Isocline [isoclinic line]

An isocline in a geophysical map is the line joining all points having the same value of magnetic inclination. More generally, the term indicates, in a cartographic representation, the line that joins all points in which a quantity assumes the same value. In this sense the term takes on the meaning of contour line in a geographical or topographical map, isobar in a meteorological map. In mathematics, in the prey-predator model of Lotka-Volterra, are called isocline of the prey and isocline of the predator the curves along which cancel, respectively, the derivative with respect to time of the variable prey and the derivative with respect to time of the variable biomass of the predator.

An isoclinic line connects points of equal magnetic dip, and an aclinic line is the isoclinic line of magnetic dip zero. Line connecting all points on the Earth’s surface characterized by the same value of magnetic inclination. The isocline of zero inclination is called the magnetic equator.

In construction science, each of the curved lines intersecting, under a constant angle, the isostatic lines relative to a certain stress regime. Their knowledge, deducible from the photoelastic model, is important because, from them, the isostatic lines are geometrically constructible. In the mechanics of plane elastic systems, any curve is said to intersect under constant angle all isostatic lines relative to an assigned stress regime. The importance of such lines is related to the fact that their configuration, for any plane stress system, can be obtained by operating on photoelastic model (photoelasticity) and that from them the tracing of isostatic lines can be deduced.

In naval architecture, each of the hulls which, in the same float, present, with respect to the waterline plane, an equal angle of inclination.

Isocline curves are often considered in mathematical models that, if linear, result in isocline straight lines. An example of isocline lines is obtained by considering a differential equation of the type \(y\;’ = -ky/x\) with \(k \in \mathbb{R}^+\), which has as general solution a family of equilateral hyperbolas, corresponding in a homotety of center the origin O of the Cartesian reference: each of the lines passing through the origin meets each curve at points whose derivatives have the same value and are, therefore, isocline lines because they meet the curves at points where they have the same inclination.

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