General coil-shape in superconducting quadrupole magnets for generating the perfect field

It is well-known that a constant current in a coil configuration generated by two perpendicular intersecting ellipses produces the perfect quadrupole field. We have found that an ideal quadrupole field can be obtained with a variety of coil shapes and the intersecting ellipse shape is only a special form in that class. We have given the general form of the coil shape for generating the ideal quadrupole field. The shape parameters of the coil can be optimized to obtain the maximum possible gradient for a given coil cross-section and aperture.