Chapter 27 — Further Reading

Each entry says what to read it for, not just the title. The two standard course references this book is benchmarked against (see _continuity.md §8) map to this chapter as follows.

Textbook section mapping

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.), Cengage. - §10.5 "Conic Sections" — the core match for §27.2–27.4 and §27.7: standard Cartesian forms of the parabola, ellipse, and hyperbola; foci, directrix, vertices, asymptotes; translated conics by completing the square. Work Stewart's §10.5 exercise set alongside Parts A–D of our exercises. - §10.6 "Conic Sections in Polar Coordinates" — the match for §27.5: the unified form $r = ed/(1 \pm e\cos\theta)$, eccentricity classification, and Kepler-orbit applications. Pairs with our Part G. - Tangents and area for these conics draw on Stewart's earlier §3.5 (implicit differentiation) and §7.3 (trig substitution), exactly our §27.8 cross-references to Chapters 8 and 16.

Strang, G., & Herman, E. Calculus, Volume 2 (OpenStax, free). - §7.5 "Conic Sections" — covers all of §27.2–27.5 and §27.7 in one section: focus–directrix definitions, standard equations, eccentricity, and the polar form together. The most efficient free companion to this chapter; its applied examples lean on orbits and reflectors just as ours do. - §7.4 "Area and Arc Length in Polar Coordinates" and §6.4 (arc length) — background for our §27.8.3 perimeter discussion.

Spivak, M. (2008). Calculus (4th ed.). Treats conics in its problem sets rather than a dedicated chapter; read it for the rigorous geometric viewpoint and for the focus–directrix property proved from scratch — the "Formal" rigor level this book reserves for Math Major Sidebars.

Apostol, T. M. (1967). Calculus, Volume I, §13. A clean axiomatic development of the conics from the focus definitions outward, good if you want the geometry derived before any calculus is applied.

Classical and historical

Apollonius of Perga, Conics (~200 BC), trans. T. L. Heath (Dover reprint). The foundational text that named and catalogued the three curves with no algebra at all. Read Book I to see the cone-slice definitions of §27.6 in their original form.

Heath, T. L. (1921). A History of Greek Mathematics, Vol. II. Oxford. The accessible guide to what Apollonius actually proved and how, for readers who will not wade through the Conics itself.

Coolidge, J. L. (1968). A History of the Conic Sections and Quadric Surfaces. Dover. Traces the two-thousand-year arc from Greek geometry through Descartes's coordinates (§27.6) to the quadric surfaces that open Part VI.

Orbital mechanics (Case Study 1)

Goldstein, H., Poole, C., & Safko, J. (2001). Classical Mechanics (3rd ed.), Ch. 3. Derives the orbit equation $r = p/(1 + e\cos\theta)$ from the inverse-square force — the Chapter 19 / Chapter 26 payoff our Mars case study only sketches. The single best place to see why orbits must be conics.

Murray, C. D., & Dermott, S. F. (1999). Solar System Dynamics, Ch. 2. The professional reference for orbital elements; shows how the chapter's $a$ and $e$ extend to the full six-element description of a real orbit.

Vallado, D. A. (2013). Fundamentals of Astrodynamics and Applications. Microcosm Press. Engineering-side companion: how the non-elementary perimeter integral of §27.8.3 is actually evaluated in mission-planning software.

Reflector optics and antennas (Case Study 2)

Hecht, E. (2017). Optics (5th ed.), Ch. 5–6. Pearson. The cleanest undergraduate treatment of the parabolic reflective property and spherical aberration, with ray diagrams that mirror §27.3.4.

Christiansen, W. N., & Högbom, J. A. (1985). Radiotelescopes (2nd ed.). Cambridge. Reflector-antenna geometry in exactly the language of our radio-telescope case study, including the Cassegrain parabola–hyperbola pairing of §27.4.5.

Baars, J. W. M. (2007). The Paraboloidal Reflector Antenna. Springer. A focused monograph on the dish geometry of Case Study 2, including surface-tolerance budgets.

Acoustics and medical applications

Lingeman, J. E. (2007). Shock Wave Lithotripsy: Principles and Practice. Humana Press. The medical companion to the ellipse's two-focus reflective property (§27.2.4): how a shock wave from one focus is concentrated on a kidney stone at the other.

A practice recommendation

The single most useful habit: sketch each conic by hand repeatedly until you can place foci, vertices, and asymptotes at sight, and say out loud which $abc$ relation you are using each time. Then, once the geometry is automatic, take one real orbit (Mars from Case Study 1, or a Voyager gravity-assist flyby — a hyperbola relative to the planet) and identify every conic parameter in it. Deriving Kepler's first law from $F = -GMm/r^2$ — using Chapter 19's differential equations and Chapter 26's polar form — is the capstone exercise and one of the great intellectual triumphs in the history of science.