ProofWiki
ProofWiki is an extensive online encyclopedia of mathematical proofs, particularly strong in abstract algebra and discrete mathematics. It aims to provide rigorously verifiable proofs for a wide range of mathematical theorems, making it a valuable resource for students and researchers seeking to understand the logical underpinnings of mathematical concepts.
概览
收录于
2026年3月17日
学科与领域
computer-science-fundamentals · discrete-mathematics
年级范围
九年级（高一）–十二年级（高四）
页面类型
Wiki
简介
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.
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