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.

Description

A logarithmic spiral changes radius exponentially with angle.

A logarithmic spiral changes radius exponentially with angle. A negative exponential rate describes shrinking motion, while zero rate reduces to a circular arc.^[1]

Compute logarithmic spiral segment length for r=a exp(bθ), retaining shrinking spirals and exact b=0 circular limit; finite output bounds exponential overflow.

When to use Logarithmic Spiral Length Calculator

Use this tool to compute logarithmic spiral segment length for r=a exp(bθ), retaining shrinking spirals and exact b=0 circular limit; finite output bounds exponential overflow.

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

How to use Logarithmic Spiral 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 referenceRadius, radialExponentialRate, startAngleRadians, endAngleRadians using the field table and calculation rules, then run the tool.
Read spiralSegmentLength. 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
referenceRadius	Reference radius	number	Required
radialExponentialRate	Radial exponential rate	number	Required
startAngleRadians	Start angle radians	number	Required
endAngleRadians	End angle radians	number	Required

Use positive reference radius, ordered finite radian angles and |b|≤100. Evaluate the exponential radius and a stable expm1 difference; preserve nonnegative length for shrinking spirals and return zero for equal endpoints.

length = a exp(b start) sqrt(1+b²) expm1(b span)/b; b=0 gives a span

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

Output field	Meaning	Value type
spiralSegmentLength	Spiral segment length	number

json

{
  "input": {
    "referenceRadius": 1,
    "radialExponentialRate": 1,
    "startAngleRadians": 0,
    "endAngleRadians": 1
  },
  "output": {
    "spiralSegmentLength": 2.4300174657860216
  }
}

Limitations and assumptions

Overflow is rejected. This describes only the supplied finite angular segment.

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 Archimedean Spiral Length Calculator: Compute arc length of monotone Archimedean r=a+bθ for nonnegative radii/angles and positive b using exact antiderivative, distinct from MRI spiral gradient timing. For a complementary task, 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.

References

Logarithmic Spiral Length Calculator — tool.io

Similar or alternative tools

Archimedean Spiral Length Calculator
Compute arc length of monotone Archimedean r=a+bθ for nonnegative radii/angles and positive b using exact antiderivative, distinct from MRI spiral gradient timing.
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.