u5728\u89c6\u56fe\u7a7a\u95f4\u7684\u5750\u6807\u8f6c\u6362\u4e3a\u88c1\u526a\u7a7a\u95f4\u7684\u5750\u6807\uff0c\u6b63\u4ea4\u6295\u5f71\u77e9\u9635\u548c\u900f\u89c6\u6295\u5f71\u77e9\u9635\u3002 \u76f8\u673a\u7684\u89c6\u91ce\u662f\u6709\u9650\u7684\uff0c\u800c\u76f8\u673a\u89c6\u91ce\uff0c\u5373\u80fd\u770b\u5230\u7684\u573a\u666f\uff0c\u662f\u7531\u89c6\u9525\u51b3\u5b9a\u7684\uff0c\u843d\u5728\u89c6\u9525\u4e2d\u7684\u5750\u6807\u624d\u80fd\u88ab\u770b\u89c1\uff0c\u4e5f\u5c31\u662f\u663e\u793a\u5230\u5c4f\u5e55\u4e0a\u3002 \u8fdb\u884c\u900f\u89c6\u6295\u5f71\u7684\u7b2c\u4e00\u6b65\u8fdb\u884c\u6324\u538b\uff0c\u5c06\u89c6\u9525\u4f53\uff0c\u5373\u7531\u8fd1\u5e73\u9762\u548c\u8fdc\u5e73\u9762\u6784\u6210\u7684\u68f1\u53f0\uff0c\u6324\u538b\u6210\u89c6\u666f\u4f53\uff0c\u5373\u4e00\u4e2a\u957f\u65b9\u4f53\u3002\u6b64\u65f6\u8fd1\u5e73\u9762\u4e0d\u53d8\uff0c\u800c\u8fdc\u5e73\u9762\u88ab\u6324\u538b\u6210\u548c\u8fd1\u5e73\u9762\u4e00\u6837\u5927\uff0c\u4e14\u8fdc\u5e73\u9762\u4e2d\u5fc3\u4fdd\u6301\u4e0d\u53d8\uff0c\u5e76\u4e14\u5185\u90e8\u4e0d\u5747\u5300\uff0c\u5f62\u6210\u8fd1\u5927\u8fdc\u5c0f\u7684\u6548\u679c\u3002<\/p>\n\n\n\n \u7b2c\u4e00\u6b65\u5f97\u5230\u7684\u89c6\u666f\u4f53\uff0c\u5c31\u548c\u6b63\u4ea4\u6295\u5f71\u4e2d\u7684\u89c6\u9525\u76f8\u7b49\u4e86\u3002\u4e8e\u662f\u7b2c\u4e8c\u6b65\u5c31\u662f\u8fdb\u884c\u6b63\u4ea4\u6295\u5f71\u3002 \u6295\u5f71\u77e9\u9635=[\u7f29\u653e]*[\u5e73\u79fb]*[\u6324\u538b]<\/strong><\/p>\n\n\n\n \u89c6\u9525\uff1a [latex]n=zNear[\/latex] [latex] \u6a21\u578b\u77e9\u9635\u7684\u89c6\u9525\u90e8\u5206\u4e3b\u8981\u662f\u5728GAMES101-L05 \u5149\u6805\u5316\uff08\u4e09\u89d2\u5f62\u7684\u79bb\u6563\u5316\uff09<\/a>\u4e2d\u8bf4\u660e\uff0c\u6b63\u4ea4\u6295\u5f71\u548c\u900f\u89c6\u6295\u5f71\u5728 GAMES101-L04 \u53d8\u6362\uff08\u6a21\u578b\u3001\u89c6\u56fe\u3001\u6295\u5f71\uff09<\/a>\u4e2d\u8bf4\u660e \u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u6982\u8ff0 MVP\u53d8\u6362\u7684\u76ee\u7684\u662f\u5c06\u4e09\u7ef4\u7684\u573a\u666f\uff0c\u4e5f\u5c31\u662f\u76f8\u673a\u770b\u5230\u7684\u573a\u666f\uff0c\u8f6c\u5316\u5230\u88c1\u526a\u7a7a\u95f4\u4e2d\uff0c\u4e3a\u5149\u6805\u5316\u5c06\u573a\u666f\u663e\u793a\u5230\u5c4f\u5e55\u4e0a\u505a\u51c6\u5907\u3002 