# Orthogonal Trajectories

### Definition and Examples

Let a family of curves be given by the equation

$g\left( {x,y} \right) = C,$

where $$C$$ is a constant. For the given family of curves, we can draw the orthogonal trajectories, that is another family of curves $$f\left( {x,y} \right) = C$$ that cross the given curves at right angles.

For example, the orthogonal trajectory of the family of straight lines defined by the equation $$y = kx,$$ where $$k$$ is a parameter (the slope of the straight line), is any circle having center at the origin (Figure $$1$$):

${x^2} + {y^2} = {R^2},$

where $$R$$ is the radius of the circle.

Similarly, the orthogonal trajectories of the family of ellipses

${\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{c^2} – {a^2}}} = 1,\;\;}\kern0pt{\text{where}\;\;}\kern-0.3pt{0 \lt a \lt c,}$

are confocal hyperbolas satisfying the equation:

${\frac{{{x^2}}}{{{b^2}}} – \frac{{{y^2}}}{{{b^2} – {c^2}}} = 1,\;\;}\kern0pt{\text{where}\;\;}\kern-0.3pt{0 \lt c \lt b.}$

Both families of curves are sketched in Figure $$2.$$ Here $$a$$ and $$b$$ play the role of parameters describing the family of ellipses and hyperbolas, respectively.

### General Method of Finding Orthogonal Trajectories

The common approach for determining orthogonal trajectories is based on solving the partial differential equation:

$\nabla f\left( {x,y} \right) \cdot \nabla g\left( {x,y} \right) = 0,$

where the symbol $$\nabla$$ means the gradient of the function $$f\left( {x,y} \right)$$ or $$g\left( {x,y} \right)$$ and the dot means the dot product of the two gradient vectors.

Using the definition of gradient, one can write:

${\nabla f\left( {x,y} \right) = \mathbf{grad}\,f\left( {x,y} \right) }={ \left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right),}$

${\nabla g\left( {x,y} \right) = \mathbf{grad}\,g\left( {x,y} \right) }={ \left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right).}$

Hence, the partial differential equation is written in the form:

${\nabla f\left( {x,y} \right) \cdot \nabla g\left( {x,y} \right) = 0,\;\;}\Rightarrow {\left( {\frac{{\partial f}}{{\partial x}},\frac{{\partial f}}{{\partial y}}} \right) \cdot \left( {\frac{{\partial g}}{{\partial x}},\frac{{\partial g}}{{\partial y}}} \right) = 0,\;\;}\Rightarrow {\frac{{\partial f}}{{\partial x}}\frac{{\partial g}}{{\partial x}} + \frac{{\partial f}}{{\partial y}}\frac{{\partial g}}{{\partial y}} = 0.}$

Solving the last PDE, we can determine the equation of the orthogonal trajectories $$f\left( {x,y} \right) = C.$$

### A Practical Algorithm for Constructing Orthogonal Trajectories

Below we describe an easier algorithm for finding orthogonal trajectories $$f\left( {x,y} \right) = C$$ of the given family of curves $$g\left( {x,y} \right) = C$$ using only ordinary differential equations. The algorithm includes the following steps:

1. Construct the differential equation $$G\left( {x,y,y’} \right) = 0$$ for the given family of curves $$g\left( {x,y} \right) = C.$$ See the web page Differential Equations of Plane Curves about how to do this.
2. Replace $$y’$$ with $$\left( { – \large\frac{1}{{y’}}\normalsize} \right)$$ in this differential equation. As a result, we obtain the differential equation of the orthogonal trajectories.
3. Solve the new differential equation to determine the algebraic equation of the family of orthogonal trajectories $$f\left( {x,y} \right) = C.$$

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Find the orthogonal trajectories of the family of straight lines $$y = Cx,$$ where $$C$$ is a parameter.

### Example 2

A family of hyperbolic curves is given by the equation $$y = {\large\frac{C}{x}\normalsize}.$$ Find the orthogonal trajectories for these curves.

### Example 3

Find the orthogonal trajectories of the family of curves given by the power function $$y = C{x^4}.$$

### Example 4

Determine the orthogonal trajectories of the family of sinusoids $$y = C\sin x.$$
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