Lemniscate Arc Length Approximation

RestrictedBernoullilemniscatequadrantarcscale×integral0…u dt/√(1−t⁴),0≤u≤.99, no endpoint-singular naive quadrature.

Description

A normalized lemniscate arc integral has an endpoint singularity as its parameter approaches one.

A normalized lemniscate arc integral has an endpoint singularity as its parameter approaches one.^[1]

This tool integrates only through 0.99 using fixed-panel quadrature and a supplied scale.

When to use Lemniscate Arc Length Approximation

This tool integrates only through 0.99 using fixed-panel quadrature and a supplied scale.

How to use Lemniscate Arc Length Approximation and how it works

Prepare normalized arc endpoint, lemniscate scale, using the input definitions and matching units below.
Enter the required fields and run Lemniscate Arc Length Approximation.
Read result.value using the units, signs and ordering explained below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualNormalizedArcEndpoint	Dimensionless endpoint u from 0 through 0.99 inclusive.	number	Required; nonnegative; finite; other constraints below
callerLemniscateScale	Positive length scale a under the stated lemniscate convention.	number	Required; strictly positive; finite; other constraints below

Require normalized endpoint u from 0 to 0.99 inclusive and positive scale a. Use 512-panel Simpson quadrature on integral from 0 to u of 1/sqrt(1-t^4), then multiply by a. Return the restricted arc quantity in the scale's unit; u=1 is rejected and no error certificate is provided.

arcApprox=a*Simpson(1/sqrt(1-t^4),0,u,512)

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.value	Geometric length in the supplied radius, side or curve-scale unit.	number

json

{
  "input": {
    "actualNormalizedArcEndpoint": 0,
    "callerLemniscateScale": 1
  },
  "output": {
    "result": {
      "value": 0
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use Nephroid Perimeter Calculator for a complementary operation: A nephroid is an epicycloid whose convention here uses a fixed-circle radius twice the rolling-circle radius.

Use Nephroid Area Calculator for a complementary operation: Nephroid area depends on the curve's radius convention.

References

Lemniscate Arc Length Approximation — tool.io

Similar or alternative tools

Nephroid Perimeter Calculator
A nephroid is an epicycloid whose convention here uses a fixed-circle radius twice the rolling-circle radius.
Nephroid Area Calculator
Nephroid area depends on the curve's radius convention.