“Logarithmic spiral”的意思、由来-开放百科全书

A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

Definition

with

being the base of natural logarithms, and

and

being arbitrary positive real constants.

The spiral has the property that the angle

between the tangent and radial line at the point

is constant. This property can be expressed in differential geometric terms as

The derivative of

is proportional to the parameter

. In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that

(

) the spiral becomes a circle of radius

. Conversely, in the limit that

approaches infinity (

) the spiral tends toward a straight half-line. The complement of

is called the pitch.

Spira mirabilis and Jacob Bernoulli

Properties

The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

Logarithmic spirals are self-similar in that the result of applying any similarity transformation to the spiral is congruent to the original untransformed spiral. Scaling by a factor

, where b is the parameter from the definition of the spiral, with the center of scaling at the origin, gives the same curve as the original; other scale factors give a curve that is rotated from the original position of the spiral. Logarithmic spirals are also congruent to their own involutes, evolutes, and the pedal curves based on their centers.

Starting at a point

and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as

goes toward

is finite. This property was first realized by Evangelista Torricelli even before calculus had been invented.^[4]

The total distance covered is

, where

is the straight-line distance from

to the origin.

The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0. (Up to adding integer multiples of

to the lines, the mapping of all lines to all logarithmic spirals is onto.) The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.

The function

, where the constant

is a complex number with a non-zero imaginary unit, maps the real line to a logarithmic spiral in the complex plane.

The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.

Logarithmic spirals in nature

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:

See also

References

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Definition

Spira mirabilis and Jacob Bernoulli

Properties

Logarithmic spirals in nature

See also

References

External links