Conic Sections

Conic sections are curves formed by intersecting a plane with a double cone. They appear throughout mathematics, physics, and engineering -- from planetary orbits to satellite dishes to bridge arches.

The Focus-Directrix Definition

All conic sections (except circles) can be defined using a single elegant property:

A conic section is the locus of points P where the ratio of the distance to a fixed point (focus) and the distance to a fixed line (directrix) is constant.

This constant ratio is called the eccentricity (e).

Explore the Conic

Start by dragging eccentricity to morph between conic types:

Eccentricity (e): 0.6 Focal parameter (p): 3

Verifying the Focus-Directrix Property

Drag the sample point angle to move point P along the conic and verify the ratio stays constant:

Sample point angle: 60

For point P on the conic:

• Distance to focus: 1.385
• Distance to directrix: 2.308
• Ratio (should equal e): 0.600

Conic Type Classification

The eccentricity 0.6 determines the shape:

Ellipse (0 < e < 1)

An ellipse is a "stretched circle" with two foci. The sum of distances from any point to both foci is constant. Planetary orbits follow ellipses -- Earth's orbit has e = 0.017 (nearly circular), while Pluto's has e = 0.25.

Standard form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Your ellipse properties:

• Semi-major axis (a): 2.812
• Semi-minor axis (b): 2.250
• Distance between foci: 3.375
• Sum of focal distances: 5.62 (constant = 2a)

Try dragging 0.6 toward 0 to watch the ellipse become more circular, or toward 1 to see it elongate. Whispering galleries exploit this shape: sound from one focus reflects off the ellipse and converges at the other focus.

Key Formulas

The Reflective Property

Ellipse: Any ray from one focus reflects off the ellipse and passes through the other focus.

This is the secret behind whispering galleries -- a whisper at one focus can be heard clearly at the other, even across a large room. Drag 0.6 to reshape the ellipse and see how the reflection geometry changes.

Polar Equation

All conics can be expressed in polar form with the focus at the origin:

where:

• $r$ is the distance from the focus
• $\theta$ is the angle from the positive x-axis (toward directrix)
• $l = pe$ is the semi-latus rectum
• $e$ is the eccentricity

Sweep 60 around the conic to trace the polar equation in action:

From Cone to Conic

Imagine a double cone (two identical cones joined at their tips). The type of conic depends on the angle of the cutting plane:

1. Circle: Plane perpendicular to the cone's axis
2. Ellipse: Plane at an angle less steep than the cone's surface
3. Parabola: Plane parallel to exactly one edge of the cone
4. Hyperbola: Plane steeper than the cone's surface (cuts both cones)

The mathematical connection: if $\alpha$ is the cone's half-angle and $\beta$ is the cutting plane's angle from horizontal:

Return to the top and drag 0.6 slowly from 0 to 2 to watch the full morphing sequence -- circle, ellipse, parabola, hyperbola -- all unified by a single parameter.

Try it in Code

import math

# Conic section properties from eccentricity and focal parameter
e = 0.6
p = 3
theta = 60

l = p * max(e, 0.01)
directrix_dist = l / max(e, 0.01)
theta_rad = theta * math.pi / 180
polar_r = l / (1 + e * math.cos(theta_rad))
a = l / max(abs(1 - e**2), 0.0001)
b = a * math.sqrt(max(abs(1 - e**2), 0.0001))
c_val = a * e

conic_type = "Circle" if e < 0.01 else "Ellipse" if e < 0.99 else "Parabola" if e < 1.01 else "Hyperbola"
print(f"Type: {conic_type} (e = {e:.3f})")
print(f"Semi-latus rectum l: {l:.3f}")
print(f"Semi-major axis a: {a:.3f}")
print(f"Semi-minor axis b: {b:.3f}")
print(f"Focus distance c: {c_val:.3f}")
print(f"Polar r at theta={theta:.0f} deg: {polar_r:.3f}")