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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