MVP\u5206\u522b\u7531Mo …<\/p>","protected":false},"author":1,"featured_media":863,"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-839","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\/839","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=839"}],"version-history":[{"count":1,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/839\/revisions"}],"predecessor-version":[{"id":1758,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/posts\/839\/revisions\/1758"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media\/863"}],"wp:attachment":[{"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/media?parent=839"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/categories?post=839"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.forillusion.com\/index.php\/wp-json\/wp\/v2\/tags?post=839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\u53c2\u6570<\/h4>\n\n\n\n
\u65cb\u8f6c\uff1a\u65cb\u8f6c\u8f74[latex]\\overrightarrow{n}=\\left(x,y,z\\right)[\/latex]\uff0c\u65cb\u8f6c\u89d2\u5ea6[latex]\\alpha[\/latex]
\u5e73\u79fb\uff1a[latex]\\left(t_x, t_y, t_z\\right)[\/latex]
<\/p>\n\n\n\n\u516c\u5f0f<\/h4>\n\n\n\n
\\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}}
[\/latex]
<\/p>\n\n\n\n
\\mathbf{Model} =
\\begin{bmatrix}
1 & 0 & 0 & t_x \\\\
0 & 1 & 0 & t_y \\\\
0 & 0 & 1 & t_z \\\\
0 & 0 & 0 & 1
\\end{bmatrix}\\mathbf{R'}(\\mathbf{n}, \\alpha)
\\begin{bmatrix}
s_x & 0 & 0 & 0 \\\\
0 & s_y & 0 & 0 \\\\
0 & 0 & s_z & 0 \\\\
0 & 0 & 0 & 1
\\end{bmatrix}
[\/latex]
<\/p>\n\n\n\nView\u89c6\u56fe\u77e9\u9635<\/h2>\n\n\n\n
\u516c\u5f0f<\/h3>\n\n\n\n
\u53c2\u6570<\/h4>\n\n\n\n
\u76f8\u673a\u7684\u89c6\u7ebf[latex]\\widehat{g}[\/latex]
\u76f8\u673a\u5411\u4e0a\u65b9\u5411[latex]\\widehat{t}[\/latex]<\/p>\n\n\n\n
<\/p>\n\n\n\n\u516c\u5f0f<\/h4>\n\n\n\n
\\mathbf{View} =
\\begin{bmatrix}
x_{\\hat{g} \\times \\hat{t}} & y_{\\hat{g} \\times \\hat{t}} & z_{\\hat{g} \\times \\hat{t}} & 0 \\\\
x_t & y_t & z_t & 0 \\\\
x_{-g} & y_{-g} & z_{-g} & 0 \\\\
0 & 0 & 0 & 1
\\end{bmatrix}
\\begin{bmatrix}
1 & 0 & 0 & -x_e \\\\
0 & 1 & 0 & -y_e \\\\
0 & 0 & 1 & -z_e \\\\
0 & 0 & 0 & 1
\\end{bmatrix}
[\/latex]
<\/p>\n\n\n\nProjection\u6295\u5f71\u77e9\u9635<\/h2>\n\n\n\n
