Explanation

Inside an ideal long solenoid the magnetic field is uniform, so the field lines take the form of parallel straight lines running along the solenoid's axis. The field strength inside is B = mu0*n*I and is nearly constant, while outside an ideal solenoid the field is approximately zero. The lines therefore do not curve into ellipses, parabolas or hyperbolas. This uniform axial field is what makes solenoids useful as electromagnets.