# determining envelope

Theorem. Let $c$ be the parameter of the family $F(x,y,c)=0$ of curves and suppose that the function $F$ has the partial derivatives^{} ${F}_{x}^{\prime}$, ${F}_{y}^{\prime}$ and ${F}_{c}^{\prime}$ in a certain domain of ${\mathbb{R}}^{3}$. If the family has an envelope $E$ in this domain, then the coordinates $x,y$ of an arbitrary point of $E$ and the value $c$ of the parameter determining the family member touching $E$ in $(x,y)$ satisfy the pair of equations

$\{\begin{array}{cc}F(x,y,c)=0,\hfill & \\ {F}_{c}^{\prime}(x,y,c)=0.\hfill & \end{array}$ | (1) |

I.e., one may in principle eliminate $c$ from such a pair of equations and obtain the equation of an envelope.

Example 1. Let us determine the envelope of the the family

$y=Cx+{\displaystyle \frac{Ca}{\sqrt{1+{C}^{2}}}}$ | (2) |

of lines, with $C$ the parameter ($a$ is a positive constant). Now the pair (1) for the envelope may be written

$F(x,y,C):=Cx-y+{\displaystyle \frac{Ca}{\sqrt{1+{C}^{2}}}}=0,{F}_{C}^{\prime}(x,y,C)\equiv x+{\displaystyle \frac{a}{(1+{C}^{2})\sqrt{1+{C}^{2}}}}=0.$ | (3) |

It’s easier to first eliminate $x$ by taking its expression from the second equation and putting it to the first equation. It follows the expression $y=\frac{{C}^{3}a}{(1+{C}^{2})\sqrt{1+{C}^{2}}}$, and so we have the parametric presentation

$$x=-\frac{a}{(1+{C}^{2})\sqrt{1+{C}^{2}}},y=\frac{{C}^{3}a}{(1+{C}^{2})\sqrt{1+{C}^{2}}}$$ |

of the envelope. The parameter $C$ can be eliminated from these equations by squaring both equations, then taking cube roots and adding both equations. The result is symmetric^{} equation

$$\sqrt[3]{{x}^{2}}+\sqrt[3]{{y}^{2}}=\sqrt[3]{{a}^{2}},$$ |

which represents an astroid. But the parametric form tells, that the envelope consists only of the left half of the astroid.

Example 2. What is the envelope of the family

$y-{\displaystyle \frac{1}{2}}{a}^{2}=-{\displaystyle \frac{1}{4}}{(x-a)}^{2},$ | (4) |

of parabolas^{}, with $a$ the parameter?

With a fixed $a$, the equation presents a parabola which is congruent (http://planetmath.org/Congruence) to the parabola $y=-\frac{1}{4}{x}^{2}$ and the apex of which is $(a,\frac{1}{2}{a}^{2})$. When $a$ is changed, the parabola is submitted to a translation^{} such that the apex draws the parabola $y=\frac{1}{2}{x}^{2}.$

The pair (1) for the envelope of the parabolas (4) is simply

$$y-\frac{1}{2}{a}^{2}+\frac{1}{4}{(x-a)}^{2}=0,x=-a,$$ |

which allows immediately eliminate $a$, giving

$y=-{\displaystyle \frac{1}{2}}{x}^{2}.$ | (5) |

Thus the envelope of the parabolas is a “narrower” parabola. One infers easily, that a parabola (4) touches the envelope (5) in the point $(-a,-\frac{1}{2}{a}^{2})$ which is symmetric with the apex of (4) with respect to the origin.

The converse of the above theorem is not true. In fact, we have the

Proposition. The curve

$x=x(c),y=y(c),$ | (6) |

given in this parametric form and satisfying the condition (1), is not necessarily the envelope of the family $F(x,y,c)=0$ of curves, but may as well be the locus of the special points of these curves, namely in the case that the functions (6) satisfy except (1) also both of the equations

$${F}_{x}^{\prime}(x,y,c)=0,{F}_{y}^{\prime}(x,y,c)=0.$$ |

Examples. Let’s look some simple cases illustrating the above proposition.

a) The family ${(x-c)}^{2}-y=0$ consists of congruent parabolas having their vertices on the $x$-axis. Differentiating the equation with respect to $c$ gives $x-c=0$, and thus the corresponding pair (1) yields the result $x=c,y=0$, i.e. the $x$-axis, which also is the envelope.

b) In the case of the family ${(x-c)}^{2}-{y}^{3}=0$ (or $y=\sqrt[3]{{(x-c)}^{2}}$) the pair (1) defines again the $x$-axis, which now isn’t the envelope but the locus of the special points (sharp vertices) of the curves.

c) The third family ${(x-c)}^{3}-{y}^{2}=0$ of the semicubical parabolas also gives from (1) the $x$-axis, which this time is simultaneously the envelope of the curves and the locus of the special points.

Title | determining envelope |
---|---|

Canonical name | DeterminingEnvelope |

Date of creation | 2013-03-22 17:10:48 |

Last modified on | 2013-03-22 17:10:48 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 53A04 |

Classification | msc 51N20 |

Classification | msc 26B05 |

Classification | msc 26A24 |

Related topic | SingularSolution |