Update

distance.md

@@ -7,7 +7,7 @@ Distance is similar/related to [length](length.md), the difference is that dista
 There are many ways to define distance within given space. Most common and implicitly assumed distance is the **[Euclidean distance](euclidean_distance.md)** (basically the "straight line from point A to point B" whose length is computed with [ Euclidean Theorem](euclidean_theorem.md)), but other distances are possible, e.g. the [taxicab distance](taxicab_distance.md) (length of the kind of perpendicular path taxis take between points A and B in Manhattan, usually longer than straight line). Mathematically a space in which distances can be measured are called [metric spaces](metric_space.md), and a distance within such space can be any [function](function.md) *dist* (called a *distance* or *metric* function) that satisfies these [axioms](axiom.md):
 
 1. *dist(p,p) = 0* (distance from identical point is zero)
-2. Values given by *dist* are never negative.
+2. *dist(p,q) > 0* if *p != q* (distance between two distinct points is always positive)
 3. *dist(p,q) = dist(q,p)* ([symmetry](symmetry.md), distance between two points is the same in both directions).
 4. *dist(a,c) <= dist(a,b) + dist(b,c)* (triangle inequality)
 
@@ -16,7 +16,7 @@ There are many ways to define distance within given space. Most common and impli
 Computing Euclidean distance requires multiplication and most importantly [square root](sqrt.md) which is usually a pretty slow operation, therefore many times we look for simpler [approximations](approximation.md). Note that a possible approach here may also lead through computing the distance normally but using a fast approximation of the square root.
 
 Two very basic and rough approximations of Euclidean distance, both in 2D and 3D, are [taxicab](taxicab.md) (also Manhattan) and [Chebyshev](chebyshev.md) distances. Taxicab distance
-simply adds the absolute coordinate differences along each principal axis (*dx*, *dy* and *dz*) while Chebyshev takes the maximum of them. In [C](c.md) (for generalization to 3D just add one coordinate of course):
+simply adds the absolute coordinate differences along each principal axis (*dx*, *dy* and *dz*) while Chebyshev takes the maximum of them (NOTE: taking minimum isn't possible due to the definition which requires two distinct points to always have positive distance). In [C](c.md) (for generalization to 3D just add one coordinate of course):
 
 ```
 int distTaxi(int x0, int y0, int x1, int y1)
@@ -38,6 +38,48 @@ int distCheb(int x0, int y0, int x1, int y1)
 
 Both of these distances approximate a [circle](circle.md) in 2D with a square or a sphere in 3D with a cube, the difference is that taxicab is an upper estimate of the distance while Chebyshev is the lower estimate. For speed of execution ([optimization](optimization.md)) it may also be important that taxicab distance only uses the operation of addition while Chebyshev may result in [branching](branch.md) (*if*) in the max function which is usually not good for performance.
 
+```
+--------------------------------------------------------------------------
+
+  Euclidean                           _____
+                                  _.''     ''._    
+   sqrt(                    B    /             \                         F
+     (B.x - A.x)^2 +     _,-+   /       C   d   \                      _-+
+     (B.y - A.y)^2) _,--'       (       +-------)                  _,-'
+               _,--'            \               /              _,-'
+          _,--'                  \_           _/           _,-'   
+  A  _,--'                         '--_____--'      E  _,-'
+  +-'                                               +-'
+
+--------------------------------------------------------------------------
+
+  Taxicab (Manhattan)                  _A_
+                            B        _/   \_                             F
+                            +      _/       \_                      ,----+
+     abs(B.x - A.x) +       |    _/     C   d \_                  ,'
+     abx(B.y - A.y)         |   <_      +------->               ,'
+                            |     \_         _/               ,'
+                            |       \_     _/               ,'
+  A                         |         \_ _/         E     ,'
+  +-------------------------'           V           +----'
+
+--------------------------------------------------------------------------
+
+  Chebyshev                      _______________
+                            B   |               |                        F
+    max(                    +   |               |                       ,+
+      abs(B.x - A.x),           |       C   d   |                     ,'      
+      abs(B.y - A.y))           |       +-------|                   ,' 
+                                |               |         ,--------'
+                                |               |       ,'
+  A                             |_______________|   E ,'
+  +--------------------------                       +'                  
+
+--------------------------------------------------------------------------
+```
+
+*Three commonly used distance measures: the distance is the length of the path illustrated between points A and B; next are shown "circles" (sets of points with distance d from point C) and "lines" (one diagonal of a rhombus in which points E and F are opposite to each other) drawn by respective measures.*
+
 A bit more accuracy can be achieved by averaging the taxicab and Chebyshev distances which in 2D approximates a circle with an 8 segment polygon and in 3D approximates a sphere with 24 sided [polyhedron](polyhedron.md). The integer-only [C](c.md) code is following:
 
 ```