[b]Circumference [/b]is the set of points of a plane whose distance to a certain point ([b]center[/b]) of that plane is equal to a certain (non-zero) distance. This distance is the [b]radius [/b]of the circumference.

Consider a plane [math]\alpha[/math], a point O of [math]\alpha[/math] and a distance r,[br] [math]\textrm{ λ(O,r)=\left\{P∈α| d_{P,O}=r \right\}[/math][br]Where [math]\textrm{ λ(O,r)}[/math] represents the circumference of center O and radius r.[br]

Consider a point X and a circumference [math]\textrm{ λ(O,r)}[/math], [br]X is internal to [math]\textrm{ λ↔\left\{d_{P,O}r \right\}[/math][br]X is within [math]\textrm{ λ↔\left\{d_{P,O}=r \right\}[/math],

[b]Circle [/b]is the set of points of a plane whose distance to a certain point ([b]center[/b]) of that plane is less than or equal to a certain (non-zero) distance.

Center circle A and radius r [br] [math]\textrm{ c(A,r)=\left\{P∈α| d_{P,O}≤r \right\}[/math][br]