ekline

Ekline is a geometric construct used in the study of scalar fields and directional derivatives. It is defined as the locus of points where the magnitude of the directional derivative of a scalar field along a fixed unit vector equals a prescribed constant.

Formally, let phi: R^2 -> R be a scalar field, u be a fixed unit vector, and k ≥ 0. The ekline Ek is the set { (x,y) in R^2 | |∇phi(x,y) · u| = k }. In three dimensions, the concept generalizes by using the appropriate vector and projection. Eklines are thus determined by the interaction of the gradient field with a chosen direction.

Eklines need not be straight; they are curves whose shape depends on the gradient of phi and the selected direction u. When k = 0, an ekline consists of points where the gradient is orthogonal to u. In some common fields, such as phi(x,y) = x^2 + y^2, eklines can simplify to lines.

Example: take phi(x,y) = x^2 + y^2 and u = (1,0). Then ∇phi = (2x, 2y) and ∇phi·u = 2x, so Ek is the pair of vertical lines x = ±k/2. This illustrates how eklines can reduce to simple lines in certain cases and demonstrates the basic method of construction by thresholding the directional projection of the gradient.

Applications include illustrating regions of uniform directional change in image processing and identifying zones of constant shear in fluid dynamics. Eklines relate to level sets and gradient lines and are typically computed by evaluating the projection ∇phi·u and applying a threshold.

See also: contour line, isocline, gradient line, directional derivative.