Bipolar coordinate system
Bipolar coordinates are an orthogonal coordinate system on a plane based on Apollonius circles . The following formulas are used to transition from bipolar coordinates to Cartesian coordinates :
- {\ displaystyle \ left \ {{\ begin {matrix} x = {\ frac {a \, \ mathrm {sh} \, \ tau} {\ mathrm {ch} \, \ tau - \ cos \ sigma}} \ \ y = {\ frac {a \ sin \ sigma} {\ mathrm {ch} \, \ tau - \ cos \ sigma}} \ end {matrix}} \ right.}
Where {\ displaystyle 0 \ leqslant \ sigma <\ pi} 
, {\ displaystyle - \ infty <\ tau <\ infty} 
.
Lame Odds :
- {\ displaystyle L _ {\ tau} = L _ {\ sigma} = {\ frac {a ^ {2}} {(\ mathrm {ch} \, \ tau - \ cos \ sigma) ^ {2}}}.}
Laplace operator in bipolar coordinates:
- {\ displaystyle \ Delta f = {\ frac {(\ mathrm {ch} \, \ tau - \ cos \ sigma) ^ {2}} {a ^ {2}}} \ left ({\ frac {\ partial ^ {2} f} {\ partial \ sigma ^ {2}}} + {\ frac {\ partial ^ {2} f} {\ partial \ tau ^ {2}}} \ right).}
In space, bipolar coordinates are generalized bispherical .
