Gregorian telescope
A parabola and an ellipse, each placed at the one pair of points it images perfectly, and a traced spot seven ten-millionths of a millimetre across.
Open in the canvas →Background
A spherical mirror does not focus. Rays near its rim cross the axis ahead of rays near its centre, and no plane anywhere along the axis catches them all — the light forms a caustic rather than a point. This is not a manufacturing defect; it is what a sphere is.
The conic sections are the cure, and each one is exact for exactly one pair of points. A parabola images infinity onto its focus. An ellipse images one of its foci onto the other. A hyperbola does the same for one real focus and one virtual one. These are not approximations that improve on the sphere — they are geometric identities, true for a ray at the rim as much as one on the axis.
James Gregory published this telescope in 1663, before anyone had built a reflecting telescope at all: a parabolic primary to collect the light, and a concave elliptical secondary placed past the primary's focus to relay that image out through a hole in the primary. Each mirror is asked to do the one job its own shape does perfectly, so the pair inherits the exactness.
The arrangement costs length — the secondary must sit beyond the prime focus, so the tube is longer than a Cassegrain of the same focal length — and repays it with an upright image and, more usefully, a real intermediate focus inside the instrument, where a field stop can sit and reject stray light before it ever reaches the detector.
What this setup demonstrates
Both mirrors are real conic surfaces, intersected analytically, so nothing here is a paraxial stand-in: the focus is as good or as bad as the geometry makes it.
The primary is a parabola (k = −1) of f = 40 mm, forming a real image at x = 460. The secondary sits 20 mm beyond that, and its prescription follows directly from requiring its two foci to land on the intermediate image and on the focal plane: R = 2pq/(p+q) = 31.35 mm and k = −((q−p)/(q+p))² = −0.3221, with p = 20 and q = 72.5. That is the mirror equation and the eccentricity, nothing more.
The traced spot at the focal plane spans 7×10⁻⁷ mm — zero, to the precision the arithmetic holds. Set either conic constant to zero and the surface becomes a sphere, and the point becomes a smear you can measure.
The central obstruction is not drawn in. The secondary is an opaque mirror sitting in the beam, so it blocks the middle of the aperture because it is genuinely in the way, and the primary is illuminated as an annulus in consequence.
This is a 2D meridional section. There is no sagittal plane, so astigmatism and field curvature are not reproduced as a real instrument shows them, and the off-axis behaviour sampled here is only one cut through a rotationally symmetric system.
Nothing is diffractive: no Airy disc, and none of the ring redistribution a central obstruction really causes, so the geometric point is sharper than any real telescope's. Reflectivity is a flat percentage with no angle, polarization or wavelength dependence, and the scale here is a teaching choice rather than any catalogue instrument.