[10分] Euclidean distance 問題

其實可以分開
只要一個函數滿足以下條件﹐都可以叫做distance function

d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y. (Distance is positive between two different points, and is zero precisely from a point to itself.)
It is symmetric: d(x,y) = d(y,x). (The distance between x and y is the same in either direction.)
It satisfies the triangle inequality: d(x,z) ≤ d(x,y) + d(y,z). (The distance between two points is the shortest distance along any path).
你寫的這個function
sqrt{ (x1-x2)^2 } + sqrt{ (y1-y2)^2 } + sqrt{ (z1-z2)^2 }
滿足以上三個條件﹐所以是distance function ﹐不過人們不叫它euclidean distance。因為要留給常用的那個[即sqrt{ (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 }]
以下是部份常用的distance (你寫的那個即是1-norm distance)




1-階范數
=

圖片參考：http://upload.wikimedia.org/math/8/7/1/871ec2923f11fb6af43901f13970ee2c.png



2-階范數
=

圖片參考：http://upload.wikimedia.org/math/9/9/9/999b7e7c5b79f528cec014fb9b624f7d.png



n-階范數
=

圖片參考：http://upload.wikimedia.org/math/4/4/4/444044eafaa19384ea32fa58e5637356.png



無窮大階范數
=
t 階范數的極限，即 n 趨向無窮大

圖片參考：http://upload.wikimedia.org/math/e/2/5/e25f07b37eab73aa02db180e81b38f11.png





=
max
圖片參考：http://upload.wikimedia.org/math/8/c/f/8cfc29f87548f0f28ba0dad1fad4f758.png