Abstract. A quadratic form f: S m --~ S n between spheres is separable if, up to isometries on the source and the range, the components of f are pure or mixed quadratic polynomials. The space parametrizing the separated quadratic eigenmaps f is shown here to fiber over a semi-algebraic set with each fiber a finite-dimensional compact convex body. For m --3, this gives a new description of the parameter space of all quadratic eigenmaps f: S 3 ~ S m as a fibration over an 'inflated tetrahedron' and generic hexagonal fibres.Mathematics Subject Classification (1991): 54--xx.