Case Study 2 — Designing a Parabolic Radio Telescope
Field: Engineering / optics (radio astronomy instrumentation) Calculus & geometry used: Parabola focus–directrix definition and standard form (§27.3), the reflective property (§27.3.4), tangent slope by implicit differentiation (§27.8.1 / Chapter 8), the Cassegrain hyperbola (§27.4.5)
Why a dish must be a paraboloid
A radio telescope faces a brutal problem: the radio signal from a distant galaxy is almost unimaginably faint — a receiver a few centimeters wide intercepts a power measured in femtowatts. The only way to detect it is to gather radio waves over a huge area and concentrate them onto that small receiver without smearing them out. Section 27.3.4 tells us exactly which shape does this perfectly: the parabola, because it is the unique curve for which every ray arriving parallel to the axis reflects through a single common point, the focus.
This case study walks through the design of a $60$-meter dish from the geometry up. We will choose its shape, locate its receiver, check the focusing with calculus, and see why a sphere would fail — and then see why the largest telescopes add a second conic, a hyperbola, to fold the light into a more usable place.
Throughout, the engineering theme of the chapter holds: a $2{,}000$-year-old curve studied by Apollonius for its beauty turns out to be the one shape that makes the instrument work, and only the parabola will do.
Setting up the standard form
Orient the dish so its axis points up the $y$-axis and its vertex sits at the origin. In cross-section the reflecting surface is the parabola
$$x^2 = 4py,$$
from §27.3.2, where $|p|$ is the focal length — the distance from the vertex to the focus, which sits at $(0, p)$. The full dish is the paraboloid of revolution obtained by spinning this curve about the $y$-axis, but every focusing fact we need lives in the two-dimensional cross-section.
A radio dish is specified by two numbers an engineer chooses: its diameter $D$ (how much area it gathers) and its focal ratio $f = p/D$ (how deep the dish is). A common choice for a steerable dish is $f = 0.4$, deep enough to be compact but shallow enough that the receiver does not block too much of the aperture. For our $D = 60$ m dish,
$$p = f \cdot D = 0.4 \times 60 = 24 \text{ m}.$$
So the design equation of the cross-section is $x^2 = 4(24)y = 96y$, and the receiver — the feed horn that the incoming waves are concentrated onto — must be placed at the focus $(0, 24)$, twenty-four meters out along the axis from the bottom of the dish, held there by a support structure.
How deep is the dish?
The rim of the dish is at $x = \pm D/2 = \pm 30$ m. Its depth is the $y$-value there:
$$y_{\text{rim}} = \frac{x^2}{96} = \frac{30^2}{96} = \frac{900}{96} = 9.375 \text{ m}.$$
So a $60$-meter dish with $f = 0.4$ is about $9.4$ meters deep — a bowl three stories deep and wider than a football field. Every point on that bowl, from the vertex to the rim, has been ground (or, for radio wavelengths, paneled) to lie on the paraboloid to a tolerance of a small fraction of the observing wavelength. At radio wavelengths of centimeters this is achievable with shaped metal panels; the same precision demand at visible wavelengths is what makes optical mirrors so famously delicate.
Verifying the focus with the reflective property
Why does the receiver belong at $(0, 24)$ and nowhere else? The focus–directrix definition of §27.3.1 is the cleanest argument. A parabola is the set of points equidistant from the focus $F = (0, p)$ and the directrix line $y = -p$. A wave arriving straight down from a distant galaxy travels along a vertical line; by the time it reaches the dish, the part of the wavefront that strikes the surface at a point $P$ has traveled a distance proportional to how high $P$ sits, and the geometry of equidistance guarantees that the remaining path from $P$ to the focus exactly compensates. Every ray, whichever point of the dish it hits, arrives at the focus having traveled the same total optical path — so they arrive in phase and add up coherently. That coherence is the entire point: it is what turns a femtowatt scattered across a $2{,}800$-square-meter dish into a detectable signal at one feed horn.
