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.
Description
A tractrix approaches an asymptote, so its complete length is infinite.
A tractrix approaches an asymptote, so its complete length is infinite. A bounded parameter interval describes a finite segment of x=a(t−tanh t), y=a sech t.1
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.
When to use Tractrix Length Calculator
Use this tool to 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.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Tractrix 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 tractrixScale, startParameter, endParameter using the field table and calculation rules, then run the tool.
- Read tractrixSegmentLength. 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 |
|---|---|---|---|
| tractrixScale | Tractrix scale | number | Required |
| startParameter | Start parameter | number | Required |
| endParameter | End parameter | number | Required |
Use positive scale and ordered parameters in [0,100]. Evaluate an equivalent log1p expression in the parameter difference to reduce cancellation; equal endpoints return zero length in the scale's units.
segment length = a[log cosh(end) − log cosh(start)] Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| tractrixSegmentLength | Tractrix segment length | number |
{
"input": {
"tractrixScale": 2,
"startParameter": 0,
"endParameter": 1
},
"output": {
"tractrixSegmentLength": 0.8675616609660545
}
}Limitations and assumptions
This measures only the supplied segment. It does not assign finite length to the full asymptotic curve.
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 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. 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
-
Tractrix Length Calculator — tool.io
Similar or alternative tools
- 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.
- 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.