TO DIVIDE A CIRCLE DRAWN IN PERSPECTIVE INTO ANY GIVEN NUMBER OF EQUAL
PARTS.
Let _AB_, Fig. 32., be the circle drawn in perspective. It is required
to divide it into a given number of equal parts; in this case, 20.
Let _KAL_ be the semicircle used in the construction. Divide the
semicircle _KAL_ into half the number of parts required; in this case,
10.
Produce the line _EG_ laterally, as far as may be necessary.
From _O_, the center of the semicircle _KAL_, draw radii through the
points of division of the semicircle, _p_, _q_, _r_, etc., and produce
them to cut the line _EG_ in _P_, _Q_, _R_, etc.
From the points _PQR_ draw the lines _PP′_, _QQ′_, _RR′_, etc.,
through the center of the circle _AB_, each cutting the circle in two
points of its circumference.
Then these points divide the perspective circle as required.
If from each of the points _p_, _q_, _r_, a vertical were raised to
the line _EG_, as in Fig. 31., and from the point where it cut _EG_ a
line were drawn to the vanishing-point, as _QQ′_ in Fig. 31., this
line would also determine two of the points of division.
[Illustration: Fig. 32.]
If it is required to divide a circle into any number of given
_un_equal parts (as in the points _A_, _B_, and _C_, Fig. 33.), the
shortest way is thus to raise vertical lines from _A_ and _B_ to the
side of the perspective square _XY_, and then draw to the
vanishing-point, cutting the perspective circle in _a_ and _b_, the
points required. Only notice that if any point, as _A_, is on the
nearer side of the circle _ABC_, its representative point, _a_, must
be on the nearer side of the circle _abc_; and if the point _B_ is on
the farther side of the circle _ABC_, _b_ must be on the farther side
of _abc_. If any point, as _C_, is so much in the lateral arc of the
circle as not to be easily determinable by the vertical line, draw the
horizontal _CP_, find the correspondent _p_ in the side of the
perspective square, and draw _pc_ parallel to _XY_, cutting the
perspective circle in _c_.
[Illustration: Fig. 33.]