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[parent] determining envelope (Topic)

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

\begin{align*}\begin{cases}F(x,\,y,\,c) = 0,\\ F'_c(x,\,y,\,c) = 0. \end{cases}\end{align*} (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

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

of lines, with $ C$ the parameter ($ a$ is a positive constant). Now the pair (1) for the envelope may be written
$\displaystyle F(x,\,y,\,C)\, := \,Cx-y+\frac{Ca}{\sqrt{1+C^2}} = 0,\quad F'_C(x,\,y,\,C) \equiv x+\frac{a}{(1+C^2)\sqrt{1+C^2}} = 0.$ (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^3a}{(1+C^2)\sqrt{1+C^2}}$, and so we have the parametric presentation
$\displaystyle x= -\frac{a}{(1+C^2)\sqrt{1+C^2}},\quad y = \frac{C^3a}{(1+C^2)\sqrt{1+C^2}}$
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
$\displaystyle \sqrt[3]{x^2}+\sqrt[3]{y^2} = \sqrt[3]{a^2},$
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

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

of parabolas, with $ a$ the parameter?

With a fixed $ a$, the equation presents a parabola which is congruent 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

$\displaystyle y-\frac{1}{2}a^2+\frac{1}{4}(x-a)^2 = 0,\quad x = -a,$
which allows immediately eliminate $ a$, giving
$\displaystyle y = -\frac{1}{2}x^2.$ (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

$\displaystyle x = x(c),\quad y = y(c),$ (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
$\displaystyle F'_x(x,\,y,\,c) = 0,\quad F'_y(x,\,y,\,c) = 0.$

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 semicubic 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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See Also: singular solution

Keywords:  envelope

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Cross-references: vertices, congruent, simple, locus, proposition, converse, origin, translation, apex, fixed, parabolas, parametric form, astroid, represents, symmetric, cube roots, parametric presentation, expression, positive, lines, equations, point, coordinates, envelope, domain, partial derivatives, function, curves, parameter
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This is version 6 of determining envelope, born on 2007-06-01, modified 2007-10-02.
Object id is 9494, canonical name is DeterminingEnvelope.
Accessed 744 times total.

Classification:
AMS MSC: 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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