# 笛卡兒座標系

## 平面笛卡兒坐標的公式

### 在兩點間的距離

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}$

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}$

### 歐幾里得變換

#### 平移

${\displaystyle (x',y')=(x+a,y+b).}$

#### 旋轉

${\displaystyle x'=x\cos \theta -y\sin \theta }$
${\displaystyle y'=x\sin \theta +y\cos \theta .}$

${\displaystyle (x',y')=((x\cos \theta -y\sin \theta \,),(x\sin \theta +y\cos \theta \,)).}$

#### 反射

${\displaystyle x'=x\cos 2\theta +y\sin 2\theta }$
${\displaystyle y'=x\sin 2\theta -y\cos 2\theta .}$

${\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}$

#### 變換的一般矩陣形式

${\displaystyle (x',y')=(x,y)A+b}$

${\displaystyle x'=xA_{11}+yA_{21}+b_{1}}$
${\displaystyle y'=xA_{12}+yA_{22}+b_{2},}$

${\displaystyle A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}.}$ [注意對點坐標使用行向量則矩陣要寫在右側。]

${\displaystyle A_{11}A_{21}+A_{12}A_{22}=0}$

${\displaystyle A_{11}^{2}+A_{12}^{2}=A_{21}^{2}+A_{22}^{2}=1.}$

${\displaystyle A_{11}A_{22}-A_{21}A_{12}=1.}$

${\displaystyle A_{11}A_{22}-A_{21}A_{12}=-1.}$

### 仿射變換

${\displaystyle {\begin{pmatrix}A_{11}&A_{21}&b_{1}\\A_{12}&A_{22}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.}$ [注意來自上式的矩陣A是轉置的。矩陣在左側並對點坐標使用列向量。]

#### 縮放

${\displaystyle (x',y')=(mx,my).}$

#### 錯切

${\displaystyle (x',y')=(x+ys,y)}$

${\displaystyle (x',y')=(x,xs+y)}$

3D笛卡兒坐標的手規則

## 向量

${\displaystyle \mathbf {r} =x{\hat {\mathbf {i} }}+y{\hat {\mathbf {j} }}+z{\hat {\mathbf {k} }}}$

## 參考文獻

1. ^ Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. Calculus : Single and Multivariable 6. John wiley. 2013. ISBN 978-0470-88861-2.
2. ^ Smart 1998，Chap. 2
3. ^ Brannan，Esplen & Gray（1998），pg. 49

## 參考目錄

• Descartes, René. Oscamp, Paul J. (trans). Discourse on Method, Optics, Geometry, and Meteorology. 2001.
• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 656. ISBN 978-0-07-043316-8.
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: p. 177.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: pp. 55–79. ASIN B0000CKZX7.
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 94.
• Moon P, Spencer DE. Rectangular Coordinates (x, y, z). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: pp. 9–11 (Table 1.01). ISBN 978-0387184302.