Orthogonal Trajectories
Definition and Examples
Let a family of curves be given by the equation
where
For example, the orthogonal trajectory of the family of straight lines defined by the equation
where
Similarly, the orthogonal trajectories of the family of ellipses
are confocal hyperbolas satisfying the equation:
Both families of curves are sketched in Figure
General Method of Finding Orthogonal Trajectories
The common approach for determining orthogonal trajectories is based on solving the partial differential equation:
where the symbol
Using the definition of gradient, one can write:
Hence, the partial differential equation is written in the form:
Solving the last PDE, we can determine the equation of the orthogonal trajectories
A Practical Algorithm for Constructing Orthogonal Trajectories
Below we describe an easier algorithm for finding orthogonal trajectories
- Construct the differential equation
for the given family of curves See the web page Differential Equations of Plane Curves about how to do this. - Replace
with in this differential equation. As a result, we obtain the differential equation of the orthogonal trajectories. - Solve the new differential equation to determine the algebraic equation of the family of orthogonal trajectories
Solved Problems
Click or tap a problem to see the solution.
Example 1
Find the orthogonal trajectories of the family of straight lines
Example 2
A family of hyperbolic curves is given by the equation
Example 1.
Find the orthogonal trajectories of the family of straight lines
Solution.
We apply the algorithm described on the previous page.
Eliminate the constant
We obtain the differential equation of the initial set of straight lines.
By replacing
Example 2.
A family of hyperbolic curves is given by the equation
Solution.
Now we eliminate the parameter
It follows from the first equation that
In the last equation we replaced