From 40fc6c367915273f9260499bd034497fb88ab8ac Mon Sep 17 00:00:00 2001 From: hwc Date: Tue, 11 Jul 2023 16:49:14 +0800 Subject: [PATCH] =?UTF-8?q?=E4=BD=9C=E4=B8=9A1=E5=8F=8A=E7=AC=94=E8=AE=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../1689059400149.png" | Bin 0 -> 116220 bytes .../1689059532544.png" | Bin 0 -> 25899 bytes ...00\344\272\233\347\254\224\350\256\260.md" | 139 ++++++++++++++++++ 3 files changed, 139 insertions(+) create mode 100644 "image/\344\270\200\344\272\233\347\254\224\350\256\260/1689059400149.png" create mode 100644 "image/\344\270\200\344\272\233\347\254\224\350\256\260/1689059532544.png" create mode 100644 "\344\270\200\344\272\233\347\254\224\350\256\260.md" diff --git "a/image/\344\270\200\344\272\233\347\254\224\350\256\260/1689059400149.png" "b/image/\344\270\200\344\272\233\347\254\224\350\256\260/1689059400149.png" new file mode 100644 index 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+\end{pmatrix}\begin{pmatrix} +1&0&0&t_x\\ +0&1&0&t_y\\ +0&0&1&t_z\\ +0&0&0&1\\ +\end{pmatrix} + +$$ + + +$$ +变换=线性变换\times平移变换 +$$ + + +### 透视投影 + +![1689059400149](image/一些笔记/1689059400149.png) + +对远端平面上的任意一点,x、y坐标的缩放如下图所示 + +n为原点与近端平面的距离 + +![1689059532544](image/一些笔记/1689059532544.png) + +因此变换后的齐次坐标为 + +$$ +\begin{pmatrix} +nx/z\\ny/z\\z'\\1 +\end{pmatrix}==\begin{pmatrix} +nx\\ny\\zz'\\z +\end{pmatrix} +$$ + +设变换矩阵的为 + +$$ +\begin{pmatrix} +n&0&0&0\\ +0&n&0&0\\ +A&B&C&D\\ +0&0&1&0\\ +\end{pmatrix} +$$ + +近端平面上任意一点的坐标不变(规定)为n + +$$ +\begin{pmatrix} +x\\y\\n\\1 +\end{pmatrix}==\begin{pmatrix} +nx\\ny\\n^2\\n +\end{pmatrix} +$$ + +因此 + +$$ +\begin{pmatrix} +A&B&C&D +\end{pmatrix}\begin{pmatrix} +x\\y\\n\\1 +\end{pmatrix}=Ax+By+Cn+D=n^2\\ +=>A=B=0 +$$ + +远端平面上任意一点的z坐标不变(规定)为f + +$$ +\begin{pmatrix} +0&0&C&D +\end{pmatrix}\begin{pmatrix} +x\\y\\f\\1 +\end{pmatrix}=Cf+D=f^2 +$$ + +联立解得 + +$$ +C=n+f\\ +D=-nf +$$ + +因此透视投影的变换矩阵为 + +$$ +\begin{pmatrix} +n&0&0&0\\ +0&n&0&0\\ +0&0&n+f&-nf\\ +0&0&1&0\\ +\end{pmatrix} +$$ + +对梯形中的任意一点,变换之后的坐标为 + +$$ +\begin{pmatrix} +n&0&0&0\\ +0&n&0&0\\ +0&0&n+f&-nf\\ +0&0&1&0\\ +\end{pmatrix}\begin{pmatrix} +x\\y\\z\\1 +\end{pmatrix}=\begin{pmatrix} +nx\\ny\\(n+f)z-nf\\z +\end{pmatrix}\\ +其中\ f\le z\le n\lt0 +$$ + +计算可知 + +$$ +\frac{(n+f)z-nf}{z}\le z +$$ + +因此变换之后,梯形体内部的z坐标相比变换前变小了,也即变远了