The Double Cone — Unified Conic Geometry

The Core Idea

The circle, ellipse, parabola, and hyperbola are not four separate curves. They are a single family — the conic sections — produced by slicing a double cone at different angles. One geometric object, one algebraic framework, one parameter (eccentricity) that smoothly morphs between them.

This note compresses everything you know about conics into a single mental model.

flowchart TB
    CONE["Double Cone<br/>(two nappes, apex to apex)"]
    CONE --> SLICE

    subgraph SLICE["Cutting Plane Angle φ (from horizontal)"]
        direction LR
        PHI0["φ = 0°"] --> CIRCLE
        PHI1["0° < φ < ψ"] --> ELLIPSE
        PHI2["φ = ψ"] --> PARABOLA
        PHI3["ψ < φ ≤ 90°"] --> HYPERBOLA
    end

    CIRCLE[Circle<br/>e = 0]
    ELLIPSE[Ellipse<br/>0 < e < 1]
    PARABOLA[Parabola<br/>e = 1]
    HYPERBOLA[Hyperbola<br/>e > 1]

The cone's half-angle ψ sets the threshold. When the cutting plane tilts past ψ (parallel to a generator), you transition from ellipse to parabola. Tilt further — you cut both nappes and get a hyperbola.

1. The Double Cone — Geometric Setup

Visualise a double cone: two identical right circular cones placed apex-to-apex, sharing a common axis.

  • Axis: the central line through both apexes
  • Generator: any straight line on the cone's surface passing through the apex
  • Half-angle ψ: the angle between the axis and any generator (fixed for a given cone)
  • Nappes: the upper and lower halves of the double cone

A plane slicing through this cone produces a conic section. The angle of the cutting plane (relative to the axis) alone determines which conic you get:

Cutting angle (from axis) Conic What you see
φ = 90° (perpendicular to axis) Circle Plane cuts one nappe horizontally
ψ < φ < 90° Ellipse Plane cuts one nappe, steeper than a generator
φ = ψ (parallel to a generator) Parabola Plane skims parallel to a generator line
0° ≤ φ < ψ Hyperbola Plane cuts both nappes

The double cone sits upstream of all four conics. One shape, four curves.

2. The Unified Algebraic Classification (Exam Essential)

Every conic section can be written as a quadratic equation in two variables:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

For your syllabus, $B = 0$ (no rotated conics). This simplifies the discriminant:

$$\boxed{\Delta = B^2 - 4AC = -4AC}$$

The signs of $A$ and $C$ alone classify the conic:

flowchart TD
    EQ["General: Ax² + Cy² + Dx + Ey + F = 0<br/>(B = 0 for your syllabus)"]
    EQ --> CHECK{"Are both A and C present?"}

    CHECK -->|"No — one of A, C = 0"| PARA["Parabola<br/>Δ = 0"]
    CHECK -->|"Yes — both present"| SAME{"Do A and C have<br/>the same sign?"}

    SAME -->|"Yes, Δ < 0"| ELLIPSE_FAM["Ellipse Family"]
    SAME -->|"No, Δ > 0"| HYPER["Hyperbola<br/>Δ > 0"]

    ELLIPSE_FAM --> EQ_CHECK{"A = C?"}
    EQ_CHECK -->|"Yes"| CIRCLE["Circle"]
    EQ_CHECK -->|"No"| ELLIPSE["Ellipse"]

Quick classification table:

Condition on A, C Δ = B² − 4AC Conic
$A = C$ (same sign) $\Delta < 0$ Circle
$A \neq C$ (same sign) $\Delta < 0$ Ellipse
$A = 0$ or $C = 0$ (but not both) $\Delta = 0$ Parabola
$A$ and $C$ opposite signs $\Delta > 0$ Hyperbola

Exam rule of thumb: Look at the $x^2$ and $y^2$ coefficients.

  • Same coefficient → Circle. Same sign → Ellipse.
  • One missing → Parabola.
  • Opposite signs → Hyperbola.

3. Eccentricity — The Master Parameter

Eccentricity $e$ is a single number that tells you everything about a conic's shape. It encodes the cutting-plane angle from the double cone:

Conic Eccentricity Cutting plane from horizontal
Circle $e = 0$ Horizontal
Ellipse $0 < e < 1$ Tilted, but not as steep as generator
Parabola $e = 1$ Parallel to a generator
Hyperbola $e > 1$ Steeper: cuts both nappes 
flowchart LR
    subgraph ECCENTRICITY["Eccentricity Continuum"]
        direction LR
        C[e=0] --> E1[e=0.5] --> E2[e=0.8] --> P[e=1] --> H1[e=1.5] --> H2[e=3]
    end

    C --- CIR[Circle]
    E1 --- ELL1[Ellipse]
    E2 --- ELL2[Ellipse]
    P --- PAR[Parabola]
    H1 --- HYP1[Hyperbola]
    H2 --- HYP2[Hyperbola]

    style C fill:#4CAF50
    style P fill:#FF9800
    style CIR fill:#4CAF50
    style PAR fill:#FF9800

Formula for $e$ (from standard equation):

  • Ellipse: $e = \dfrac{c}{a}$ where $c^2 = a^2 - b^2$
  • Hyperbola: $e = \dfrac{c}{a}$ where $c^2 = a^2 + b^2$
  • Parabola: $e = 1$ by definition
  • Circle: $e = 0$ (ellipse with $a = b$, so $c = 0$)

4. The Unified Formula Table (Exam Cheat Sheet)

All four conics tabled side-by-side. Same structure, different numbers. Memo rise the pattern, not each row separately.

