THE CONIC INVARIANT rotate the axes all you like — ellipse stays ellipse
A conic Ax² + Bxy + Cy² + … looks different in every rotated coordinate frame: the coefficients A, B, C all change. But the combination B² − 4AC — the discriminant — does not. Its sign is a rotation invariant that classifies the curve for good: negative is an ellipse, zero a parabola, positive a hyperbola. No choice of axes can turn an ellipse into a hyperbola, because no rotation can change the sign of B²−4AC. This is invariant theory in miniature: find the polynomial in the coefficients that the symmetry group leaves fixed, and it tells you what the object is.
THE TECHNIQUE B²−4AC unchanged by a rotation of the axes
A conic's discriminant, then the same conic after a 90° rotation of the axes (which swaps A and C). The demo re-measures — the classifier is unchanged: live demo
“Whether a curve is an ellipse depends on how you look at it.” — the sign of B²−4AC is rotation-invariant, so ellipse-ness is intrinsic. The frame changes the coefficients; it cannot change what the curve is. cited
1748–1801 · Euler (conic classification, Introductio 1748) and Lagrange, Gauss (discriminant-invariance of a quadratic form, Disquisitiones 1801) — the conic invariant itself, in the classical analytic tradition. 1841–51 · Boole, Cayley & Sylvester — the general theory of invariants that subsumes B²−4AC as one instance; Sylvester coined invariant and discriminant (1851). now · algebraic geometry, tensor invariants in physics, moment invariants in vision.
Change coordinates and the coefficients scramble; the invariant polynomial in them stays fixed, and its sign names the curve. Invariant theory: what the symmetry group cannot rewrite. Cayley-Sylvester
RECOMMEND FOR I-13 discriminant before and after a rotation, computed
On the canonical compiler, the ellipse 2x²+y² has discriminant −8; after a 90° rotation to x²+2y² it is −8 again — still an ellipse:
$ i13 run inv_conic.i13 # B^2-4AC, then after a 90-degree rotation
disc_before = -8
disc_after = -8
diff = 0 -- negative -> ellipse, in any frame
Recommend: the conic invariant is the same enacted-invariance move as darts 278–279, from invariant theory. i13 computes B²−4AC, rotates the axes, recomputes, and the number holds — the rotation group's fixed polynomial, evaluated in f64. Its sign (here negative) is the classification, and no transformation i13 applies can flip it. This is the third foreign shape in a row that i13 carries cleanly: projective (cross-ratio), linear (determinant), and now the invariant-theory discriminant — each a quantity a transformation group is defined to leave alone, and each one i13 can enact.