5.10 ENVELOPES
his made arbitrarily small, so thatP→P 1 , the three defining equations reduce
to two, which define the envelope pointP 1 :
f(x, y, α 1 )=0 and∂f(x, y, α 1 )
∂α=0. (5.42)In (5.42) both the function and the gradient are evaluated atα=α 1. The equation
of the envelopeg(x, y) = 0 is found by eliminatingα 1 between the two equations.
As a simple example we will now solve the problem which when posed mathe-matically reads ‘calculate the envelope appropriate to the family of straight lines
in thexy-plane whose points of intersection with the coordinate axes are a fixed
distance apart’. In more ordinary language, the problem is about a ladder leaning
against a wall.
A ladder of lengthLstands on level ground and can be leaned at any angle against a
vertical wall. Find the equation of the curve bounding the vertical area below the ladder.We take the ground and the wall as thex-andy-axes respectively. If the foot of the ladder
isafrom the foot of the wall and the top isbabove the ground then the straight-line
equation of the ladder is
x
a+
y
b=1,
whereaandbare connected bya^2 +b^2 =L^2. Expressed in standard form with only one
independent parameter,a, the equation becomes
f(x, y, a)=x
a+
y
(L^2 −a^2 )^1 /^2−1=0. (5.43)
Now, differentiating (5.43) with respect toaand setting the derivative∂f/∂aequal to
zero gives
−x
a^2+
ay
(L^2 −a^2 )^3 /^2=0;
from which it follows that
a=Lx^1 /^3
(x^2 /^3 +y^2 /^3 )^1 /^2and (L^2 −a^2 )^1 /^2 =Ly^1 /^3
(x^2 /^3 +y^2 /^3 )^1 /^2.
Eliminatingaby substituting these values into (5.43) gives, for the equation of the
envelope of all possible positions on the ladder,
x^2 /^3 +y^2 /^3 =L^2 /^3.This is the equation of an astroid (mentioned in exercise 2.19), and, together with the wall
and the ground, marks the boundary of the vertical area below the ladder.
Other examples, drawn from both geometry and and the physical sciences, areconsidered in the exercises at the end of this chapter. The shell trajectory problem
discussed earlier in this section is solved there, but in the guise of a question
about the water bell of an ornamental fountain.