[Geometric diagram]
Fig.28.

Let AB, Fig.28., be the curve.

Inclose it in a rectangle, CDEF.

Fix the position of the point C or D, and draw the rectangle. (ProblemVIII. Coroll.I.)[Footnote 20]

Let CDEF, Fig.29., be the rectangle so drawn.

[Geometric diagram]
Fig.29.

If an extremity of the curve, as A, is in a side of the rectangle, divide the side CE, Fig.29., so that AC shall be (in perspective ratio) to AE as AC is to AE in Fig.28. (Prob.V. Cor.II.)

Similarly determine the points of contact of the curve and rectangle e, f, g.

If an extremity of the curve, as B, is not in a side of the rectangle, let [p39] fall the perpendiculars Ba, Bb on the rectangle sides. Determine the correspondent points a and b in Fig.29., as you have already determined A, B, e, and f.

From b, Fig.29., draw bB parallel to CD,[Footnote 21] and from a draw aB to the vanishing-point of DF, cutting each other in B. Then B is the extremity of the curve.

Determine any other important point in the curve, as P, in the same way, by letting fall Pq and Pr on the rectangle’s sides.

Any number of points in the curve may be thus determined, and the curve drawn through the series; in most cases, three or four will be enough. Practically, complicated curves may be better drawn in perspective by an experienced eye than by rule, as the fixing of the various points in haste involves too many chances of error; but it is well to draw a good many by rule first, in order to give the eye its experience.[Footnote 22]

COROLLARY.

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.

[Geometric diagram]
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.

[p40]
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.

[Geometric diagram]
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 ProblemXI.

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 [p41] 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.