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
Enveloping Circular Arcs
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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
Enveloping Circular Arcs
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2r^2 = sqrt[r^2 - x^2] { sin(1/2r) - 2cos(1/2r) } (1)
This says that for any speci