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Eccentricity Classifies Them All

Parabola, ellipse, hyperbola — three names, three shapes, three definitions so far. This guide collapses them into one: a single rule with one dial, the eccentricity e, whose value alone decides which conic you get.

Three Shapes, One Suspicion

By now you have met the three curves separately and on their own terms. Guide 2 built the parabola from one focus and one directrix: every point equidistant from a fixed point and a fixed line. Guide 3 built the ellipse and the hyperbola from two foci, using sums and differences of distances. Those definitions look genuinely different — one line versus two points, equality versus a constant sum — and it is fair to ask whether these three curves have anything deeper in common than the accident of all being slices of a cone.

There is, and it is startlingly simple. Take the parabola's recipe — measure a point's distance to a focus, and its distance to a directrix — but stop insisting the two be equal. Instead, ask only that their ratio be a fixed number. Call that ratio the eccentricity, written e. The astonishing fact is that this single rule, with nothing changed but the value of e, generates the parabola, the ellipse, and the hyperbola in turn. One definition, one number, all three conics.

The Focus–Directrix Rule, Stated Once

Here is the rule in full. Fix a point F, the focus, and a line d not through it, the directrix. For any point P, let |PF| be its distance to the focus and let dist(P, d) be its perpendicular distance to the directrix. The focus–directrix property says: P lies on the conic exactly when the ratio |PF| / dist(P, d) equals a fixed positive constant e. That constant is the eccentricity, and it is the same for every point on the curve.

|PF| = e * dist(P, d)

    e = 1        ->  parabola
    0 < e < 1    ->  ellipse
    e > 1        ->  hyperbola
    e = 0        ->  circle  (the limiting, degenerate case)
One equation, four outcomes — the value of e alone decides the shape.

Read that table slowly, because it is the whole point of the guide. When e = 1, distance-to-focus equals distance-to-directrix exactly — that is word-for-word the parabola of guide 2. Push e below 1 and the curve must hug the focus more tightly than the directrix, closing up into the bounded oval of an ellipse. Push e above 1 and the curve is allowed to wander far from the focus relative to the directrix, opening out into the two unbounded branches of a hyperbola. The parabola is not a separate species; it is the exact knife-edge e = 1 between the closed and the open.

Feeling Each Case in Your Hand

Let us make the three cases concrete with a tiny worked sketch. Put the focus at F = (1, 0) and the directrix at the vertical line x = 4, so dist(P, d) for a point P = (x, y) is just |x - 4|. The rule |PF| = e * dist(P, d) then reads sqrt((x - 1)^2 + y^2) = e * |x - 4|. Watch what happens to one test point on the x-axis as we turn the dial e, the point where the curve crosses between focus and directrix.

  1. Set e = 1 (parabola). On the x-axis the rule is |x - 1| = |x - 4|, solved by the midpoint x = 2.5 — the vertex sits exactly halfway between focus and directrix, just as a parabola's vertex always does.
  2. Set e = 0.5 (ellipse). Now |x - 1| = 0.5 * |x - 4|. Between focus and directrix this gives x - 1 = 0.5(4 - x), so 1.5x = 3, x = 2 — closer to the focus. The curve will close up on both sides, an ellipse.
  3. Set e = 2 (hyperbola). Now |x - 1| = 2 * |x - 4|. Between the two this gives x - 1 = 2(4 - x), so 3x = 9, x = 3 — closer to the directrix. The curve cannot close; it flies off to infinity in two branches, a hyperbola.

Notice how the single vertex on the x-axis slides — from x = 2 to x = 2.5 to x = 3 — as e grows from 0.5 to 1 to 2. Below e = 1 the curve stays nearer the focus and turns back on itself; above e = 1 it leans toward the directrix and escapes to infinity; at e = 1 it balances precisely between. The continuous dial e really does morph one curve smoothly into the next, and the qualitative jump — closed versus open — happens exactly as e crosses 1.

Eccentricity Has a Meaning, Not Just a Value

For the ellipse and hyperbola you met in guide 3, eccentricity also measures something you can see: how far the curve departs from a perfect circle. For an ellipse with semi-axes a (the long one) and b (the short one), the two foci sit a distance c from the centre, where c^2 = a^2 - b^2, and the eccentricity is e = c / a. When the foci crowd together at the centre, c is near 0, so e is near 0 and the ellipse is nearly circular. As the foci spread toward the ends, c approaches a, e approaches 1, and the ellipse stretches into a long thin cigar. So small e means round; e near 1 means flattened.

The hyperbola obeys the same formula e = c / a, but now c^2 = a^2 + b^2 (a plus, not a minus — recall the sign flip from guide 3), which forces c > a and hence e > 1 always. And e controls the opening of the branches through the asymptotes: an eccentricity just barely above 1 gives a narrow, sharply bent hyperbola, while a large e gives wide-open branches that flatten toward a straight line. In every case the number e is not an arbitrary label — it is a faithful readout of the curve's shape.

One Polar Equation to Rule Them All

The deepest payoff of the focus–directrix view is that all three conics share a single equation — provided you place the origin at the focus and use polar coordinates (r, theta). With the focus at the pole, the polar equation of a conic is r = (e * L) / (1 + e cos theta), where L is the focus-to-directrix distance and e is the eccentricity. Change nothing but e and this one formula traces an ellipse, a parabola, or a hyperbola — exactly the unification the whole guide has been building toward, now in a single line of algebra.

You can read the three cases straight off the denominator 1 + e cos theta. If e < 1 it never reaches 0, so r stays finite for every angle and the curve closes — an ellipse. If e = 1 the denominator hits 0 at theta = 180 degrees, where r blows up: the single direction in which a parabola runs off to infinity. If e > 1 the denominator is 0 at two angles, marking the two directions of the hyperbola's asymptotes, beyond which r turns negative and the second branch appears. The arithmetic of when that denominator vanishes is the whole open-versus-closed story told once more.

This is why the focus-at-the-origin polar form is the natural language of orbits in physics: a planet, comet, or spacecraft moving under gravity travels on a conic with the Sun at one focus, and a single number e tells you whether it is bound (ellipse), barely escaping (parabola), or unbound (hyperbola). The next and final guide of this rung returns to the cone itself, where the elegant Dandelin spheres prove that these focus–directrix conics and the plane sections of a cone are one and the same family — and shows how the locus you have been tracing falls out of the general second-degree equation.