Distance measurement - Euclidean distance

Python Learning Series ： Catalog

summary

Euclidean distance , Also known as Euclid distance , We started from primary school 、 Junior high school 、 Distance measurement, which is used in high school and even now .

“ The shortest line between two points ” Everyone has learned it , Here is just a big English name , It's the formula that we use to calculate the distance on the junior and advanced examination paper

Calculation formula

① Euclidean distance on a two-dimensional plane

hypothesis Two dimensional plane There are two points inside ：$a(x_1,y_1)$ And $b(x_2,y_2)$

Then the distance formula of the two-dimensional plane is ：

$$d_{12}=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2}$$

for instance , Like the one above $A(2,2)$ And $B(6,6)$ At two o 'clock , Calculation $AB$ The distance between two points is ：
$$\begin{array}{rl}{d}_{AB}& =\sqrt{(6-2{)}^2+(6-2{)}^2}\\ & =\sqrt{4^2+4^2}\\ & =4\sqrt{2}\end{array}$$

② Euclidean distance in three-dimensional space

hypothesis three-dimensional space There are two points inside ：$a(x_1,y_1,z_1)$ And $b(x_2,y_2,z_2)$

Then the distance formula of three-dimensional space is ：

$$d_{12}=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2+(z_1-z_2{)}^2}$$

for instance , Like the one above $A(0,0,4)$ And $B(0,2,0)$ At two o 'clock , Calculation $AB$ The distance between two points is ：

$$\begin{array}{rl}{d}_{AB}& =\sqrt{(0-0{)}^2+(0-2{)}^2+(4-0{)}^2}\\ & =\sqrt{0+4+16}\\ & =2\sqrt{5}\end{array}$$

③ n Euclidean distance in dimensional space

hypothesis n Dimensional space There are two points inside ：$a(x_{11},x_{12},...,x_{1n})$ And $b(x_{21},y_{22},...,z_{2n})$

be n The distance formula of dimensional space is ：

$$d_{12}=\sqrt{\sum _{k=1}^n(x_{1k}-x_{2k}{)}^2}$$

Empathy ,n Dimensional space is also , Do the above operations on the corresponding vector .（n Wei's painting doesn't come out , Need to be expressed in other forms , Just like the picture below ）.