What are orthogonal trajectories of a curve?

orthogonal trajectory, family of curves that intersect another family of curves at right angles (orthogonal; see figure). Two lines are orthogonal, or perpendicular, if their slopes are negative reciprocals of each other.

What are trajectories in differential equations?

A sketch of a particular solution in the phase plane is called the trajectory of the solution. This will give a vector that represents \(\vec x’\)at that particular solution. As with the single differential equation case this vector will be tangent to the trajectory at that point.

What do you mean by orthogonal curve?

Two (or more) intersecting curves that are perpendicular at their intersection are said to be orthogonal.

What are the orthogonal trajectories of the family of curves?

The orthogonal trajectories are the curves that are perpendicular to the family everywhere. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in original family.

How do you know if two curves are orthogonal?

Two curves are said to be orthogonal if their tangent lines are perpendicular at every point of intersection. Two families of curves are said to be orthogonal if every curve in one family is orthogonal to every curve in the other family.

Where are orthogonal trajectories used?

Orthogonal trajectories are used in mathematics for example as curved coordinate systems (i.e. elliptic coordinates) or appear in physics as electric fields and their equipotential curves. If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory.

What is the orthogonal trajectories of the given family of straight lines?

How do you find orthogonal trajectories of the family of curves?

Procedure to find orthogonal trajectory

1. Let f(x,y,c)=0 be the equation of the given family of curves, where c is an arbitrary parameter.
2. Differentiate f=0; w.r.t. ‘x’ and eliminate c,ie, form a differential equation.
3. Substitute −dydx for dxdy in the above differential equation.

What is an example of an orthogonal trajectory?

For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram). Suitable methods for the determination of orthogonal trajectories are provided by solving differential equations.

What is an isogonal trajectory in physics?

Orthogonal trajectories are used in mathematics for example as curved coordinate systems (i.e. elliptic coordinates) or appear in physics as electric fields and their equipotential curves. If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory.

What is an orthogonal curve in math?

In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves orthogonally. For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram).

What are orthogonal curves in electrostatics?

It consists of two mutually orthogonal curves, flow lines and equipotential curves. In electrostatics, equipotential and electric field lines are two curves that are orthogonal to each other. Consider two families of curves, circles and lines as shown in the figure.