What is the equation of the Tschirnhausen cubic?

Cartesian equation: 27ay^2 = (a - x)(8a + x)^2; Polar equation: r = a sec^3 θ/3; Parametric equations: x = a(1 - 3t^2), y = a t(3 - t^2). a sets the size: the loop runs from x = −8a to x = a.

What is the area of the Tschirnhausen cubic?

Area (inside the loop): A = ((72√3)/5)a^2. a sets the size: the loop runs from x = −8a to x = a.

Classic Cartesian curves

Tschirnhausen cubic

27ay² = (a − x)(8a + x)²: a cubic with a loop, the caustic of a parabola lit at right angles to its axis.

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Formulas and key values

Cartesian equation: $27ay^2 = (a - x)(8a + x)^2$
Polar equation: $r = a\sec^3\frac{\theta}{3}$
Parametric equations: $x = a(1 - 3t^2)$ $y = a\,t(3 - t^2)$
Area · inside the loopIn the graph ≈ 24.94: $A = \frac{72\sqrt3}{5}a^2$

a sets the size: the loop runs from x = −8a to x = a.

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01 · Definition

What is the Tschirnhausen cubic?

Ehrenfried Walther von Tschirnhaus studied this cubic as a caustic: shine light at a parabola perpendicular to its axis, and the reflected rays envelope this curve. It also goes by l’Hôpital’s cubic and the trisectrix of Catalan.

Its polar form r = a·sec³(θ/3) and the parametrisation x = a(1 − 3t²), y = a·t(3 − t²) turn into each other with t = tan(θ/3). The curve crosses itself at (−8a, 0), closing a loop that reaches x = a and encloses area 72√3a²/5.

02 · Properties

Key properties

Self-intersection at (−8a, 0); the loop reaches (a, 0).
The loop encloses area 72√3a²/5.
Polar form r = a·sec³(θ/3); parametrisation x = a(1 − 3t²), y = a·t(3 − t²).
Catacaustic of a parabola for light perpendicular to its axis.
Also called l’Hôpital’s cubic and the trisectrix of Catalan.

03 · Questions