Let a b, Fig-. 28., exist the curve. Enclose it in a rectangle, c d e f. Fix the position of the signal c or d, and describe the rectangle. (Prob. Eight. Cor. I.)i

Problem XI To Draw Any Curve In A Horizontal Or Ve Perspective Elements 84

Fig. 28.

Let c D East F, Fig. 29., be the rectangle so drawn.

If an extremity of the curve, as A, is in a side of the rectangle, divide the side c E, Fig. 29., then that A c shall be (in perspective ratio) to A e equally a c is to a e in Fig. 28. (Prob. V. Cor. II.)

Similarly make up one's mind the points of contact of the bend and rectangle e, f, thousand.

1 Or if the bend 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 bend will in either case be regulated by convenience. See the Exercises on this Prob em, in the Appendix, p. 289.

Problem Xi - Corollary

If an extremity of the curve, as B, is not in a side of the rectangle, allow fall the perpendiculars B a, b b on the rectangle sides. Determine the contributor points a and b in Fig. 29., as you take already determined a, B, eastward, and f.

From b, Fig. 29., draw b B paralled to C D1, and from a draw a b to the vanishing-bespeak of D F, 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, past letting fall p q and p r on the rectangle sides.

Any number of points in the curve may be thus determined, and the curve drawn through the series; in nigh cases, iii or four will be enough. Practically, complicated curves may be better fatigued in perspective past an experienced eye than by rule, as the fixing of the various points in haste involves as well many chances of error; but it is well to describe a good many by rule get-go, in order to give the heart its experience.two

Corollary

If the curve required be a circle, Fig. 30., the rectangle which encloses information technology will become a square, and the curve will have four points of contact, A B c d, in the centre of the sides of the square.

Draw the square, and every bit a square may be drawn

Corollary Perspective Elements 85

Fig. 29.

ane Or to its vanishing-point, if c D has one.

2 Of course, by dividing the original rectangle into any number of equal rectangles, and dividing the perspective rectangle similarly, the bend may be approximately fatigued without whatsoever problem; simply, when accuracy is required, the points should be fixed, as in the problem.About a circle in whatsoever position, draw it with its nearest side, East Chiliad, parallel to the sight-line.

Let e f, Fig. 31., exist the square and then drawn.

Draw its diagonals E F, Thou h; and through the center of the square (determined by their intersection) draw A B to the vanishing-point of G F, and c D parallel to e chiliad. Then the points a b c d are the 4 points of the circle's contact.

On due east g describe a half square, e l; draw the semicircle k A l; and from its middle, r, the diagonals r e, r g, cut the circle in ten, y.

Corollary Perspective Elements 86

Fig. thirty.

Corollary Perspective Elements 87

Fig. 31.

From the points x, y, where the circle cuts the diagonals, raise perpendiculars, p x, q y, to east m.

From p and Q depict p p', Q Q', to the vanishing-point of G F, cutting the diagonals in chiliad, n, and 0, p.

Then g, n, o, p are four other points in the circle.

Through these eight points the circle may be drawn past the hand accurately enough for full general purpos s; but whatever number of points required may, of class, be adamant, equally in Problem Eleven.

The distance e p is approximately one seventh of E g, and may be assumed to be and so in quick practice, as the mistake involved is not greater than would be incurred in the hasty functioning of drawing the circle and diagonals.

It may frequently happen that, in result of associated constructions, it may be inconvenient to depict E G parallel to the sight-line, the square existence perhaps first constructed in some oblique direction. In such cases, Q Grand and E P must be determined in perspective ratio past the dividing-signal, the line Due east G existence used as a measuring-line.

[0bs. In drawing Fig. 31. the station-point has been taken much nearer the paper than is usually advisable, in guild to show the graphic symbol of the curve in a very singled-out class.

If the student turns the book so that E G may be vertical, Fig. 31. will represent the structure for drawing a circle in a vertical airplane, the sight-line being then of class parallel to Grand L; and the semicircles A D B, A c B, on each side of the diameter A B, will represent ordinary semicircular arches seen in perspective. In that case, if the volume be held and then that the line Due east H is the tiptop of the square, the upper semicircle will stand for a semicircular arch, above the centre, drawn in perspective. But if the book be held and then hat the line Yard F is the pinnacle of the square, the upper se. icircle volition represent a semicircular arch, below the heart, drawn in per pective.

If the book be turned upside downwards, the figure volition stand for a circumvolve drawn on the ceiling, or any other horizontal plane above the heart; and the construction is, of course, accurate in every case.]

Problem 11

It is seldom that any complicated curve, except occasionally a spiral, needs to be fatigued in perspective; just the student will practice well to practice for some time any fantastic

Problem XI Perspective Elements 119

Fig. 63.

shapes which he can detect drawn on flat surfaces equally on wall-papers, carpets, c, in society to accustom himself to the strange and great changes which perspective causes in them.

The curves most required in architectural drawing, after the circle, are those of pointed arches; in which, however, all that volition be generally needed is to fix the apex, and two points in the sides. Thus if nosotros have to draw a range of pointed arches, such every bit A P B, Fig. 63., draw the measured arch to its sight-magnitude kickoff neatly in a rectangle, A B C D; then depict the diagonals A D and B C; where they cut the curve draw a horizontal line (equally at the level E in the figure), and behave it along the range to the vanishing-point, fixing the points where the arches cut their diagonals all along. If the curvation is cusped, a line should be drawn at F to mark the top of the cusps, and verticals raised at G and H, to determine the interval betwixt them.

Problem XI Perspective Elements 120

Fig. 64.

Any other points may be similarly determined, just these will usually be enough. Figure 63. shows the perspective construction of a square niche of good Veronese Gothic, with anuncusped curvation of like size and curve beyond.

In Fig. 64. the more distant arch only is lettered, as the construction of the nearer explains itself more conspicuously to the eye without letters. The more distant arch shows the general structure for all arches seen underneath, as of bridges, cathedral aisles, c. The rectangle A B C D is beginning drawn to contain the exterior curvation; then the depth of the arch, A a, is determined by the measuring-line, and the rectangle, a b c d, drawn for the inner curvation.

a a, Bb, c, go to 1 vanishing-point; A B, a b, c, to the contrary ane.

In the nearer arch another narrow rectangle is drawn to make up one's mind the cusp. The parts which would actually come into sight are slightly shaded.