This calculator plots the ideal Archimedean-spiral relationship between radius and angle. It does not establish the accuracy, tolerances or construction of a particular historical drawing instrument.
Helicograph Interactive Calculator
Explore an ideal Archimedean spiral from entered starting radius, radial pitch per revolution and turn count, plus a clearly labelled comparison with a fixed 0.05 mm value.
Equation Used
- Radius increases linearly with angle as an ideal Archimedean spiral.
- Pitch and starting radius remain exact and constant over the entered turns.
- Backlash is user-entered and is not propagated into the spiral geometry.
- No specific instrument mechanism, accuracy, fit, calibration, material, pen or paper behavior is modeled.
Ideal geometry only. The signed backlash result is 0.05 mm minus the entered value; 0.05 mm is not established here as an allowable limit. No instrument error, compliance, lost motion, pen behavior or drawing accuracy is predicted.
Ideal polar construction
An Archimedean spiral has radius r=a+bθ. If radial pitch P is the increase in radius per full revolution, b=P/(2π). A conceptual rotating arm and radial feed can therefore place a pen at the required polar coordinate as the arm turns.
The animation illustrates that relationship only. It does not claim the gearing, leadscrew, tolerances or accuracy of any named historical instrument.
What the geometry can represent
The equation can support drawing, layout and educational exploration of planar Archimedean spirals. Whether a physical instrument, plotter or fabrication process can reproduce the curve within a required tolerance needs a separate error budget and verified equipment data.
Archimedean-spiral equations
With starting radius a and radial pitch P per revolution, b=P/(2π) and r(θ)=a+bθ. After N revolutions, θ=2πN, radial advance is PN and final radius is a+PN.
The calculator also shows 0.05 mm minus the entered backlash. This signed difference compares the entered value with an illustrative 0.05 mm reference; it is not an allowable backlash or accuracy margin.
Transparent example
For a=20 mm, P=12 mm/rev and N=3, radial advance is 36 mm, final radius is 56 mm and b=12/(2π)≈1.910 mm/rad.
If entered backlash is 0.08 mm, the final tile shows -0.03 mm because 0.05-0.08=-0.03. That sign does not determine whether an instrument is acceptable.
Scope and limitations
The ideal curve omits pitch error, backlash direction and reversal, bearing play, arm flexure, feed-screw error, carriage constraint, pen offset and pressure, paper movement and setup error. Those effects require measurements and a mechanism-specific model.
Questions
Is an Archimedean spiral a helix viewed end-on?
No. An end-on orthographic projection of a circular helix is a circle. A planar Archimedean spiral has radius that changes with angle.
Does negative 0.05 mm difference mean the drawing is unusable?
No. The fixed comparison is not a validated tolerance. Determine allowable error from the actual drawing requirement and instrument.
Does the final radius include backlash error?
No. The spiral outputs use ideal entered geometry; backlash is displayed separately.
Calculation basis
The polar equation r = a + bθ defines the ideal Archimedean spiral used here. With radial pitch P per revolution, b = P/(2π). OpenStax: Polar Coordinates explains how radius and angle locate points in a plane. A mechanism-specific drawing and error model would be needed to describe an actual historical instrument.
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