We can confirm the reflection direction with calculus. Differentiate $x^2 = 96y$ implicitly (the technique of §27.8.1 and Chapter 8):
$$2x = 96\,\frac{dy}{dx} \quad\Longrightarrow\quad \frac{dy}{dx} = \frac{x}{48}.$$
Take a point partway up the dish, say $P = (24, 6)$ — check it lies on the surface: $24^2 = 576 = 96 \times 6$. ✓ The tangent slope there is $\frac{dy}{dx} = 24/48 = 1/2$. A wave coming straight down has direction "vertical." The law of reflection about the tangent line of slope $1/2$ sends that vertical incoming ray off toward the point $(0, 24)$ — the focus — and the same computation at any other point of the dish lands on the same focus. This is the §27.3.4 reflective property made arithmetic: the slope changes from point to point, but the reflected rays all converge on $(0, p)$.
Why not a sphere?
A spherical dish is far cheaper to build — a sphere looks the same from every direction, so its panels are interchangeable. But a sphere does not focus parallel rays to a single point. Rays striking near the rim cross the axis closer to the mirror than rays striking near the center; the focus is smeared into a short segment instead of a point. This is spherical aberration, and §27.3.4 names the parabola as the unique cure. For a radio telescope the consequence is direct: a blurred focus means the faint signal is spread over a region larger than the feed horn, and most of it is simply lost. The Arecibo telescope famously was spherical — a $305$-meter bowl fixed in a Puerto Rican sinkhole — and paid for its sphericity with a complicated line-feed and later a Gregorian sub-reflector system specifically engineered to correct the aberration the sphere introduced. The parabola needs no such correction; that is its engineering value.
Folding the light: the Cassegrain hyperbola
Placing the receiver at the prime focus, $24$ m up in the air, is awkward: the feed and its amplifiers are heavy, hard to access, and partly block the aperture. The standard fix uses a second conic, and it is the hyperbola of §27.4.5. A small hyperbolic secondary mirror is mounted near the prime focus. The key property is that a hyperbola has two foci, and a ray heading toward one of them reflects so as to head toward the other. The secondary is positioned with one of its foci coinciding with the parabola's focus; it intercepts the converging cone of radio waves before they reach that focus and relays them cleanly down to the hyperbola's other focus, placed behind the dish through a hole at the vertex.
This Cassegrain arrangement — parabolic primary plus hyperbolic secondary — puts the heavy receiver electronics in a stable, accessible cabin behind the dish instead of dangling at the prime focus. It is the configuration of most large modern radio dishes (and, with the same conic pairing, the Hubble and James Webb space telescopes). Two conic sections, chosen so that they share a focus, hand the light from one to the other exactly as their reflective properties guarantee. The chapter's unity is literally bolted together here: the parabola gathers, the hyperbola relays, and the geometry that Apollonius proved on a cone now steers radio waves from the edge of the universe into an amplifier.
Discussion questions
- We chose $f = 0.4$, giving $p = 24$ m. If an engineer instead picked $f = 0.25$ (a deeper dish), recompute $p$, the design equation $x^2 = 4py$, and the dish depth at the $30$-m rim. What practical trade-off does a deeper dish buy?
- Show, using $x^2 = 4py$, that the dish width at the height of the focus (set $y = p$) equals $4p$ — the latus rectum. For our dish, what is that width, and why might a designer care about it when sizing the secondary mirror?
- The reflective verification used the point $(24, 6)$. Repeat the implicit-differentiation argument at the rim point $(30, 9.375)$: find the tangent slope and confirm qualitatively that the reflected vertical ray still heads toward $(0, 24)$.
- A Cassegrain design requires the secondary hyperbola to share one focus with the primary parabola. In one or two sentences, explain why the focus–difference definition of the hyperbola (§27.4.1) is what makes this "relay one focus to the other" behavior possible.
A short annotated reading list
- Christiansen, W. N., & Högbom, J. A. (1985). Radiotelescopes (2nd ed.), Cambridge. The standard text on reflector antenna geometry; Chapter 4 treats the paraboloid and Cassegrain optics directly in the language of this chapter.
- Wilson, T. L., Rohlfs, K., & Hüttemeister, S. (2013). Tools of Radio Astronomy (6th ed.), Springer, Ch. 6–7. Connects the focal geometry to antenna gain and beam shape — the "why the focus must be sharp" story quantified.
- Hecht, E. (2017). Optics (5th ed.), Pearson, Ch. 5–6. The cleanest undergraduate treatment of the parabolic reflective property and spherical aberration, with ray diagrams that mirror §27.3.4.
- Baars, J. W. M. (2007). The Paraboloidal Reflector Antenna in Radio Astronomy and Communication. Springer. A focused monograph on exactly the dish geometry designed in this case study, including surface-tolerance budgets.