The Asteroid is the envelope of a family of lines...

Dicembre 21st, 2020 | by Marcello Colozzo |

asteroid,curve,envelope
Fig. 1

Exercise

Determine the envelope of the lines on which the coordinate axes detach a segment of constant length a.


Solution
We write the equation of the generic line of the assigned family. For this purpose, let's look at the figure:


Here we denote by φ the angle that the aforementioned straight line forms with the x axis. His equation is therefore:


The family equation follows:


where


By deriving with respect to φ


We have to solve the system of equations


From the second


Substituting in the first


Then the following parametric representation of the envelope:

It is a known curve: the asteroid (fig. 1).

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