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 (x, y) = 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.

The orthogonal trajectory of the family of straight lines y=kx is any circle centered at the origin — Figure 1.

Similarly, the orthogonal trajectories of the family of ellipses

\[\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{c^2} - {a^2}}} = 1,\;\; \text{where}\;\; 0 \lt a \lt c,\]

are confocal hyperbolas satisfying the equation:

\[\frac{{{x^2}}}{{{b^2}}} - \frac{{{y^2}}}{{{b^2} - {c^2}}} = 1,\;\; \text{where}\;\; 0 \lt c \lt b.\]

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.

The orthogonal trajectories of the family of ellipses are confocal hyperbolas — Figure 2.

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:

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.
Replace \(y'\) with \(\left( { - \frac{1}{{y'}}} \right)\) 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 \(f\left( {x,y} \right) = C.\)

Solved Problems

Example 1.

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

Solution.

We apply the algorithm described on the previous page.

\(1)\) First, we construct the differential equation for the family of straight lines \(y = Cx.\) By differentiating the last equation with respect to \(x,\) we get:

\[y' = C = \text{const}.\]

Eliminate the constant \(C\) from the system of equations:

\[\left\{ \begin{array}{l} y = Cx\\ y' = C \end{array} \right.,\;\; \Rightarrow y' = \frac{y}{x}.\]

We obtain the differential equation of the initial set of straight lines.

\(2)\) Replace \(y'\) with \(\left( { - \frac{1}{{y'}}} \right).\) This gives the differential equation of the orthogonal trajectories:

\[ - \frac{1}{{y'}} = \frac{y}{x},\;\; \Rightarrow y' = - \frac{x}{y}.\]

\(3)\) Now we solve the last differential equation to find the algebraic equation of the family of orthogonal trajectories:

\[y' = - \frac{x}{y},\;\; \Rightarrow \frac{{dy}}{{dx}} = - \frac{x}{y},\;\; \Rightarrow ydy = - xdx,\;\; \Rightarrow \int {ydy} = - \int {xdx} ,\;\; \Rightarrow \frac{{{y^2}}}{2} = - \frac{{{x^2}}}{2} + C,\;\; \Rightarrow \frac{{{x^2}}}{2} + \frac{{{y^2}}}{2} = C,\;\; \Rightarrow {x^2} + {y^2} = 2C.\]

By replacing \(2C\) with \({R^2}\) we see that the orthogonal trajectories for the family of straight lines are concentric circles (Figure \(1\)):

\[{x^2} + {y^2} = {R^2}.\]

Example 2.

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

Solution.

\(1)\) Determine the differential equation for the given family of hyperbolas. Differentiating the equation with respect to \(x\) gives:

\[y' = - \frac{C}{{{x^2}}}.\]

Now we eliminate the parameter \(C\) from the system of two equations:

\[\left\{ \begin{array}{l} y = \frac{C}{x}\\ y' = - \frac{C}{{{x^2}}} \end{array} \right..\]

It follows from the first equation that \(C = xy.\) Substituting into the second equation yields:

\[y' = - \frac{{xy}}{{{x^2}}} = - \frac{y}{x}.\]

\(2)\) Replace \(y'\) with \(\left( { - \frac{1}{{y'}} } \right):\)

\[ - \frac{1}{{y'}} = - \frac{y}{x},\;\; \Rightarrow y' = \frac{x}{y}.\]

\(3)\) Now we integrate the differential equation of the orthogonal trajectories:

\[y' = \frac{x}{y},\;\; \Rightarrow \frac{{dy}}{{dx}} = \frac{x}{y},\;\; \Rightarrow ydy = xdx,\;\; \Rightarrow \int {ydy} = \int {xdx} ,\;\; \Rightarrow \frac{{{y^2}}}{2} = \frac{{{x^2}}}{2} + C,\;\; \Rightarrow {x^2} - {y^2} = C.\]

In the last equation we replaced \(2C\) with just a constant \(C.\) Thus, we have obtained the equation of the family of orthogonal trajectories. As it can be seen, these orthogonal trajectories are also hyperbolas. Both the families of hyperbolas are shown schematically in Figure \(3.\)

Orthogonal trajectories for a family of hyperbolic curves — Figure 3.

See more problems on Page 2.