#### Orthogonal trajectories

Let S: \(\vec{r}=\vec{r}(u,v) \) be a surface with family of curves

\( \phi(u,v)=c\) (i)

\( \psi(u,v)=c_1\) (ii)

Now, family (i) and (ii) are called orthogonal trajectories, if the curves of the both families are orthogonal at each of their intersection.

#### Examples

As we know, family of curves \( xy=c;c\neq 0 \) and \( x^2-y^2=c;c\neq 0\) of hyperbolas are orthogonal to each other, thus these families are orthogonal trajectories to each other.

Similarly, \(uv=c;c\neq 0 \) and \( u^2-v^2=c;c\neq 0\) are orthogonal to each other on the surface, so these families are orthogonal trajectories to each other.

Next, the family of circles \(x^2+y^2=c\) and that of lines \(y=mx\) are orthogonal trajectories to each other.

Similarly, \(u^2+v^2=c \) and \( v=mu\) are orthogonal to each other on the surface, so these families are orthogonal trajectories to each other.

#### Differential equation of orthogonal trajectories

Let S: \(\vec{r}=\vec{r}(u,v) \) be a surface. If

\( \phi(u,v)=c\) (i) be a family of curves with directions (du,dv)

\( \psi(u,v)=c_1\) (ii)be another family of curves with directions (-Q,P)

Then, the families (i) and (i) are orthogonal trajectories if (du,dv) and (-Q,P) are orthogonal.

Then, differential equation of orthogonal trajectories is

\(Ell'+F(lm'+l'm))+Gmm'=0\)

or
\(Edu(-Q)+F(Pdu-Qdv)+GPdv=0\)

or
\((FP-EQ)du+(GP-FQ)dv=0 \)

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