立体几何

斜二测画法

$y$轴顺时针旋转$45^{\circ} $，已知图形中平行于$x$轴的线段在直观图中长度保持不变，平行于$y$轴的线段长度变成原来的一半

$i=\left[ {\begin{array}{*{20}c}1\\0\end{array} } \right] \rightarrow\left[ {\begin{array}{*{20}c}1\\0\end{array} } \right]$ $j=\left[ {\begin{array}{*{20}c}0\\1\end{array} } \right]\rightarrow\left[ {\begin{array}{*{20}c}\frac{\sqrt{2} }{4} \\\frac{\sqrt{2} }{4}\end{array} } \right]$

斜二测画法的变换矩阵

$\left[ {\begin{array}{*{20}c}x'\\y'\end{array} } \right]=\left[{\begin{array}{*{20}c}1&\frac{\sqrt{2} }{4}\\0&\frac{\sqrt{2} }{4}\end{array} } \right]\left[ {\begin{array}{*{20}c}x\\y\end{array} } \right]$

变换前后的面积比

$\frac{S'}{S}=\left|{\begin{array}{*{20}c}1&\frac{\sqrt{2} }{4}\\0&\frac{\sqrt{2} }{4}\end{array} }\right|=\frac{\sqrt{2} }{4}$

三维空间内的

$\left[ {\begin{array}{*{20}c}x'\\y'\\z'\end{array} } \right]=\left[ \begin{array}{*{20}c} { 1 } & { \frac { \sqrt { 2 } } { 4 } } & { 0 } \\ { 0 } & { \frac { \sqrt { 2 } } { 4 } } & { 1 } \\ { 0 } & { 0 } & { 0 } \end{array} \right]\left[ {\begin{array}{*{20}c}x\\y\\z\end{array} } \right]$

特殊法求解

$\text{ 该变换写为}\begin{cases} { x ^ { \prime } = a _ { 1 } x + b _ { 1 } y + c _ { 1 } z } \\ { y ^ { \prime } = a _ { 2 } x + b _ { 2 } y + c _ { 2 } z } \end{cases}$ $\text{带入可得}\begin{cases} { x ^ { \prime } = x + \frac { \sqrt { 2 } } { 4 } y } \\ { y ^ { \prime } = z + \frac { \sqrt { 2 } } { 4 } y } \end{cases}$

公理与定理

公理一

$A\in l,B\in l, C\notin l \Rightarrow \exists \alpha \quad s.t.\quad A,B,C\in\alpha$

公理二

$A,B\in\alpha,A,B\in l \Rightarrow l\subset\alpha$

公理三

$P\in\alpha,P\in\beta \Rightarrow \alpha \cap\beta=l\quad and\quad P\in l$ $A\in l,B\in l, C\notin l \Rightarrow \exists \alpha \quad s.t.\quad A,B,C\in\alpha\\ A,B\in\alpha,A,B\in l \Rightarrow l\subset\alpha\\ P\in\alpha,P\in\beta \Rightarrow \alpha \cap\beta=l\quad \text{and}\quad P\in l\\\begin{cases}{a\nsubseteqq \beta},a\parallel b,b\subset\beta\Rightarrow a\parallel \beta\\a\parallel \beta,a\subset \alpha ,\alpha \cap\beta=l\Rightarrow a\parallel l\end{cases}$

平行

$\begin{cases}{a\nsubseteqq \beta},a\parallel b,b\subset\beta\Rightarrow a\parallel \beta\\a\parallel \beta,a\subset \alpha ,\alpha \cap\beta=l\Rightarrow a\parallel l\end{cases}$ $\begin{cases}a\parallel \alpha,b\parallel \alpha,{a\cap b=P},a,b\subset\beta\Rightarrow\alpha\parallel \beta\\ \alpha\parallel \beta,\gamma\cap\alpha=a,\gamma\cap\beta=b\Rightarrow a\parallel b \end{cases}$

垂直

$\begin{cases}l\perp m,l\perp n\;,{m\cap m=P},m,n\subset\beta\Rightarrow l\perp \beta\\ a\perp\beta,b\perp\beta\Rightarrow a\parallel b \end{cases}$ $\begin{cases}l\perp \beta,l\subset \alpha \Rightarrow \alpha\perp \beta\\ \alpha\perp\beta,a\subset\beta,\alpha \cap\beta=l,a\perp l\Rightarrow a\perp \alpha \end{cases}$ $\begin{cases}a\parallel \alpha,b\parallel \alpha,{a\cap b=P},a,b\subset\beta\Rightarrow\alpha\parallel \beta\\ \alpha\parallel \beta,\gamma\cap\alpha=a,\gamma\cap\beta=b\Rightarrow a\parallel b \end{cases}\\ \begin{cases}l\perp m,l\perp n\;,{m\cap m=P},m,n\subset\beta\Rightarrow l\perp \beta\\ a\perp\beta,b\perp\beta\Rightarrow a\parallel b \end{cases}\\ \begin{cases}l\perp \beta,l\subset \alpha \Rightarrow \alpha\perp \beta\\ \alpha\perp\beta,a\subset\beta,\alpha \cap\beta=l,a\perp l\Rightarrow a\perp \alpha \end{cases}$

