They move all other points away from the South pole and toward the North pole ( or vice versa ), along a family of curves called "'loxodromes " '.
12.
A loxodrome is defined as a curve on the sphere that has a constant angle \ beta with all meridians of longitude, and therefore must be parallel to the unit vector \ hat \ beta.
13.
The angle that the loxodrome subtends relative to the lines of longitude ( i . e . its slope, the " tightness " of the spiral ) is the argument of " k ".
14.
He was the first to propose the idea of a loxodrome and was also the inventor of several measuring devices, including the nonius ( from which Vernier scale was derived ), named after his Latin surname.
15.
But theoretically a loxodrome can extend beyond the right edge of the map, where it then continues at the left edge with the same slope ( assuming that the map covers exactly 360 degrees of longitude ).
16.
Finding the loxodromes between two given points can be done graphically on a Mercator map, or by solving a nonlinear system of two equations in the two unknowns m = \ cot \ beta and \ lambda _ 0.
17.
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 M�bius transformation makes.
18.
It applied equally well to the windrose lines as it did to loxodromes because the term only applied locally and only meant whatever a sailor did in order to sail with constant bearing, with all the imprecision that that implies.
19.
O'Brien published several other papers and books . " The Wandlebury-Hatfield Heath Astronomical Complex " described his surveying and discovering what he calls the Wandlebury Enigma or Line A Loxodrome, a claim which has not gained much acceptance.
20.
The distance between two points S, measured along a loxodrome, is simply the absolute value of the secant of the bearing ( azimuth ) times the north-south distance ( except for circles of latitude for which the distance becomes infinite ):
