Conic Sections Overview

Learning Goals

• Objective is to illustrate different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases.

Historical Context

• Conic sections date back to Ancient Greece, discovered by Menaechmus around 360-350 B.C.
• Initially explored to address the problem of doubling a cube.
• Solutions involved mean proportions and the construction of a cone.

Definition of Conic Sections

• A conic section is a curve created by the intersection of a plane and a double right circular cone.

Types of Conic Sections

• Conic sections include circles, parabolas, ellipses, and hyperbolas.
• Circles are considered a special case of ellipses.
• These curves are prevalent in nature and various applications.

Characteristics and Applications

• Circle: Formed when the intersecting plane is horizontal.
• Parabola: Appears when the plane intersects one cone, creating an unbounded curve (e.g., trajectory of a thrown ball).
• Ellipse: Formed by a tilted plane intersecting one cone, resulting in a bounded curve (e.g., planetary orbits around the sun).
• Hyperbola: Created when the plane intersects both cones, giving rise to two unbounded curves (e.g., applications in telescope design and navigation systems).

Degenerate Conics

• Degenerate conics arise when the intersection includes:
  • A point (one intersection).
  • One line (tangential intersection).
  • Two lines (crossing intersection).

Architecture and Conic Sections

• Conic sections have applications in design and architecture, as seen in structures like:
  • Eiffel Tower
  • Farmer’s Cottage Deluxe Summer House
  • Tycho Brahe Planetarium