determining envelope

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

′

,

F

y

′

and

F

c

′

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

{

F

⁢

(

x

,

y

,

c

)

=

0

,

F

c

′

⁢

(

x

,

y

,

c

)

=

0

.

(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

=

C

⁢

x

+

C

⁢

a

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

)

:=

C

⁢

x

-

y

+

C

⁢

a

1

+

C

2

=

0

,

F

C

′

⁢

(

x

,

y

,

C

)

≡

x

+

a

(

1

+

C

2

)

⁢

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

=

C

3

⁢

a

(

1

+

C

2

)

⁢

1

+

C

2

,  and so we have the parametric presentation

x

=

-

a

(

1

+

C

2

)

⁢

1

+

C

2

,

y

=

C

3

⁢

a

(

1

+

C

2

)

⁢

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

x

2

3

+

y

2

3

=

a

2

3

,

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

-

1

2

⁢

a

2

=

-

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

=

-

1

4

⁢

x

2

and the apex of which is

(

a

,

1

2

⁢

a

2

)

.  When

a

is changed, the parabola is submitted to a

translation

such that the apex draws the parabola

y

=

1

2

⁢

x

2

.

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

y

-

1

2

⁢

a

2

+

1

4

⁢

(

x

-

a

)

2

=

0

,

x

=

-

a

,

which allows immediately eliminate

a

, giving

y

=

-

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

,

-

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

′

⁢

(

x

,

y

,

c

)

=

0

,

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

=

(

x

-

c

)

2

3

) 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

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