How do you find the envelope of a family of curves?

Dicembre 21st, 2020 | by Marcello Colozzo |

How do you find the envelope of a family of curves?
Fig. 1

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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