Catenary Length Calculator
Exact catenary y=a cosh(x/a) arc length between supplied horizontal coordinates; not sailing chain sag under force or its parabolic approximation.
Description
A catenary y=a cosh(x/a) describes a specific mathematical curve.
A catenary y=a cosh(x/a) describes a specific mathematical curve. Its segment length is determined by scale and horizontal endpoint coordinates.1
Exact catenary y=a cosh(x/a) arc length between supplied horizontal coordinates; not sailing chain sag under force or its parabolic approximation.
When to use Catenary Length Calculator
Use this tool to exact catenary y=a cosh(x/a) arc length between supplied horizontal coordinates; not sailing chain sag under force or its parabolic approximation.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Catenary Length Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter catenaryScale, startX, endX using the field table and calculation rules, then run the tool.
- Read catenarySegmentLength. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| catenaryScale | Catenary scale | number | Required |
| startX | Start x | number | Required |
| endX | End x | number | Required |
Use positive scale, ordered x bounds and |each x/a|≤100. Evaluate the equivalent midpoint-cosh times half-span-sinh expression to reduce cancellation; equal endpoints give zero.
length = a[sinh(end/a)−sinh(start/a)] Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| catenarySegmentLength | Catenary segment length | number |
{
"input": {
"catenaryScale": 2,
"startX": -2,
"endX": 2
},
"output": {
"catenarySegmentLength": 4.7008047745752055
}
}Limitations and assumptions
This geometric calculation does not solve loading, tension or chain sag from forces.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Tractrix Length Calculator: Compute finite tractrix segment x=a(t−tanh t), y=a sech t over nonnegative t from a log-cosh primitive; whole asymptotic tractrix has infinite length. For a complementary task, use Logarithmic Spiral Length Calculator: Compute logarithmic spiral segment length for r=a exp(bθ), retaining shrinking spirals and exact b=0 circular limit; finite output bounds exponential overflow.
References
-
Catenary Length Calculator — tool.io
Similar or alternative tools
- Tractrix Length Calculator
Compute finite tractrix segment x=a(t−tanh t), y=a sech t over nonnegative t from a log-cosh primitive; whole asymptotic tractrix has infinite length.
- Logarithmic Spiral Length Calculator
Compute logarithmic spiral segment length for r=a exp(bθ), retaining shrinking spirals and exact b=0 circular limit; finite output bounds exponential overflow.