|
Bipolar coordinates are a two-dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates.〔(Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 1999 )〕 The first is based on the Apollonian circles. The curves of constant ''σ'' and of ''τ'' are circles that intersect at right angles. The coordinates have two foci ''F''1 and ''F''2, which are generally taken to be fixed at (−''a'', 0) and (''a'', 0), respectively, on the ''x''-axis of a Cartesian coordinate system. The second system is two-center bipolar coordinates. There is also a third coordinate system that is based on two poles (biangular coordinates). The term "bipolar" is sometimes used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term ''bipolar coordinates'' is reserved for the coordinates described here, and never used to describe coordinates associated with those other curves, such as elliptic coordinates. == Definition == The most common definition of bipolar coordinates (''σ'', ''τ'') is : : where the ''σ''-coordinate of a point ''P'' equals the angle ''F''1 ''P'' ''F''2 and the ''τ''-coordinate equals the natural logarithm of the ratio of the distances ''d''1 and ''d''2 to the foci : (Recall that ''F''1 and ''F''2 are located at (−''a'', 0) and (''a'', 0), respectively.) Equivalently :〔〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bipolar coordinates」の詳細全文を読む スポンサード リンク
|