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Premium experiment • one eccentricity family

MATHEMATICS • GEOMETRY

Conic Sections Explorer

Tilt a cutting plane through a double cone and watch eccentricity connect circles, ellipses, parabolas and hyperbolas as one geometric family.

● CONE + CUTTING PLANEConic geometry
Conic
Eccentricity
Cone half-angle
Plane-axis angle

Reasoning inspector

live model state
Conic type—
Eccentricity e—
Cone half-angle—
Plane-axis angle—
Angle gap—
Major scale—
Minor scale—
Parameter p—

One family, four familiar curves

Changing the plane orientation moves eccentricity from zero through one and beyond, linking the closed and open conics through a continuous geometric construction.

LEARN WITH THE SIMULATION

Understand Conic Sections Explorer by experimenting.

START HERE

Circle, ellipse, parabola and hyperbola are one geometric family. When a plane cuts a double cone, the relationship between the plane orientation and the cone generator determines an eccentricity e. This laboratory connects that 3D cutting geometry to the normalized 2D conic and keeps eccentricity, plane angle, cone half-angle and curve scales visible together. The parabolic case becomes the precise boundary e = 1 between closed ellipses and open hyperbolas.

BUILD THE IDEA

What you should understand first

  • Classify conics from eccentricity: circle e≈0, ellipse 0<e<1, parabola e=1 and hyperbola e>1.
  • Connect the plane-axis angle with the cone half-angle through e = cosβ/cosα in the model.
  • Explain why the parabola occurs when the cutting plane is parallel to a cone generator.
  • Relate eccentricity to qualitative curve openness rather than memorizing four unrelated equations.

TRY IT YOURSELF

What happens to eccentricity as the cutting plane approaches the cone generator?

  1. Fix a cone half-angle and start with the cutting plane close to perpendicular to the axis.
  2. Decrease the plane-axis angle toward the cone half-angle while monitoring e.
  3. At the boundary, verify that e approaches 1 and the curve becomes parabolic.
  4. Continue past the boundary and watch the model switch to a hyperbola with e > 1.
Read the visual:

The generator boundary is not an arbitrary classification line. When the plane angle equals the cone half-angle, the model gives e = 1, exactly the focus-directrix eccentricity of a parabola. Angles on opposite sides produce ellipse or hyperbola families.

WHY IT WORKS 01

Eccentricity is a single classifier for the whole family

In focus-directrix language, a conic is the set of points whose distance to a focus is e times their distance to a directrix. Values below one produce closed ellipses, exactly one produces a parabola and values above one produce hyperbolas. The circle appears as the limiting ellipse with e = 0.

WHY IT WORKS 02

The cone-plane geometry generates the same eccentricity boundary

In this normalized model e = cosβ/cosα, where α is the cone half-angle and β is the plane angle to the cone axis. If β > α then cosβ < cosα and e < 1, producing an ellipse. If β = α, e = 1 and the plane is parallel to a generator. If β < α, e > 1 and the cut becomes hyperbolic.

WHY IT WORKS 03

Scale changes size while eccentricity controls family shape

The parameter p changes the normalized size of the plotted conic without changing the angle-derived eccentricity. This separation is useful: one variable controls geometric family and openness, while another sets scale. Two conics can therefore share the same eccentricity but differ in size.

DO NOT MIX THESE UP

Common misconception

The four conics are not separate unrelated curves. A circle is a special ellipse with eccentricity zero, and the parabola is the transition between closed and open members of the family. The displayed curve is normalized for teaching and is not a dimensional projection of a particular physical cone section.

PRACTICE CHALLENGE

Can you deliberately produce each conic family by reasoning from angle relationships before checking the classification?

  • Choose a target conic family.
  • Compare plane-axis angle with cone half-angle and predict whether e should be below, equal to or above one.
  • Adjust the plane angle until the target is reached.
  • Change p afterward and verify that scale changes while the eccentricity classification remains fixed.
Reveal the reasoning

For this model, compare β with α. If β > α, e < 1 and the cut is elliptical; if β = α, e = 1 and the cut is parabolic; if β < α, e > 1 and the cut is hyperbolic. The circle is the limiting case as the cutting plane becomes perpendicular to the axis and e approaches zero.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • A right circular double cone is represented by a cone half-angle and a cutting-plane angle measured relative to the cone axis.
  • The teaching eccentricity relation uses e = cos(plane angle)/cos(cone half-angle), giving the expected circle/ellipse/parabola/hyperbola boundaries.
  • The displayed 2D conic is normalized from eccentricity and a scale parameter rather than projected from a full 3D mesh.

Model limitations

  • The visualization does not compute the exact dimensional intersection for arbitrary plane offsets or rotations.
  • Degenerate conics through the cone vertex are excluded.

CONNECT IT

Where this matters

  • Orbital geometry
  • Reflectors and antennas
  • Analytic geometry
  • Optical and mechanical conic surfaces

WORDS THAT MATTER

Key vocabulary

Eccentricity
Dimensionless parameter e that classifies the conic family and its departure from circularity.
Generator
Straight line lying on the cone surface and extending from the vertex.
Nappe
One of the two halves of a double cone.
Parabolic boundary
Case e = 1 where the cutting plane is parallel to a cone generator.
Scale parameter
Parameter that changes normalized curve size without changing the eccentricity family.