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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 $F'_x$ , $F'_y$ and $F'_c$ 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
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(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
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(2) |
of lines, with $C$ the parameter ($a$ is a positive constant). Now the pair (1) for the envelope may be written
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(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 $y = \frac{C^3a}{(1+C^2)\sqrt{1+C^2}}$ , and so we have the parametric presentation $$x= -\frac{a}{(1+C^2)\sqrt{1+C^2}},\quad y = \frac{C^3a}{(1+C^2)\sqrt{1+C^2}}$$ 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
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(4) |
of parabolas, with $a$ the parameter?
With a fixed $a$ , the equation presents a parabola which is congruent 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 $$y-\frac{1}{2}a^2+\frac{1}{4}(x-a)^2 = 0,\quad x = -a,$$ which allows immediately eliminate $a$ , giving
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(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
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(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 $$F'_x(x,\,y,\,c) = 0,\quad F'_y(x,\,y,\,c) = 0.$$
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.
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