Editing Bipolar coordinates (section)

== Curves of constant ''σ'' and ''τ'' ==

[[File:Bipolar sigma isosurfaces.png|right|280px]]

[[File:Bipolar tau isosurfaces.png|right|280px]]

The curves of constant ''σ'' correspond to non-concentric circles 

{{NumBlk|:|<math>
x^2 +
\left( y - a \cot \sigma \right)^2 = \frac{a^{2}}{\sin^2 \sigma}
= a^2(1+\cot^2\sigma)</math>|(1)|RawN=.}}

that intersect at the two foci.  The centers of the constant-''σ'' circles lie on the ''y''-axis at <math>a\cot \sigma</math> with radius <math>\tfrac{a}{\sin\sigma}</math>.  Circles of positive ''σ'' are centered above the ''x''-axis, whereas those of negative ''σ'' lie below the axis.  As the magnitude |''σ''| − ''π''/2 decreases, the radius of the circles decreases and the center approaches the origin (0,&nbsp;0), which is reached when |''σ''| = ''π''/2.  (From elementary geometry, all triangles on a circle with 2 vertices on opposite ends of a diameter are right triangles.)

The curves of constant <math>\tau</math> are non-intersecting circles of different radii

{{NumBlk|:|<math>
\left( x - a \coth \tau \right)^2 + y^2 = \frac{a^2}{\sinh^2 \tau} = a^2(\coth^2\tau-1)</math>|(2)|RawN=.}}

that surround the foci but again are not concentric. The centers of the constant-''τ'' circles lie on the ''x''-axis at <math>a \coth\tau</math> with radius <math>\tfrac{a}{\sinh\tau}</math>.  The circles of positive ''τ'' lie in the right-hand side of the plane (''x''&nbsp;>&nbsp;0), whereas the circles of negative ''τ'' lie in the left-hand side of the plane (''x''&nbsp;<&nbsp;0).  The ''τ''&nbsp;=&nbsp;0 curve corresponds to the ''y''-axis (''x''&nbsp;=&nbsp;0).  As the magnitude of ''τ'' increases, the radius of the circles decreases and their centers approach the foci.