Feature Circle Ellipse Parabola Hyperbola
Eccentricity $e = 0$ $0 < e < 1$ $e = 1$ $e > 1$
Standard eqn (centre origin) $x^2 + y^2 = r^2$ $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $x^2 = 4ay$ or $y^2 = 4ax$ $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$
Standard eqn (centre $h,k$) $(x-h)^2 + (y-k)^2 = r^2$ $\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$ $(x-h)^2 = 4a(y-k)$ or $(y-k)^2 = 4a(x-h)$ $\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1$
Centre $(h,k)$ $(h,k)$ Vertex at $(h,k)$ (not centre) $(h,k)$
Foci Centre itself $(\pm c, 0)$ or $(0, \pm c)$ $(0, a)$ or $(a, 0)$ $(\pm c, 0)$ or $(0, \pm c)$
Vertices All points at distance $r$ $(\pm a, 0)$ or $(0, \pm b)$ $(h,k)$ $(\pm a, 0)$ or $(0, \pm a)$
$a,b,c$ relation $a = b = r$ $c^2 = a^2 - b^2$ (or $b^2 - a^2$) Not applicable $c^2 = a^2 + b^2$
Latus rectum $2r$ $\dfrac{2b^2}{a}$ $4a$ $\dfrac{2b^2}{a}$
Key distance property Constant distance from centre $d_1 + d_2 = 2a$ (sum) $d_1 = d_2$ (equal) $\vert d_1 - d_2 \vert = 2a$ (diff)
Parametric $(h + r\cos t,; k + r\sin t)$ $(h + a\cos t,; k + b\sin t)$ $(h + at^2,; k + 2at)$ or $(h + 2at,; k + at^2)$ $(h + a\sec t,; k + b\tan t)$ or $(h + a\tan t,; k + b\sec t)$
Asymptotes None None None $y - k = \pm \dfrac{b}{a}(x - h)$

Pattern recognition:

  • Ellipse and hyperbola share the same latus rectum formula $\dfrac{2b^2}{a}$ — only the $a,b,c$ relation differs ($-$ vs $+$).
  • Parabola is the odd one out — only one squared term, no centre, $e=1$ fixed.
  • Circle is the ellipse limit $a = b$.

5. From General Form → Full Analysis (Exam Workflow)

The same 4-step workflow works for every conic:

Step 1: Classify

Look at $A$ and $C$ in $Ax^2 + Cy^2 + Dx + Ey + F = 0$.

Step 2: Complete the square

Bring to standard form.

Step 3: Read off features

Centre/vertex, $a$, $b$, $c$ from the standard form.

Step 4: Find vertices, foci, asymptotes (if applicable)

Use the formulas from the unified table.

Worked Example 1 — Ellipse

$$4x^2 + 9y^2 - 24x + 36y + 36 = 0$$

Step 1: $A = 4$, $C = 9$, same sign, $A \neq C$ → Ellipse ($\Delta < 0$).

Step 2: Complete squares. $$4(x^2 - 6x) + 9(y^2 + 4y) = -36$$ $$4(x^2 - 6x + 9) + 9(y^2 + 4y + 4) = -36 + 36 + 36$$ $$4(x - 3)^2 + 9(y + 2)^2 = 36$$ $$\frac{(x - 3)^2}{9} + \frac{(y + 2)^2}{4} = 1$$

Step 3: Centre $(3,-2)$, $a = 3$, $b = 2$. Horizontal major axis ($a > b$). $c^2 = a^2 - b^2 = 9 - 4 = 5$, so $c = \sqrt{5}$.

Step 4:

  • Vertices: $(3 \pm 3, -2) = (6, -2)$ and $(0, -2)$
  • Foci: $(3 \pm \sqrt{5}, -2)$
  • $e = \dfrac{c}{a} = \dfrac{\sqrt{5}}{3} \approx 0.745$
  • Latus rectum: $\dfrac{2b^2}{a} = \dfrac{2(4)}{3} = \dfrac{8}{3}$

Worked Example 2 — Hyperbola

$$9x^2 - 16y^2 - 36x + 64y - 44 = 0$$

Step 1: $A = 9$, $C = -16$, opposite signs → Hyperbola ($\Delta > 0$).

