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A classical Cassegrain reflector uses a concave parabolic primary and a convex hyperbolic secondary before the prime focus. The secondary returns the converging beam through a hole in the primary. It corrects on-axis spherical aberration but retains off-axis coma.

A Newtonian telescope uses a concave parabolic primary and a flat diagonal secondary to send the converging beam to a side focus. The primary avoids on-axis spherical aberration, but has off-axis coma.

The Ritchey–Chrétien reflector follows the same folded path as a Cassegrain reflector, but both the concave primary and convex secondary are hyperbolic. Their conic constants and separation are chosen to cancel third-order spherical aberration and coma. The focal plane is behind the perforated primary.

The absence of third-order coma gives a much more useful wide field than a classical Cassegrain reflector, making this design attractive for research imaging. It retains astigmatism and field curvature, so a large flat detector generally needs corrective optics; higher-order optical aberrations are not all removed. Both aspheric mirrors are more demanding to manufacture and align, and the usual secondary obstruction remains. The sketch illustrates the beam routing and mirror types; its conics are not an optimized aplanatic prescription.

In a classical Cassegrain reflector, a concave parabolic primary sends light toward an intermediate focus, but a convex hyperbolic secondary intercepts it before it reaches that focus. The secondary returns the beam through a central hole in the primary to a focal plane behind the primary. The two relevant foci of the secondary’s hyperbola are the primary’s would-be focus and the final focus.

The secondary magnifies the effective focal length, giving a compact tube and convenient rear-mounted instruments. The classical conics correct on-axis spherical aberration, but not the off-axis coma, astigmatism or field curvature. There is secondary obscuration, diffraction from its supports, and sensitivity to mirror alignment. A long effective focal length is useful for a small angular image scale per detector pixel, but yields a small field for a fixed detector size.

A Newtonian reflector uses a concave primary generated by a parabola, followed by a flat secondary inclined at $45^{\circ }$ to the optical axis. Parallel marginal rays converge toward the primary focus; the secondary intercepts that converging beam and folds it sideways to an accessible focal plane.

The Newtonian reflector needs only one powered mirror, so it is relatively simple to fabricate and inexpensive. A parabolic primary has no on-axis spherical aberration. Its principal wide-field limitation is coma, accompanied by field curvature and off-axis astigmatism. The diagonal and its supports obstruct the entrance pupil and introduce diffraction; the tube is long compared with a folded two-powered-mirror design, and heavy instruments at the side focus can be awkward to support. These comments also answer the unheaded design-comparison clause on the next PDF page.

A classical Schmidt telescope combines a concave spherical primary with a thin aspheric corrector plate at its centre of curvature. The plate is also near the aperture stop. It supplies just the extra ray bending needed to cancel the mirror’s spherical aberration; rays then return toward an internal focus roughly halfway between the plate and primary. The native best-focus surface is curved, as indicated in red.

The spherical primary is easy to manufacture, and the stop at its centre of curvature gives a nearly symmetric optical geometry that suppresses coma and astigmatism over a large field. Fast Schmidt cameras are consequently excellent survey instruments. The costs are a long enclosed optical assembly, a large precision corrector, an internal camera that obstructs the beam, and a curved focal surface. A flat detector needs field flattening. The transmissive corrector introduces wavelength-dependent effects and transmission losses, unlike an entirely reflecting system.

For the unnormalized Zernike polynomial with $n=4,m=0$, the three radial terms areSince $\cos 0\varphi =1$,Along a central chord with signed coordinate $s\in [-1,1]$, replace $\rho$ by $|s|$. The profile is even, equals one at the center and both edges, and has minima $-1/2$ at $s=\pm 1/\sqrt{2}$. This Zernike spherical mode corresponds to primary spherical aberration, balanced by defocus and piston terms rather than simply the raw quartic Seidel term.

A Ritchey–Chrétien telescope has hyperbolic primary and secondary mirrors chosen to cancel third-order spherical aberration and coma. It retains astigmatism and field curvature; wide-field instruments may require additional correction.

A Schmidt camera combines a spherical primary mirror with a thin aspheric corrector at the mirror’s centre of curvature. The corrector compensates spherical aberration; the symmetric stop location gives a wide useful field. The native focal surface is curved and lies inside the instrument.

The five Seidel aberrations of a rotationally symmetric optical system are spherical aberration, coma, astigmatism, field curvature, and optical distortion.

The unnormalized rotationally symmetric mode is $Z_4^0(\rho )=6{\rho }^4-6{\rho }^2+1$. It describes balanced primary spherical aberration. The lower-degree terms make it orthogonal to constant piston and quadratic defocus under the disk area measure: direct integration gives ${\int }_0^1{Z}_4^0\rho d\rho =0$ and ${\int }_0^1{Z}_4^0(2{\rho }^2-1)\rho d\rho =0$.