logarithmic spiral

The equation of the logarithmic spiralMathworldPlanetmath in polar coordinatesMathworldPlanetmath r,φ is

r=Cekφ (1)

where C and k are constants (C>0).  Thus the position vector of the point of this curve as the coordinate vector is written as


which is a parametric form of the curve.

Perhaps the most known of the logarithmic spiral is that any line emanating from the origin the curve under a constant angle ψ.  This is seen e.g. by using the vector r and its derivativedrdφ=r,  the latter of which gives the direction of the tangent lineMathworldPlanetmath (see vector-valued function):


One obtains




It follows that  k=cotψ.  The angle ψ is called the polar tangential angle.

The logarithmic spiral (1) goes infinitely many times round the origin without to reach it; in the case  k>0  one may state that

limφ-Cekφ= 0butCekφ 0φ

(the exponential function never vanishes).

The arc lengthMathworldPlanetmath s of the logarithmic spiral is expressible in closed form; if we take it for the intervalMathworldPlanetmathPlanetmath[φ1,φ2],  we can calculate in the case  k>0  that




Letting  φ1-  one sees that the arc length from the origin to a point of the spiral is finite.

Other properties

  • Any curve with constant polar tangential angle is a logarithmic spiral.

  • All logarithmic spirals with equal polar tangential angle are similarMathworldPlanetmathPlanetmath.

  • A logarithmic spiral rotated about the origin is a spiral homotheticMathworldPlanetmath to the original one.

  • The inversionMathworldPlanetmathz1z  causes for the logarithmic spiral a reflexion against the imaginary axis and a rotation around the origin, but the image is congruent to the original one.

  • The evolute of the logarithmic spiral is a congruent logarithmic spiral.

  • The catacausticMathworldPlanetmath of the logarithmic spiral is a logarithmic spiral.

  • The families  r=C1eφ  and  r=C2e-φ  are orthogonal curves to each other.

Title logarithmic spiral
Canonical name LogarithmicSpiral
Date of creation 2013-03-22 19:02:26
Last modified on 2013-03-22 19:02:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 26
Author pahio (2872)
Entry type Topic
Classification msc 14H45
Synonym Bernoulli spiral
Related topic AngleBetweenTwoCurves
Related topic EvoluteOfCycloid
Related topic PolarTangentialAngle2
Related topic AngleBetweenTwoLines