curvature of Nielsen’s spiral

Nielsen’s spiral is the plane curveMathworldPlanetmath defined in the parametric form

x=acit,y=asit (1)

where a is a non-zero constant, “ci” and “si” are the cosine integralDlmfDlmfDlmfMathworld ( and the sine integral ( and t is the parameter ( (t>0).

We determine the curvature ( κ of this curve using the expression

κ=xy′′-yx′′[(x)2+(y)2]3/2. (2)

The first derivativesMathworldPlanetmath of (1) are

x=ddt(atcosuu𝑑u)=acostt, (3)
y=ddt(atsinuu𝑑u)=asintt, (4)

and hence the second derivatives


Substituting the derivatives in (2) yields


which is easily simplified to

κ=ta. (5)

The arc lengthMathworldPlanetmath ( of Nielsen’s spiral can also be obtained in a closed formMathworldPlanetmath (; using (3) and (4) we get:



s=alnt. (6)

Note.  The expressions for x and y allow us determine as well


which says that the sense of the parameter t is the slope angle of the tangent line of the Nielsen’s spiral.

Figure 1: Plot of Nielsen’s spiral for 2t50. Axis scaling is in units of a. (Octave / MATLAB program for plot; in format)
Title curvature of Nielsen’s spiral
Canonical name CurvatureOfNielsensSpiral
Date of creation 2015-02-06 12:53:54
Last modified on 2015-02-06 12:53:54
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 22
Author pahio (2872)
Entry type Example
Classification msc 53A04
Synonym arc length of Nielsen’s spiral
Related topic CosineIntegral
Related topic SineIntegral
Related topic FamousCurvesInThePlane
Related topic DerivativeForParametricForm
Defines Nielsen’s spiral