\u6295\u5f71\u77e9\u9635\u7531\u4e09\u90e8\u5206\u6784\u6210\uff1a\u6324\u538b\u77e9\u9635\uff0c\u5e73\u79fb\u77e9\u9635\uff0c\u7f29\u653e\u77e9\u9635\u3002<\/p>\n\n\n\n
\u89c6\u9525\u67094\u4e2a\u53c2\u6570\uff1a\u5782\u76f4\u53ef\u89c6\u89d2\uff0c\u957f\u5bbd\u6bd4\uff0c\u8fd1\u5e73\u9762\uff0c\u8fdc\u5e73\u9762\u3002<\/p>\n\n\n\n
\u5728\u8fdb\u884c\u6b63\u4ea4\u6295\u5f71\u524d\uff0c\u9700\u8981\u83b7\u53d6\u89c6\u666f\u4f53\u76846\u4e2a\u4fe1\u606f\uff0c\u5373n,f,l,r,t,b\uff0c\u8fd9\u4e9b\u4fe1\u606f\u53ef\u4ee5\u5229\u7528\u89c6\u9525\u76844\u4e2a\u53c2\u6570\u8ba1\u7b97\u5f97\u51fa\u3002
\u6b63\u4ea4\u6295\u5f71\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u5c06\u89c6\u666f\u4f53\u7684\u4e2d\u5fc3\u79fb\u5230\u5750\u6807\u539f\u70b9\u4e0a\uff0c\u7b2c\u4e8c\u6b65\u662f\u8fdb\u884c\u7f29\u653e\uff0c\u5c06\u89c6\u666f\u4f53\u7f29\u653e\u5230[latex][-1,1]^3[\/latex]\u5927\u7684\u6807\u51c6\u7acb\u65b9\u4e2d\u3002
\u7531\u4e8e\u6709\u4e4b\u524d\u7684\u89c6\u56fe\u53d8\u6362\uff0c\u89c6\u666f\u4f53n\u3001f\u5e73\u9762\u7684\u4e2d\u5fc3\u5c31\u5728Z\u8f74\u4e0a\uff0c\u6240\u4ee5\u5e73\u79fb\u65f6\u53ea\u9700\u8981\u5c06\u89c6\u666f\u4f53\u6cbfZ\u8f74\u5e73\u79fb\u4f7f\u4e2d\u5fc3\u5728\u5750\u6807\u539f\u70b9\u4e0a\u3002<\/p>\n\n\n\n\u516c\u5f0f<\/h3>\n\n\n\n
\u53c2\u6570<\/h4>\n\n\n\n
\u8fd1\u5e73\u9762\uff1a[latex]zNear[\/latex]
\u8fdc\u5e73\u9762\uff1a[latex]zFar[\/latex]
\u5782\u76f4\u53ef\u89c6\u89d2\uff1a[latex]fovY[\/latex]
\u5bbd\u9ad8\u6bd4\uff1a[latex]aspect[\/latex] <\/p>\n\n\n\n\u516c\u5f0f<\/h4>\n\n\n\n
[latex]f=zFar[\/latex]
[latex]t=tan(\\frac{fovY}{2} )\\times n[\/latex]
[latex]b=-t[\/latex]
[latex]r=aspect\\times t[\/latex]
[latex]l=-r[\/latex]
<\/p>\n\n\n\n
\\mathbf{Projection} =
\\begin{bmatrix}
\\frac{2}{r-l} & 0 & 0 & 0 \\\\
0 & \\frac{2}{t-b} & 0 & 0 \\\\
0 & 0 & \\frac{2}{n-f} & 0 \\\\
0 & 0 & 0 & 1
\\end{bmatrix}
\\begin{bmatrix}
1 & 0 & 0 & 0 \\\\
0 & 1 & 0 & 0 \\\\
0 & 0 & 1 & -\\frac{n+f}{2} \\\\
0 & 0 & 0 & 1
\\end{bmatrix}
\\begin{bmatrix}
n & 0 & 0 & 0 \\\\
0 & n & 0 & 0 \\\\
0 & 0 & n+f & -nf \\\\
0 & 0 & 1 & 0
\\end{bmatrix}=\\begin{bmatrix}
\\frac{2n}{r-l} & 0 & 0 & 0 \\\\
0 & \\frac{2n}{t-b} & 0 & 0 \\\\
0 & 0 & \\frac{n+f}{n-f} & \\frac{-2nf}{n-f} \\\\
0 & 0 & 1 & 0
\\end{bmatrix}
[\/latex]
<\/p>\n\n\n\n