Fig. 164. Toledo.
[ LXXXVIII]
The Circle
If we inscribe a circle in a square we find that it touches that square at four points which are in the middle of each side, as at a b c d. It will also intersect the two diagonals at the four points o (Fig. 165). If, then, we put this square and its diagonals, &c., into perspective we shall have eight guiding points through which to trace the required circle, as shown in Fig. 166, which has the same base as Fig. 165.
| Fig. 165. | Fig. 166. |
[ LXXXIX]
The Circle in Perspective a True Ellipse
Although the circle drawn through certain points must be a freehand drawing, which requires a little practice to make it true, it is sufficient for ordinary purposes and on a small scale, but to be mathematically true it must be an ellipse. We will first draw an ellipse (Fig. 167). Let ee be its long, or transverse, diameter, and db its short or conjugate diameter. Now take half of the long diameter eE, and from point d with cE for radius mark on ee the two points ff, which are the foci of the ellipse. At each focus fix a pin, then make a loop of fine string that does not stretch and of such a length that when drawn out the double
thread will reach from f to e. Now place this double thread round the two pins at the foci ff· and distend it with the pencil point until it forms triangle fdf·, then push the pencil along and right round the two foci, which being guided by the thread will draw the curve, which is a true ellipse, and will pass through the eight points indicated in our first figure. This will be a sufficient proof that the circle in perspective and the ellipse are identical curves. We must also remember that the ellipse is an oblique projection of a circle, or an oblique section of a cone. The difference between the two figures consists in their centres not being in the same place, that of the perspective circle being at c, higher up than e the centre of the ellipse. The latter being a geometrical figure, its long diameter is exactly in the centre of the figure, whereas the centre c and the diameter of the perspective are at the intersection of the diagonals of the perspective square in which it is inscribed.