The name is absent

Rectilinear Drawing         55

On considering the above figure it is clear that the area dA
of the small curvilinear quadrilateral ABCD is given by
pdτdp. Likewise the amount of lead in ABCD is fpdτdp,
since the length of each segment of the straight lines
crossing ABCD is dp while the number of these lines is
fds=fpdτ. Hence the limiting density at a point P is fρ∕p∙.
F ≈jp∣p.

Suppose now that we are given a drawing, and desire to
determine whether or not it arises from such a one-parameter
family of straight lines. According to what has just been
shown, it is clearly necessary for this: (1) that the equation
F = ∞ defines the convex arc; and (2) that along any tangent
to this arc C, F varies inversely as the distance from the
point of tangency. If these two conditions are satisfied and
if p(>0) denotes the radius of curvature at the point of
tangency, the desired distribution function /(j) is clearly
Ppl p∙

Before proceeding further it is worth while to note the
practical use of such one-parameter families for obtaining
outline drawings of curvilinear arcs C. In fact since the
density F is infinite along the arc C, this curve appeared as
clearly etched. Similarly any set of arcs can be simultane-
ously etched, although there is no reason to believe that the
shading of masses will then be as desired. The pumpkin-head
drawing of Mr. Middleton herewith shown (figure 4) affords
a simple and amusing illustration of the possibilities.

It is interesting to observe that in such an outline drawing
the density function F is given as a sum of terms of the form
f(s)p∣p referred to above. For example, in his figure Mr.
Middleton has taken f(s)p to be the same along every circle
and circular arc; this requirement means that the tangents
are drawn at equal angular intervals along all of the circles
and circular arcs, inasmuch as/ is inversely proportional to

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