Logarithmic spiral

A logarithmic spiral is changed, the distance from this center to the same factor, a spiral, in which with each turn about its center ( center pole). Each straight line through the pole intersects the logarithmic spiral is always at the same angle. Because of this property is also called an equiangular spiral. Due to this property, the logarithmic spiral is clearly characterized.

Occupation is that 1638 René Descartes discussed in letters to Marin Mersenne first time the logarithmic spiral.

Mathematical representation

Just can specify any logarithmic spiral in polar coordinates. For describing the equation

A function, and by means of polar coordinates interpreting a logarithmic spiral in the Euclidean plane. The parameter is referred to as the pitch of the helix. Can be expressed by, and then the pitch angle is called.

In Cartesian coordinates, we have:

The name derives from the representation in which the angle is expressed as a function of radius:

In the complex plane, each logarithmic spiral can even display even easier.

Because is the polar form of, shall apply

So this is done with the two ( analytical ) and bijections, since by assumption.

Another simple representation of the differential geometry of plane curves is:

Properties

The logarithmic spiral has a number of unique properties, which is why it has been labeled by one of her biggest fans, Jakob Bernoulli, also known as spiral mirabilis ( " miraculous spiral" ):

• The sign of is the ideological direction of rotation of the spiral resist in the plane.
• All passing through the pole lines intersect the curve - ie their tangents - under the same tangent angle and therefore ( see figure). You can even call this as a property and as logarithmic spirals define (see their representation in the form of a differential equation ).
• The spiral encircles the origin infinitely often without ever reaching it ( asymptotic point).
• Although the curve the pole, they "infinity " is often encircled reached for any finite angle value, the arc length of each curve point to the pole is finite and is.
• With each turn of the radius increases by a constant factor:

• The logarithmic spiral is - in generalization of the above derivation - self-similar ( invariant ) relative to a central extension by a factor with simultaneous rotation through the angle.

• The curve is its own evolute.
• The curve is its own focal line ( caustic ).
• The curve is their own Fußpunktkurve.
• An inversion of the curve ( ) results in rotation and mirroring of the curve in the Y axis ( only for the mirror); from a left-handed logarithmic spiral is a right-handed and vice versa.
• All spirals of the same pitch are similar.
• For the spiral approaches more and more to a circle with a radius that satisfies the curve equation for ( cutting angle 90 ° =). Therefore, one can also allow the definition of the spiral and consider the circle as a special case of logarithmic spiral, which in the spherical geometry is particularly significant.

Special cases and approximations

The Golden Spiral is a special case of the logarithmic spiral. This spiral can be by means of recursive division of a Golden Rectangle construct each in a square and another, smaller golden rectangle ( see figure below). With her thus applies to the value of the golden ratio.

Each logarithmic spiral can be approximated by a polygon. For the construction of triangles with an equal pitch angle, and each of the shorter side are lined up as long as the longer side of the last triangle. An extension of this idea also applies to certain irregular polygons that can be put together. This design principle is widely distributed in nature, and generally provides more consistent spirals.

Irregular polygon spiral

Formulas

See also: formulary geometry

Logarithmic spiral and rhumb line

Starting from a logarithmic spiral in the plane with the coordinate origin as their asymptotic point, a rhumb line to be constructed on a spherical surface. This curve is projected onto a spherical surface by a sphere with arbitrary radius is set to the origin. This contact point denote the south pole of the sphere. From the points of the logarithmic spiral in the plane of rays considered by this sphere through the north pole of the sphere. These rays define then when they were first cut of the sphere there a new spherical curve. Straight lines that go into the plane through the origin, are by this figure to meridians of longitude on the sphere and the plane logarithmic spiral describes on the sphere surface, a rhumb line. Conversely, a right (north pole and south pole are the asymptotic points of the rhumb line ) projection of a rhumb lines of the sphere in the plane there is a logarithmic spiral. This type of conformal projection from sphere to plane is called stereographic projection.

Occurrence

In the living world, there are numerous examples of logarithmic spirals with different slopes, as created for example by growth snail shells or the arrangement of nuclei in the flower of a sunflower.

A flying insect is based on a night flight at the level of the ( remote ) moon by keeping the angle constant to him. By ( selective near) street lamp the trajectory is, however, regularly corrected so that it becomes a logarithmic spiral in the center of the street lamp is located.

In addition, approximately logarithmic spiral structures found in dynamic multi-body systems and fluid dynamic systems ( vortex formation at sufficiently large velocity gradient ) as well as in the art ( eg background turning).

Section of a nautilus shell

Whirlpool Galaxy, a typical spiral galaxy

Low-pressure systems over Iceland in September Photographed from about 700 km altitude in 2003

According to legend, it was a desire of the mathematician Jacob Bernoulli, who was much occupied with the logarithmic spiral that his beloved logarithmic spiral, with the inscription eadem mutata resurgo ( " Transforms I return to be the same again ") should be inscribed on his grave stone. The competent stonemason carved after the death of Bernoulli's indeed a spiral on his grave, but it was (presumably out of ignorance or in order to save work ) applies to an Archimedean spiral, for none of the above properties. Bernoulli's grave stone can be seen in the cloister of the cathedral to Basel today.