面积体积公式

$圆锥\begin{cases} S_{侧}=\pi rl \\S_{底}=\pi r^2\\V=\frac13Sh\end{cases}$ $台\begin{cases} S_{侧}=\pi (r+r' )l \\V=\frac13(S+\sqrt {SS'}+{S'})h\end{cases}$ $正四面体\begin{cases} h=\frac{\sqrt 6}{3}a \\ r_{外}=\frac{\sqrt 6}{4}a \\r_{内}=\frac{\sqrt 6}{12}a\end{cases}$ $球\begin{cases} S=4\pi R^2 \\V=\frac43\pi R^3\end{cases}$

空间向量

线线角

$\vec{a},\vec{b}$分别为直线$m,n$的方向向量

$\theta=\begin{cases}\arccos|\frac{|\vec{a}\cdot\vec{b} | }{|\vec{a}|\cdot|\vec{b}|}|,<\vec{a},\vec{b}>\in[0,\frac\pi2]\\ \pi-\arccos|\frac{|\vec{a}\cdot\vec{b} | }{|\vec{a}|\cdot|\vec{b}|}|,<\vec{a},\vec{b}>\in(\frac{\pi}{2},\pi ] \end{cases}$ $\cos\theta=\left |\frac{|\vec{a}\cdot\vec{\rm{n} }| }{|\vec{a}|\cdot|\vec{\rm{n} }|}\right |$

线面角

$\sin\theta=\cos<\vec{a},\vec{\rm{n} }>=\frac{|\vec{a}\cdot\vec{\rm{n} }| }{|\vec{a}|\cdot|\vec{\rm{n} }|}$

面面角

$\begin{cases}\vec{\rm{n} }\cdot\vec{a}=0 \\\vec{\rm{n} }\cdot\vec{b}=0\end{cases}$ $\cos\theta=\left|\frac{\vec{\rm{n_1} }\cdot\vec{\rm{n_2} } }{|\vec{\rm{n_1} }|\cdot|\vec{\rm{n_2} }|}\right |=\left|\frac{x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2} }{\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2} } \sqrt{x_{2}^{2}+y_{2}^{2}+z_{2}^{2} } }\right |$

二面角

同进同出

$\cos\theta=-\frac{\vec{\rm{n_1} }\cdot\vec{\rm{n_2} } }{|\vec{\rm{n_1} }|\cdot|\vec{\rm{n_2} }|}=-\frac{x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2} }{\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2} } \sqrt{x_{2}^{2}+y_{2}^{2}+z_{2}^{2} } }$

一进一出

$\cos\theta=\frac{\vec{\rm{n_1} }\cdot\vec{\rm{n_2} } }{|\vec{\rm{n_1} }|\cdot|\vec{\rm{n_2} }|}=\frac{x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2} }{\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2} } \sqrt{x_{2}^{2}+y_{2}^{2}+z_{2}^{2} } }\\\vec{a} \times \vec{b}=\left(y_{1} z_{2}-z_{1} y_{2}, z_{1} x_{2}-x_{1} z_{2}, x_{1} y_{2}-y_{1} x_{2}\right)\\ d=|AA'|=\sqrt{\left |\vec{PA}\right |^2-\left |\frac{\vec{PA}\cdot\vec{s} }{|\vec{s}|}\right |}=\left |\vec{PA}\cdot \vec{n_0}\right |$

距离

$\begin{array}{l}d=|AA'|&=\sqrt{\left |\vec{PA}\right |^2-\left |\frac{\vec{PA}\cdot\vec{s} }{|\vec{s}|}\right |}\\&=\sqrt{\left |\vec{PA}\right |^2-\left |{\vec{PA}\cdot\vec{s_0} }\right |} \\&=\left |\vec{PA}\cdot \vec{n_0}\right |\end{array}$

叉积

$\vec{a} \times \vec{b}=\left(y_{1} z_{2}-z_{1} y_{2}, z_{1} x_{2}-x_{1} z_{2}, x_{1} y_{2}-y_{1} x_{2}\right)$

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