Bernoulli Lemniscate Area Calculator

Compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.

Description

The Bernoulli lemniscate r²=a² cos(2θ) has two lobes.

The Bernoulli lemniscate r²=a² cos(2θ) has two lobes. Its scale here is the maximum polar radius, which differs from focal half-distance.^[1]

Compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.

When to use Bernoulli Lemniscate Area Calculator

Use this tool to compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Bernoulli Lemniscate Area 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 lemniscateScale using the field table and calculation rules, then run the tool.
Read totalBothLobesArea, oneLobeArea. 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
lemniscateScale	Lemniscate scale	number	Required

Enter positive lemniscate scale in a chosen length unit. Return total and individual-lobe areas in squared units.

both-lobe area = a²; one-lobe area = a²/2

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
totalBothLobesArea	Total both lobes area	number
oneLobeArea	One lobe area	number

json

{
  "input": {
    "lemniscateScale": 2
  },
  "output": {
    "totalBothLobesArea": 4,
    "oneLobeArea": 2
  }
}

Limitations and assumptions

Use the declared polar-radius convention when converting geometric measurements.

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 Rose Area Calculator: Compute distinct rose petals for r=a cos(kθ), integer k; odd k repeats the curve over 2π, so repeated traces are not counted as extra area. For a complementary task, use Limacon Area Calculator: Compute one 0–2π polar trace area for r=a+b cos θ; when b>a inner/outer loops are both counted by polar integral, not a geometric union area.

References

Bernoulli Lemniscate Area Calculator — tool.io

Similar or alternative tools

Rose Area Calculator
Compute distinct rose petals for r=a cos(kθ), integer k; odd k repeats the curve over 2π, so repeated traces are not counted as extra area.
Limacon Area Calculator
Compute one 0–2π polar trace area for r=a+b cos θ; when b>a inner/outer loops are both counted by polar integral, not a geometric union area.