🔗 Catenary Curve Calculator
Calculate hanging cable/chain shape (catenary curve)
Higher a = flatter curve, Lower a = steeper curve
Option 1: Calculate at Point x
Option 2: Find a from Endpoints
How to Use This Calculator
Enter Parameter a
Input the shape parameter a (must be positive). This controls how flat or steep the catenary curve is.
Choose Calculation Method
Either: (A) Enter an x-coordinate to find y, slope, and length at that point, or (B) Enter two endpoints to estimate parameter a.
Click Calculate
Press "Calculate" to get the catenary curve values or parameter estimate.
Formula
y = a × cosh(x/a)
where cosh is the hyperbolic cosine function
Key Formulas:
- Curve equation: y = a × cosh(x/a)
- Slope: dy/dx = sinh(x/a)
- Arc length from origin: s = a × sinh(x/a)
- Lowest point: (0, a) - the vertex
Where:
- a = shape parameter (controls curve steepness)
- x = horizontal coordinate
- y = vertical coordinate
- cosh(t) = (eᵗ + e⁻ᵗ)/2
- sinh(t) = (eᵗ - e⁻ᵗ)/2
Example: For a = 1, find y at x = 2
y = 1 × cosh(2/1) = cosh(2) ≈ 3.762
Slope = sinh(2) ≈ 3.627
About Catenary Curve Calculator
The Catenary Curve Calculator finds values for the mathematical curve that describes the shape of a hanging chain or cable under uniform gravity. This curve is called a catenary, from the Latin word "catena" meaning chain.
When to Use This Calculator
- Engineering: Design suspension bridges and hanging cables
- Architecture: Calculate the shape of arches and catenary structures
- Physics: Model hanging chains, ropes, and flexible cables
- Mathematics: Study hyperbolic functions and their applications
- Construction: Plan power lines, suspension bridges, and cables
- Design: Create catenary-shaped structures and arches
Why Use Our Calculator?
- ✅ Hyperbolic Functions: Uses cosh and sinh for accurate calculations
- ✅ Multiple Methods: Calculate at a point or find parameter from endpoints
- ✅ Comprehensive Results: Shows y-value, slope, and arc length
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Educational: Helps understand catenary curves
- ✅ Completely Free: No registration required
Understanding Catenary Curves
A catenary is the curve formed by a perfectly flexible, uniform chain hanging under gravity. Unlike a parabola (which would form if the weight were uniform per horizontal distance), a catenary has uniform weight per arc length.
- Shape: y = a × cosh(x/a)
- Parameter a controls steepness (higher a = flatter curve)
- Lowest point is at (0, a) - the vertex
- Symmetric about the y-axis
- Uses hyperbolic cosine (cosh) function
- Arc length from origin: s = a × sinh(x/a)
Real-World Applications
Suspension Bridges: The main cables of suspension bridges follow catenary curves, ensuring even weight distribution and structural stability.
Power Lines: Overhead electrical cables hang in catenary shapes. Engineers use catenary equations to determine cable sag and tension.
Architecture: The Gateway Arch in St. Louis is an inverted catenary, providing optimal structural strength for its height and span.
Frequently Asked Questions
What is a catenary curve?
A catenary is the mathematical curve formed by a hanging chain or cable under uniform gravity. The equation is y = a × cosh(x/a), where a is a shape parameter and cosh is the hyperbolic cosine function.
Is a catenary the same as a parabola?
No! A parabola has uniform weight per horizontal distance. A catenary has uniform weight per arc length. They look similar but are different curves with different equations.
What does parameter a represent?
Parameter a controls the steepness of the catenary. Higher a values produce flatter curves (wider hang), while lower a values produce steeper curves (narrower hang). The vertex is always at (0, a).
How do I find a from two endpoints?
Given two endpoints, you need to solve the catenary equation system. This calculator provides an approximation. For exact solutions, use numerical methods or iterative algorithms.
What is cosh (hyperbolic cosine)?
cosh(x) = (eˣ + e⁻ˣ)/2, where e is Euler's number (~2.718). It's similar to cosine but uses exponentials instead of circles. cosh(0) = 1, and cosh(x) grows exponentially as |x| increases.
Can catenary be used for arches?
Yes! An inverted catenary (upside down) is often used for arches because it distributes weight optimally. The Gateway Arch in St. Louis is an inverted catenary.
What's the difference between catenary and parabola?
Parabola: y = ax² (uniform weight per horizontal distance). Catenary: y = a × cosh(x/a) (uniform weight per arc length). They look similar but have different mathematical properties.