Enveloping Circular Arcs

Jeremy Dunn asked for an equation defining the envelope of the family
of circles centered on the y axis with an arc of length 1 above
the x axis.  To begin the construction we start with a circle of
unit circumference tangent to the x axis, and then we increase the
radius of the circle and lower the center point so that the length
of the perimeter above the x axis remains constant, as shown below:

  

In the left hand figure each circle entirely encloses the preceeding 
circles, and the central height has been monotonically increasing.  
In the right hand figure, as we continue to increase the radius of 
the circle and lower its central point, the top points begin to fall 
but the sides widen out to extend outside the previous circles.

Continuing to increase the radius and lower the center, we have

         

   

In the limit we have a complete envelope that contains this entire
family of circles

    

Let's see if we can find an explicit representation for this bell-
shaped curve.  By definition, the radius r of each circle is related 
to the vertical height y0 of the circle's center according to

                y0 = -r cos(1/(2r))

Thus, for any specified radius r, the points of the circle are given 
by
              x = r cos(q)    
              y = r sin(q) - r cos(1/(2r))

where q is the angle between the positive x axis and the line from 
the center of the circle to the point x,y.  Now, for any particular 
value of x, we want the maximum possible value of y.  Substituting 
for sin(q) = sqrt[1 - (x/r)^2] into the equation for y gives

             y = sqrt[r^2 - x^2] - r cos(1/(2r))

Setting the derivative of this to zero gives the condition

   2r^2  =  sqrt[r^2 - x^2] { sin(1/2r) - 2cos(1/2r) }       (1)

This says that for any specified value of x, the value of r for 
the limiting circle must satisfy this equation.  Solving for x 
and substituting into the expression for y, we arrive at an exact 
parametric expression for the curve representing the minimum 
envelope of the family of circles.  For ease of typing, I'll express 
this in terms of the parameter u = 1/(2r)

                 ____________________________
         +  1   /     /        1          \2
  x(u) = - --  / 1 - (  -----------------  )
           2u /       \ u sin(u) + cos(u) /

              _                            _
           1 |         1                    |
  y(u) =  -- |  -----------------  - cos(u) |
          2u |_ u sin(u) + cos(u)          _|


The envelope curve is governed by the circles with radii ranging 
from 0.214488958... to infinity, which corresponds to values of 
the parameter u ranging from 0 to 2.331122332...  (Obviously the 
values of x for this curve extend only from -1/2 to +1/2, since 
an arc of length 1 can't reach any further.)  Here is a plot of
the envelope curve

    

Incidentally, dividing  dy/du  by  dx/du  gives the derivative of y 
with respect to x:

       dy        __________________________
       --  =  - / [cos(u) + u sin(u)]^2 - 1
       dx

Here's a plot of this derivative:

    

The maximum point occurs when this slope is zero, so the parameter
u must satisfy 1 = cos(u) + u sin(u), which is transcendental, and
has the solution u0 = 2.33112237...  To find the maximum height of
the envelope, we can insert this u0 back into the full parametric 
equation for y(u), which under these conditions reduces to simply
y(u0) = sin(u0)/2 = 0.362305677...  

Those parametric equations are probably the simplest analytical
expression for the envelope curve.  On the other hand, if you prefer 
to work numerically, you could let R(x) denote the root of equation 
(1) for any given x, and let Y(x) denote the corresponding value 
of y.  Then the envelope curve can be expressed as

    Y(x)  =  sqrt[R(x)^2 - x^2]  -  R(x)cos(1/(2R(x)))

Evaluating Y(x) numerically, we note that the max height is 
Y(0) = 0.36230567..., which is achieved by a circle whose 
center is at y=0.1478167... with radius r = 0.214488954...
Also, the curve is closely approximated by a cosine shape, 
i.e., [Y(0)/2]{1 + cos(2pi x)}, but it's slightly fatter.  
The total area under the curve is 0.1921152402... (as opposed
to 0.181152 for the cosine curve).  

The total length of the curve can be found by integrating 
sqrt[dx/du)^2 + (dy/du)^2]du from u=0 to 2.33112237, although 
this is a somewhat ill-conditioned integral, so it works better 
with the substitution u = ln(n) and then integrating over n.  
The result is about 1.272305...  

By the way, it might seem as if the upper part of the curve is a
pure circular arc for some distance, but in fact no finite part of 
the curve is a circular arc, because as soon as the upper part of 
the circles start to drop, the circles are also expanding horizontally, 
which pushes out to the side while leaving the upper point unchanged.  
This can be seen by looking at 1/R(x) versus x:

    

Notice that the radius for the limiting circles is never constant, 
even at x=0.  Also, note that all the circles with radius less than 
0.2144 are contained entirely inside the uppermost circle, so they 
don't contribute to the curve.