Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves

Equiangular SpiralLogarithmic Spiral

Pursuit Curve

Persuit curves are the trace of an object chasing another. Suppose there are n mice each at a corner of a n sided regular polygon. Each bug crawls towards its next neighbor with uniform speed. The trace of these mice are equiangular spirals of (n-2)/n * Pi/2 radians (half the angle of the polygon's corner). The Figure 1 on the left shows the trace of n mice mice, resulting n equiangular spirals of 45 degree. In the second part of the Figure 1, we may see some whirls which are figures constructed by nesting a sequence of polygons (each haing the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the n mice and form n logarithmic spirals.



Figure 1

Persuit Curves



Catacaustic of an equiangular spiral with light source at pole is an equal spiral. Proof: Let O be the pole of the curve. Let O' be the reflection of O through the normal of a variable point P on the curve. The locus of O' is then an equal spiral since distance[O,O']/distance[O,P] is constant for any P and equiangular spiral remain unchanged by scaling. Now the reflected ray PO' is just the tangent of O'. End of proof.


Figure 2


The evolute of an equiangular spiral is an equal spiral, so is its involute. In Figure 3, The left figure shows osculating circles of the curve and the locus (red curve) of their centers (Q).
The right figure shows the curve's envelope of normals.

Figure 3a

Figure 3b



The radial of an equiangular spiral is itself scaled. The Figure 4a shows a 70 degree equiangular spiral and its radial. The Figure 4b shows its involute, which is another equiangular spiral.


Figure 4a

Figure 4b



The inversion of an equiangular spiral with respect to its pole is an equal spiral (See Figure 5).



Figure 5


The pedal of an equiangular spiral with respect to its pole is an equal spiral. In the Figure 6b, the lines from pole to the red dots is perpendicular to the tangents (blue lines). The blue curve is an equiangular spiral. The red dots forms its pedal.

Figure 6a



Figure 6b


Negative Pedal The negative pedal of an equiangular spiral is shown at Figure 7.

negative pedal

Figure 7



Parallel curves can be defined as the locus of points Q1, Q2 on the normals of the curve, where Q1 and Q2 are k distant from the curve (See Figure 8).




Figure 8

Geometric Sequence

If any part of the curve is scale up or down, it becomes congruent to other parts of the curve. Lengths of segments (red lines) cut by equally spaced radii (brown lines) is a geometric sequence. Segments of any radius vector cut by the curve is also a geometric sequence, with a multiplier of E^(2 Pi Cot[alpha]).

geometric sequence

Figure 9

Sources and References:

1. http://www.best.com/~xah/SpecialPlaneCurves_dir/EquiangularSpiral_dir/equiangularSpiral.html

2. http://www.treasure-troves.com/math/whirl.html

3. http://www.treasure-troves.com/math/MiceProblem.html

This page created December 4, 1999

This page last modified December4, 1999