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Determining Envelope

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

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

the parameter (

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

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

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

the parameter?

With a fixed , 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 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

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

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

c) The third family of the semicubical parabolas also gives from (1) the -axis, which this time is simultaneously the envelope of the curves and the locus of the special points.

1	Representable Functor
1	Polygon
1	Curve
1	Incidence Geometry
1	Point
1	Symmetry
1	Euclidean Transformation
1	Frame
1	Simple Algebraic System
1	Expression
1	Positive
1	Parameter
1	Boolean Valued Function
1	Operations On Relations
1	Fix
1	Converse Theorem
1	Relation Theory
1	Equation
2	Origin
2	Congruence
3	Partial Derivative
3	Cone In Mathbb R
4	Locus
4	Parabola
9	Envelope
9	Cube Root
13	Clairauts Equation
17	Semicubical Parabola