What Are Orthogonal Trajectories?

What Are Orthogonal Trajectories?

Orthogonal trajectories are curves in a given geometric plane that intersect another family of curves perpendicularly. A classic example of orthogonal trajectories involves the Cartesian coordinate system's surfaces defined by the coordinates x, y and z, where each surface meets at right angles with the others.

The term "orthogonal" is an adjective that defines right angles or perpendicular orientations, and so orthogonal trajectories are merely curves oriented at right angles with another set of curves on another plane. The simplest type of orthogonal trajectories involves the array of straight lines on the x-y coordinate plane that follow the formula y = kx, where k is a numerical constant. For this range of lines, the orthogonal trajectory is any circle that has the origin as its center and follows the formula x^2 + y^2 = r^2, where r is the circle's radius.

The orthogonal trajectory of a family of curves may be determined through partial differential equations involving the gradient of two functions and the dot product of two gradient vectors.

Orthogonal trajectories are useful in the topic of force field lines, particularly in ensuring that for the family of curves, each intersection point should yield a perpendicular orientation between the given vector field curves and the orthogonal trajectories. In force fields, the trajectories are often called equipotential curves.

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