ProofWiki

ProofWiki: Cycloid Arc Length Derivation

• Project Overview: ProofWiki is an online collaborative compendium dedicated to the collection, classification, and creation of mathematical proofs.
• Subject Matter: The provided text details a historical and mathematical derivation of the arc length of a cycloid.
• Mathematical Context:
  • The cycloid is defined by parametric equations.
  • The goal is to prove that the length of one arc of a cycloid is $8a$.
• Methodology:
  • The proof utilizes Descartes' method of approximating the cycloid using a polygon inscribed in a generating circle.
  • The derivation involves calculating the lengths of "arms" (chords) formed by the polygon's sides as it rotates.
  • Key mathematical tools used include the Inscribed Angle Theorem, Lagrange's Sine Identity, and trigonometric limits.
• Key Result: The total arc length is derived by summing the chord lengths and taking the limit as the number of sides ($n$) approaches infinity.