constant traces out a meridian of longitude, and together they generate a plane tangent to the sphere. The unit vector has a constant angle with the unit vector for any and , since their scalar product is A loxodrome is defined as a curve on the sphere that has a constant angle with all meridians of longitude, and therefore must be parallel to the unit vector . As a result, a differential length along the loxodrome will produce a differential displacement where and are the Gudermannian function and its inverse, and is the inverse hyperbolic sine. With this relationship between and , the radius vector becomes a parametric function of one variable, tracing out the loxodrome on the sphere: where is the isometric latitude. In the Rhumb line, as the latitude tends to the poles, , , the isometric latitude , and longitude increases without bound, circling the sphere ever so fast in a spiral towards the pole, while tending to a finite total arc length Δ given by Connection to the Mercator projection Let be the longitude of a point on the sphere, and its latitude. Then, if we define the map coordinates of the Mercator projection as a loxodrome with constant bearing from true north will be a straight line, since (using the expression in the previous section) with a slope 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 and . There are infinitely many solutions; the shortest one is that which covers the actual longitude difference, i.e. does not make extra revolutions, and does not go "the wrong way around". The distance between two points , 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): where is one of the earth average radii. Application Its use in navigation is directly linked to the style, or projection of certain navigational maps. A rhumb line appears as a straight line on a Mercator projection map. The name is derived from Old French or Spanish respectively: "rumb" or "rumbo", a line on the chart which intersects all meridians at the same angle. On a plane surface this would be the shortest distance between two points. Over the Earth's