If the curve required be a circle, Fig. 30., the rectangle which
incloses it will become a square, and the curve will have four points
of contact, _ABCD_, in the middle of the sides of the square.
[Illustration: Fig. 30.]
Draw the square, and as a square may be drawn about a circle in any
position, draw it with its nearest side, _EG_, parallel to the
sight-line.
Let _EF_, Fig. 31., be the square so drawn.
Draw its diagonals _EF_, _GH_; and through the center of the square
(determined by their intersection) draw _AB_ to the vanishing-point of
_GF_, and _CD_ parallel to _EG_. Then the points _ABCD_ are the four
points of the circle’s contact.
[Illustration: Fig. 31.]
On _EG_ describe a half square, _EL_; draw the semicircle _KAL_; and
from its center, _R_, the diagonals _RE_, _RG_, cutting the circle in
_x_, _y_.
From the points _x_ _y_, where the circle cuts the diagonals, raise
perpendiculars, _Px_, _Qy_, to _EG_.
From _P_ and _Q_ draw _PP′_, _QQ′_, to the vanishing-point of _GF_,
cutting the diagonals in _m_, _n_, and _o_, _p_.
Then _m_, _n_, _o_, _p_ are four other points in the circle.
Through these eight points the circle may be drawn by the hand
accurately enough for general purposes; but any number of points
required may, of course, be determined, as in Problem XI.
The distance _EP_ is approximately one-seventh of _EG_, and may be
assumed to be so in quick practice, as the error involved is not
greater than would be incurred in the hasty operation of drawing the
circle and diagonals.
It may frequently happen that, in consequence of associated
constructions, it may be inconvenient to draw _EG_ parallel to the
sight-line, the square being perhaps first constructed in some oblique
direction. In such cases, _QG_ and _EP_ must be determined in
perspective ratio by the dividing-point, the line _EG_ being used as a
measuring-line.
[_Obs._ In drawing Fig. 31. the station-point has been taken much
nearer the paper than is usually advisable, in order to show the
character of the curve in a very distinct form.
If the student turns the book so that _EG_ may be vertical,
Fig. 31. will represent the construction for drawing a circle in a
vertical plane, the sight-line being then of course parallel to
_GL_; and the semicircles _ADB_, _ACB_, on each side of the
diameter _AB_, will represent ordinary semicircular arches seen in
perspective. In that case, if the book be held so that the line
_EH_ is the top of the square, the upper semicircle will represent
a semicircular arch, _above_ the eye, drawn in perspective. But if
the book be held so that the line _GF_ is the top of the square,
the upper semicircle will represent a semicircular arch, _below_
the eye, drawn in perspective.
If the book be turned upside down, the figure will represent a
circle drawn on the ceiling, or any other horizontal plane above
the eye; and the construction is, of course, accurate in every
case.]
[20] Or if the curve is in a vertical plane, Coroll. to Problem IX.
As a rectangle may be drawn in any position round any given
curve, its position with respect to the curve will in either
case be regulated by convenience. See the Exercises on this
Problem, in the Appendix, p. 85.
[21] Or to its vanishing-point, if _CD_ has one.
[22] Of course, by dividing the original rectangle into any number
of equal rectangles, and dividing the perspective rectangle
similarly, the curve may be approximately drawn without any
trouble; but, when accuracy is required, the points should be
fixed, as in the problem.