This is a small collection of images of orthogonal families of two-dimensional curves.
Two curves are said to be orthogonal if their tangent lines are perpendicular at every point of intersection.
Two families of curves are said to be orthogonal if every curve in one family is orthogonal to every curve in the other family.
The classic example of this is the family of circles centered at the origin, together with the family of lines through the origin.
Every member of the family of circles centered at the origin has equation
![\[
x^2+y^2=k
\]](ortho01_1.gif)
for some k.
Consider any point (x,y) in the plane. It lies on exactly one circle centered at the origin, and exactly one line through the origin. The slope of the tangent line to the circle at (x,y) is
![\[
\frac{dy}{dx} = - \frac{x}{y}
\]](ortho01_2.gif)
while the slope of the line through the origin is
![\[
\frac{y}{x}.
\]](ortho01_3.gif)
Since
![\[
\left( -\frac{x}{y} \right) \left( \frac{y}{x} \right) = -1
\]](ortho01_4.gif)
the tangent line to the circle is perpendicular (or, orthogonal) to the line (which, of course, is its own tangent line).
Given a pair of families, it can be a straight-forward task to determine whether the families are orthogonal or not. This is at the level of first-quarter calculus.
To find a family of curves that are orthogonal to a given family, a differential equation must be solved. Many are at the level of late second-quarter calculus (i.e., the equations can be separable, and thus solvable by integration).
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