Exercise
Determine the envelope of the family of plane curves:
Solution
The system of equations that interests us is
where
for which
We immediately see that (0,0) is a solution for every &lamba;, so that (0,0) is a base point for &Fphi;, i.e. all the curves of the family pass through the origin of the coordinates. Discarding this solution, we proceed:
which replaced in the first of the system
We thus obtained the following parametric representation of the envelope curve:
By eliminating the parameter λ, the Cartesian representation is obtained:
The envelope is drawn in fig. 1.
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