Lemniscate Perimeter Approximation

Bernoulli lemniscate full perimeter scale×√πΓ(1/4)/Γ(3/4), scale conventionr²=a²cos2θ; not ellipse perimeter.

Description

The Bernoulli lemniscate perimeter depends on an explicit polar scale convention.

The Bernoulli lemniscate perimeter depends on an explicit polar scale convention.^[1]

This tool evaluates the gamma-function expression for the full perimeter under r squared equals a squared times cos two theta.

When to use Lemniscate Perimeter Approximation

This tool evaluates the gamma-function expression for the full perimeter under r squared equals a squared times cos two theta.

How to use Lemniscate Perimeter Approximation and how it works

Prepare lemniscate scale, using the input definitions and matching units below.
Enter the required fields and run Lemniscate Perimeter 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
callerLemniscateScale	Positive length scale a under the stated lemniscate convention.	number	Required; strictly positive; finite; other constraints below

Require positive lemniscate scale a for r^2=a^2*cos(2*theta). Evaluate a*sqrt(pi)*Gamma(1/4)/Gamma(3/4) using the shared gamma approximation. Return full perimeter in the scale's length unit; no endpoint coordinates or arbitrary precision are produced.

fullPerimeter=a*sqrt(pi)*Gamma(1/4)/Gamma(3/4)

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": {
    "callerLemniscateScale": 1
  },
  "output": {
    "result": {
      "value": 5.244115108584234
    }
  }
}

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 Perimeter 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.