Results are presented of measurements of Doppler-shifted cyclotron resonance (DSCR) of helicon waves in single-crystal aluminum with the magnetic field and helicon propagation along the  and  directions. The Gaussian radius of curvature of the Fermi surface in the second zone is measured at the point where the surface normal is along . The experimental value for the radius of curvature is 0.99±0.03 times the free-electron value, in agreement with the expected value. For , it is shown that DSCR absorption edges at an elliptic limiting point on an arm of the third-zone Fermi-surface "monster," and on finite orbits involving a large fraction of the electrons on the second-zone surface, are expected to occur almost simultaneously. The difficulties associated with interpretation of the data when finite-orbit absorption edges exist are discussed. It is shown that beyond an estimate of the  curvature, which is in surprisingly good agreement with the calculated value, a more precise interpretation of the  data cannot be justified. The merits of the helicon DSCR method for Fermi-surface studies are discussed, and it is concluded that in general useful information can be obtained only on very pure samples with ωcτ100, where ωc is the cyclotron frequency and τ the conduction-electron relaxation time.