Catenary Curve Calculator
Calculate catenary curve values for hanging ropes, chains, and suspension structures
Calculate Catenary Curve
Controls the curve's sag (lower values = more sag)
Horizontal distance from center
Catenary Point Results
Formula used: y = a × cosh(x/a)
Hyperbolic cosine: cosh(x) = (e^x + e^(-x)) / 2
Parameters: a = 1, x = 0
Curve Analysis
Example Calculation
Suspension Bridge Cable
Scenario: Main cable of a suspension bridge
Sag parameter (a): 2.0
Point of interest: x = 1.5 (150 units from center)
Curve type: Standard catenary
Step-by-Step Calculation
1. Formula: y = a × cosh(x/a)
2. Substitute: y = 2.0 × cosh(1.5/2.0)
3. Calculate: y = 2.0 × cosh(0.75)
4. cosh(0.75) = (e^0.75 + e^(-0.75))/2 ≈ 1.295
5. Result: y ≈ 2.59
Catenary Properties
Shape
U-shaped curve formed by hanging chains or ropes
Mathematics
Based on hyperbolic cosine function
Applications
Bridges, power lines, architecture
Formula Reference
Standard Catenary
y = a × cosh(x/a)
Weighted Catenary
y = b × cosh(x/a)
Hyperbolic Cosine
cosh(x) = (e^x + e^(-x))/2
Arc Length
s = a × sinh(x/a)
Understanding Catenary Curves
What is a Catenary Curve?
A catenary curve is the shape formed by a rope, chain, or cable hanging freely under its own weight between two support points. The name comes from the Latin word "catēna," meaning chain.
Key Characteristics
- •Symmetric U-shaped curve
- •Minimum point at x = 0
- •Steepness increases exponentially from center
- •Often confused with parabolas but mathematically different
Applications
Architecture
Gateway Arch, ancient domes, suspension bridge cables
Engineering
Power transmission lines, cable structures
Nature
Spider webs, egg shells, erosion patterns
Fun Fact: The Gateway Arch in St. Louis is an inverted weighted catenary, often mistaken for a parabola!