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[parent] orthogonal curves (Result)

If a family of plane curves (with one free parameter) satisfies the differential equation

$\displaystyle F(x,\,y,\,y') = 0,$
where $ y' = \frac{dy}{dx}$, then the family of curves intersecting orthogonally all the first curves satisfies the differential equation
$\displaystyle F\left(x,\,y,\,-\frac{1}{y'}\right) = 0.$

Example. Let's consider the family of rectangular hyperbolas

$\displaystyle x^2-y^2 = c$
with the parameter $ c$ taking any real value. Derivating with respect to $ x$ gives the differential equation of this family,
$\displaystyle x-yy' = 0,$
and by replacing here $ y'$ with $ -\frac{1}{y'}$ we obtain the differential equation
$\displaystyle x+\frac{y}{y'} = 0$
of the orthogonal curves. Integrating its form
$\displaystyle \frac{dy}{y} = -\frac{dx}{x}$
gives the solution
$\displaystyle xy = C,$
which represents another family of rectangular hyperbolas.

In the picture below (by drini), there are four hyperbolas of the first family (blue) given by the values $ c = -1,\,-2,\,-4,\,-8$ and four hyperbolas of the orthogonal family (red) given by the values $ C = 1,\,2,\,4,\,8$.


\begin{pspicture*}(-5.4,-5.4)(5.2,5.4) \psaxes[labels=none,ticks=none](0,0)(-5,-... ... x div } \psplot[linecolor=red,linewidth=1pt]{0.1}{5}{4 x div } \end{pspicture*}



"orthogonal curves" is owned by pahio.
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See Also: condition of orthogonality, harmonic conjugate function, convex angle, isocline, tilt curve, hyperbolas orthogonal to ellipses

Keywords:  family of curves

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orthogonal circles (Topic) by pahio
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Cross-references: orthogonal, hyperbolas, solution, real, rectangular hyperbolas, curves, differential equation, parameter, plane curves
There are 7 references to this entry.

This is version 11 of orthogonal curves, born on 2004-11-20, modified 2006-04-26.
Object id is 6504, canonical name is OrthogonalCurves.
Accessed 3110 times total.

Classification:
AMS MSC: 34C05 (Ordinary differential equations :: Qualitative theory :: Location of integral curves, singular points, limit cycles)
 34C99 (Ordinary differential equations :: Qualitative theory :: Miscellaneous)
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