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Equiangular Spiral

above: A equiangular spiral and its secants.

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History

The investigation of spirals began at least with the ancient Greeks. The famous Equiangular Spiral was discovered by Rene Descartes↗, its properties of self-reproduction by Jacob Bernoulli↗ (1654-1705) (aka James or Jacques) who requested that the curve be engraved upon his tomb with the phrase “Eadem mutata resurgo” (“I shall arise the same, though changed.”) [Source: Robert C Yates (1952)]

The equiangular spiral was first considered in 1638 by Descartes, who started from the property s = a.r. Evangelista Torricelli↗, who died in 1647, worked on it independently and used for a definition the fact that the radii are in geometric progression if the angles increase uniformly. From this he discovered the relation s = a.r; that is to say, he found the rectification of the curve. Jacob Bernoulli, some fifty years later, found all the “reproductive” properties of the curve; and these almost mystic properties of the “wonderful” spiral made him wish to have the curve incised on his tomb: Eadem mutata resurgo — “Though changed I rise unchanged”. [source: E H Lockwood (1961)]

Description

Equiangular spiral describes a family of spirals of one parameter. It is defined as a curve that cuts all radial line at a constant angle.

It also called logarithmic spiral, Bernoulli spiral, and logistique.

Explanation:

Let there be a spiral (that is, any curve r==f[θ] where f is a monotonic↗ inscreasing function)
From any point P on the spiral, draw a line toward the center of the spiral. (this line is called the radial line)
If the angle formed by the radial line and the tangent for any point P is constant, the curve is a equiangular spiral.

above: A example of equiangular spiral with angle 80°.

A special case of equiangular spiral is the circle, where the constant angle is 90°.

above: Equiangular spirals with 40°, 50°, 60°, 70°, 80° and 85°. (left to right)

Equiangular Spiral

Formulas

Let α be the constant angle.

Polar: r == E^(θ * Cot[α]) gcf icon equiangular_spiral.gcf

Parametric: E^(t * Cot[α]) {Cos[t],Sin[t]}

Cartesian: x^2 + y^2 == E^(ArcTan[y/x] Cot[α] )

Properties

Point Construction and Geometric Sequence

Length of segments of any radial ray cut by the curve is a geometric sequence↗, with a multiplier of E^(2 π Cot[α]).

Lengths of segments of the curve, cut by equally spaced radial rays, is a geometric sequence.

above: The curve cut by radial rays. The length of any green ray's segments is geometric sequence. The lengths of red segments is also a geometric sequence. In the figure, the dots are points on a 85° equiangular spiral.

Catacaustic

Catacaustic of a equiangular spiral with light source at center is a equal spiral.

Proof: Let O be the center of the curve. Let α be the curve's constant angle. Let Q be the reflection of O through the tangent normal of a point P on the curve. Consider Triangle[O,P,Q]. For any point P, Length[Segment[O,P]]==Length[Segment[P,Q]] and Angle[O,P,Q] is constant. (Angle[O,P,Q] is constant because the curve's constant angle definition.) Therefore, by argument of similar triangle, then for any point P, Length[Segment[O,Q]]==Length[Segment[O,P]]*s for some constant s. Since scaling and rotation around its center does not change the curve, thus the locus of Q is a equiangular spiral with constant angle α, and Angle[O,Q,P] == α. Line[P,Q] is the tangent at Q.

Equiangular Spiral Caustic

Curvature

The evolute of a equiangular spiral is the same spiral rotated.

The involute of a equiangular spiral is the same spiral rotated.

above: left: Tangent circles of a 80° equiangular spiral. The white dots are the centers of tangent circles, the lines are the radiuses. Right: Lines are the tangent normals, forming the evolute curve by envelope.

Equiangular Spiral Evolute

Radial

The radial of a equiangular spiral is itself scaled. The figure on the left shows a 70° equiangular spiral and its radial. The figure on the right shows its involute, which is another equiangular spiral.

Inversion

The inversion of a equiangular spiral with respect to its center is a equal spiral.

Pedal

The pedal of a equiangular spiral with respect to its center is a equal spiral.

above: pedal of a equiangular spiral. The lines from center to the red dots is perpendicular to the tangents (blue lines). The blue curve is a 60° equiangular spiral. The red dots forms its pedal.

Pursuit Curve

Persuit curves are the trace of a object chasing another. Suppose there are n bugs each at a corner of a n sided regular polygon. Each bug crawls towards its next neighbor with uniform speed. The trace of these bugs are equiangular spirals of (n-2)/n * π/2 radians (half the angle of the polygon's corner).

above: left: shows the trace of four bugs, resulting four equiangular spirals of 45°. Above right: six objects forming a chasing chain. Each line is the direction of movement and is tangent to the equiangular spirals so formed.