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 ${ℝ}^3$.  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