loxodromic

Finally, if the square is greater than 4, the transformation is loxodromic.

A transformation is loxodromic if and only if .

The transform is said to be loxodromic if is not in [0,4].

If both ρ and α are nonzero, then the transformation is said to be "loxodromic".

But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes.

Note that in the elliptical and loxodromic images, the α value is 1/10 .

(for any ) generates a one-parameter subalgebra of loxodromic transformations.

(The loxodromic transformations are not present because we are working with real numbers.)

Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities.

Schottky groups are finitely generated free groups such that all non-trivial elements are loxodromic.

Rhumb lines which cut meridians at oblique angles are loxodromic curves which spiral towards the poles.

In Mercator's Projection (q.v.) the Loxodromic lines are evidently straight."

And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms.

Sailings: Rhumbs, Loxodromic, Orthodromic, Meridional parts...

Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real conics.

If we take the one-parameter group generated by any loxodromic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the 'same' two points.

Non-identity Möbius transformations are commonly classified into four types, parabolic, elliptic, hyperbolic and loxodromic, with the hyperbolic ones being a subclass of the loxodromic ones.

It follows that by conjugating with an arbitrary element of SL(2,C), we can obtain from the following examples arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively.

In geometry, Coxeter's loxodromic sequence of tangent circles is an infinite sequence of circles arranged such that any four consecutive circles in the sequence are pairwise mutually tangent.

Historically, navigation by loxodrome or rhumb line refers to a path of constant bearing; the resulting path is a logarithmic spiral, similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes.

Use of this projection also has the advantage that straight lines drawn on the chart represent lines of constant bearing although in reality such lines are not straight but segments of a three dimensional "loxodromic" spiral known as a rhumb line.

Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), we will show how to determine the effect of our example of a parabolic Lorentz transformation on Minkowski spacetime, leaving the other examples as exercises for the reader.