Step 2: Complete squares. $$9(x^2 - 4x) - 16(y^2 - 4y) = 44$$ $$9(x^2 - 4x + 4) - 16(y^2 - 4y + 4) = 44 + 36 - 64$$ $$9(x - 2)^2 - 16(y - 2)^2 = 16$$ $$\frac{(x - 2)^2}{\frac{16}{9}} - \frac{(y - 2)^2}{1} = 1$$

So $a^2 = \frac{16}{9}$, $b^2 = 1$ → $a = \frac{4}{3}$, $b = 1$. $c^2 = a^2 + b^2 = \frac{16}{9} + 1 = \frac{25}{9}$, so $c = \frac{5}{3}$.

Step 3: Horizontal transverse axis. Centre $(2, 2)$.

Step 4:

  • Vertices: $\left(2 \pm \frac{4}{3}, 2\right)$
  • Foci: $\left(2 \pm \frac{5}{3}, 2\right)$
  • Asymptotes: $y - 2 = \pm \frac{b}{a}(x - 2) = \pm \frac{3}{4}(x - 2)$
  • $e = \dfrac{c}{a} = \dfrac{5/3}{4/3} = \dfrac{5}{4} = 1.25$
  • Latus rectum: $\dfrac{2b^2}{a} = \dfrac{2(1)}{4/3} = \dfrac{3}{2}$

Worked Example 3 — Parabola

$$y^2 + 6y + 1 + 4x = 0$$

Step 1: Only $y^2$ term → Parabola ($\Delta = 0$).

Step 2: Complete square in $y$. $$y^2 + 6y + 9 = -4x - 1 + 9$$ $$(y + 3)^2 = -4x + 8$$ $$(y + 3)^2 = -4(x - 2)$$

So $4a = -4$ → $a = -1$. Since $a < 0$, the parabola opens left.

Step 3: Vertex $(2, -3)$.

Step 4:

  • Focus: $(2 + a, -3) = (2 - 1, -3) = (1, -3)$
  • Directrix: $x = 2 - a = 2 - (-1) = 3$
  • Latus rectum: $4|a| = 4$

Worked Example 4 — Circle

$$x^2 + y^2 - 4x + 6y - 12 = 0$$

Step 1: $A = C = 1$, same sign and equal → Circle ($\Delta < 0$, $A = C$).

Step 2: Complete squares. $$(x^2 - 4x + 4) + (y^2 + 6y + 9) = 12 + 4 + 9$$ $$(x - 2)^2 + (y + 3)^2 = 25$$

Step 3: Centre $(2, -3)$, radius $r = 5$.

6. Common Exam Traps Caught by the Unified View

Trap What they give you What it really is Why the unified view saves you
Degenerate circle $x^2 + y^2 + Dx + Ey + F = 0$ where $r^2 < 0$ No real curve Always check $r^2 > 0$ after completing the square
Ellipse ≠ circle $4x^2 + 9y^2 + \dots = 0$ Ellipse, not circle $A = C$ test catches this instantly
Missing sign check $\frac{x^2}{9} - \frac{y^2}{4} = 1$ Hyperbola, not ellipse Sign of $y^2$ term = key, not just "both present"
Parabola hiding $y^2 + 6y + 4x = 0$ Parabola (no $x^2$ term) $\Delta = 0$ test: one squared term missing
Hyperbola's c vs ellipse's c Both use $c^2 = a^2$ — but one is $+b^2$, the other $-b^2$ Opposite signs on $b^2$ Discriminant $\Delta > 0$ reminds you it's $+$
Wrong latus rectum Rectangle through focus Length depends on orientation Ellipse: minor axis direction. Hyperbola: same formula, check $b$ carefully

7. The Big Picture — Why This Framework Works

flowchart TB
    subgraph GEOMETRY["3D Geometry — The Double Cone"]
        CONE[Double Cone]
        PLANE[Cutting Plane]
        ANGLE["Angle φ determines the conic"]
    end

    subgraph ALGEBRA["2D Algebra — The Quadratic"]
        EQ["Ax² + Cy² + Dx + Ey + F = 0"]
        DISCRIM["Δ = -4AC classifies the conic"]
    end

    subgraph PARAMETER["The Unifier — Eccentricity"]
        ECENT["e = c/a or e = 1"]
        SLIDER["e = 0 → 0<e<1 → e=1 → e>1"]
    end

    GEOMETRY -->|"produces"| ALGEBRA
    GEOMETRY -->|"encodes"| PARAMETER
    ALGEBRA -->|"describes"| CONICS[The Four Conics]
    PARAMETER -->|"morphs between"| CONICS

    CONICS --- CIRC[Circle]
    CONICS --- ELLIP[Ellipse]
    CONICS --- PARAB[Parabola]
    CONICS --- HYPER[Hyperbola]

Three ways to see the same family:

  1. Geometric (double cone + cutting plane) — why they're the same family
  2. Algebraic (discriminant of the quadratic) — how to classify instantly in an exam
  3. Parametric (eccentricity $e$) — how they morph into one another

Learn any one of these, and you can reconstruct the rest. The four conics are not separate topics — they are four snapshots of the same object at different angles.

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