The equiangular spiral
The polar equation of the spiral to equiangular is:
It receives the name of equiangular because the angle formed by the radius vector and the tangent is constant.
If
is that angle, the equiangular spiral can be expressed with the equation
It is the locus of the transform of (a,0) by a dilative rotation (Coxeter).
In the applet different equiangular spirals can be seen. For example, dragging the indicated points.
The property that gives name to the equiangular spiral can be verified pressing the bottons "Step +" or "Step -".
This spiral was called "spiral mirabilis" by Jacob Bernouilli.
REFERENCES
Coxeter - Introduction to Geometry (John Whiley and sons)
Steinhaus - Mathematical Snapshots.
D'Arcy Thompson - On Growth and Form. (Cambridge University Press)
LINKS
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Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.
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Equiangular spiral
In an equiangular spiral the angle between the position vector and the tangent is constant.
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The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
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Multiplying two complex numbers
We can see it as a dilatative rotation.
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