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Analytic Geometry2D

Special Plane Curves - Theory & Concepts - Analytic Geometry Catenary Cable

Analysis of advanced planar curves including cycloids, epicycloids, hypocycloids, and lemniscates.

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Catenary vs Parabolic Cable Comparison

Span (Width S)10.0 m
Sag (Depth H)4.0 m
Cable Comparisons
Catenary Length:13.442 m
Parabola Length:13.337 m
Max Separation:0.147 m
Catenary Param a:3.646
Parabola Coeff k:0.1600

Catenary: Curve formed by a uniform hanging chain under its own weight. Governing equation: y=a(cosh⁡(x/a)−1)y = a(\cosh(x/a) - 1).

Parabola: Curve formed by a cable supporting a uniform horizontal load (like suspension bridge decks). Governing equation: y=kx2y = k x^2.

x = -S/2x = S/2CatenaryParabola
Mathematical Equations
Catenary: y=3.65(cosh⁡(x3.65)−1)\text{Catenary: } y = 3.65 \left( \cosh\left( \frac{x}{3.65} \right) - 1 \right)Parabola: y=0.1600x2\text{Parabola: } y = 0.1600 x^2

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