Orthogonal Trajectories
Consider the two families of curves, Feb. 16, 2011
and The first family, , contains all lines that pass through the origin.
The second family, , contains all circles centered at the origin.
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Orthogonal Trajectories
Feb. 16, 2011
Placing these two families together we get:
Two families of curves F1 and F2 are said to be orthogonal if whenever any curve from F1 intersects a curve in F2, the two curves are orthogonal at the point of intersection.
Given a family of curves F, is it possible to find a family
of curves that is orthogonal to F?
Steps to find an orthogonal family:
1. For a given family of curves F, we first find a DE for which the family is a general solution, say
2. For any curve in F, the slope of a tangent line to this curve at a point (x, y) is given by f (x, y). The line perpendicular to this line will have slope: 3. Next we need to solve the DE,
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Orthogonal Trajectories
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Orthogonal Trajectories
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Example: Let’s start with and derive its orthogonal trajectory (the other family).
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Orthogonal Trajectories
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