Theorem. Let be the parameter of the family of curves and suppose that the function has the partial derivatives , and in a certain domain of . If the family has an envelope in this domain, then the coordinates of an arbitrary point of and the value of the parameter determining the family member touching in satisfy the pair of equations
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
of lines, with the parameter ( is a positive constant). Now the pair (1) for the envelope may be written
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 , and so we have the parametric presentation
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
of parabolas, with the parameter?
With a fixed , the equation presents a parabola which is congruent (http://planetmath.org/Congruence) to the parabola and the apex of which is . When is changed, the parabola is submitted to a translation such that the apex draws the parabola
The pair (1) for the envelope of the parabolas (4) is simply
which allows immediately eliminate , giving
Thus the envelope of the parabolas is a “narrower” parabola. One infers easily, that a parabola (4) touches the envelope (5) in the point 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
given in this parametric form and satisfying the condition (1), is not necessarily the envelope of the family 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
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 , i.e. the -axis, which also is the envelope.
b) In the case of the family (or ) 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.
|Date of creation||2013-03-22 17:10:48|
|Last modified on||2013-03-22 17:10:48|
|Last modified by||pahio (2872)|