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 E P is approximately one-seventh of E G, 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 E G parallel to the sight-line, the square being perhaps first constructed in some oblique direction. In such cases, Q G and E P must be determined in perspective ratio by the dividing-point, the line E G 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 E G 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 G 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 book be held so that the line E H 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 G F 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.]
[Footnote 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].] [Return to text]
[Footnote 21: ] Or to its vanishing-point, if C D has one.] [Return to text]