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.

Description

An Archimedean spiral r=a+bθ grows radius linearly with angle.

An Archimedean spiral r=a+bθ grows radius linearly with angle. Its segment length includes both radial growth and angular motion.^[1]

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.

When to use Archimedean Spiral Length Calculator

Use this tool to 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.

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

How to use Archimedean 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 initialRadius, radialSlopePerRadian, 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
initialRadius	Initial radius	number	Required
radialSlopePerRadian	Radial slope per radian	number	Required
startAngleRadians	Start angle radians	number	Required
endAngleRadians	End angle radians	number	Required

Use nonnegative initial radius and ordered nonnegative radian angles, with positive radial slope. Evaluate an equivalent stable closed antiderivative using hypot and log1p; return length in radius units.

length = integral[start..end]sqrt((a+bθ)²+b²)dθ

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": {
    "initialRadius": 0,
    "radialSlopePerRadian": 1,
    "startAngleRadians": 0,
    "endAngleRadians": 1
  },
  "output": {
    "spiralSegmentLength": 1.147793574696319
  }
}

Limitations and assumptions

Shrinking spirals, zero slope and coordinate-to-parameter inference are excluded.

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 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. 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

Archimedean Spiral Length Calculator — tool.io

Similar or alternative tools

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.
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.