An algebraic surface is said to be bispherical if the umbilics (i.e. the points at infinity of the complex sphere with zero radius ) form a double line, in the complex projective completion of the space.
The necessary and sufficient condition is that the polynomial composed of the terms with highest degree n of its Cartesian equation be divisible by  and that the polynomial composed of the terms of with degree n - 1 be divisible by .

The bispherical quartics are the Darboux cyclides.
 
 

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© Robert FERRÉOL 2017