The Theory and Practice of Perspective Part 18

[Ill.u.s.tration: Fig. 166.]

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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 pa.s.s 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.

[Ill.u.s.tration: Fig. 167.]

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FURTHER ILl.u.s.tRATION OF THE ELLIPSE

In order to show that the ellipse drawn by a loop as in the previous figure is also a circle in perspective we must reconstruct around it the square and its eight points by means of which it was drawn in the first instance. We start with nothing but the ellipse itself. We have to find the points of sight and distance, the base, &c. Let us start with base _AB_, a horizontal tangent to the curve extending beyond it on either side. From _A_ and _B_ draw two other tangents so that they shall touch the curve at points such as _TT_ a little above the transverse diameter and on a level with each other. Produce these tangents till they meet at point _S_, which will be the point of sight. Through this point draw horizontal line _H_. Now draw tangent _CD_ parallel to _AB_. Draw diagonal _AD_ till it cuts the horizon at the point of distance, this will cut through diameter of circle at its centre, and so proceed to find the eight points through which the perspective circle pa.s.ses, when it will be found that they all lie on the ellipse we have drawn with the loop, showing that the two curves are identical although their centres are distinct.

[Ill.u.s.tration: Fig. 168.]

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HOW TO DRAW A CIRCLE IN PERSPECTIVE WITHOUT A GEOMETRICAL PLAN

Divide base _AB_ into four equal parts. At _B_ drop perpendicular _Bn_, making _Bn_ equal to _Bm_, or one-fourth of base. Join _mn_ and transfer this measurement to each side of _d_ on base line; that is, make _df_ and _df_ equal to _mn_. Draw _fS_ and _fS_, and the intersections of these lines with the diagonals of square will give us the four points _o o o o_.

[Ill.u.s.tration: Fig. 169.]

The reason of this is that _ff_ is the measurement on the base _AB_ of another square _o o o o_ which is exactly half of the outer square. For if we inscribe a circle in a square and then inscribe a second square in that circle, this second square will be exactly half the area of the larger one; for its side will be equal to half the diagonal of the larger square, as can be seen by studying the following figures. In Fig.

170, for instance, the side of small square _K_ is half the diagonal of large square _o_.

[Ill.u.s.tration: Fig. 170.]

[Ill.u.s.tration: Fig. 171.]

In Fig. 171, _CB_ represents half of diagonal _EB_ of the outer square in which the circle is inscribed. By taking a fourth of the base _mB_ and drawing perpendicular _mh_ we cut _CB_ at _h_ in two equal parts, _Ch_, _hB_. It will be seen that _hB_ is equal to _mn_, one-quarter of the diagonal, so if we measure _mn_ on each side of _D_ we get _ff_ equal to _CB_, or half the diagonal. By drawing _ff_, _ff_ pa.s.sing through the diagonals we get the four points _o o o o_ through which to draw the smaller square. Without referring to geometry we can see at a glance by Fig. 172, where we have simply turned the square _o o o o_ on its centre so that its angles touch the sides of the outer square, that it is exactly half of square _ABEF_, since each quarter of it, such as EoCo, is bisected by its diagonal _oo_.

[Ill.u.s.tration: Fig. 172.]

[Ill.u.s.tration: Fig. 173.]

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HOW TO DRAW A CIRCLE IN ANGULAR PERSPECTIVE

Let _ABCD_ be the oblique square. Produce _VA_ till it cuts the base line at _G_.

[Ill.u.s.tration: Fig. 174.]

Take _mD_, the fourth of the base. Find _mn_ as in Fig. 171, measure it on each side of _E_, and so obtain _Ef_ and _Ef_, and proceed to draw _fV_, _EV_, _fV_ and the diagonals, whose intersections with these lines will give us the eight points through which to draw the circle. In fact the process is the same as in parallel perspective, only instead of making our divisions on the actual base _AD_ of the square, we make them on _GD_, the base line.

To obtain the central line _hh_ pa.s.sing through _O_, we can make use of diagonals of the half squares; that is, if the other vanishing point is inaccessible, as in this case.

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HOW TO DRAW A CIRCLE IN PERSPECTIVE MORE CORRECTLY, BY USING SIXTEEN GUIDING POINTS

First draw square _ABCD_. From _O_, the middle of the base, draw semicircle _AKB_, and divide it into eight equal parts. From each division raise perpendiculars to the base, such as _2 O_, _3 O_, _5 O_, &c., and from divisions _O_, _O_, _O_ draw lines to point of sight, and where these lines cut the diagonals _AC_, _DB_, draw horizontals parallel to base _AB_. Then through the points thus obtained draw the circle as shown in this figure, which also shows us how the circ.u.mference of a circle in perspective may be divided into any number of equal parts.

[Ill.u.s.tration: Fig. 175.]

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HOW TO DIVIDE A PERSPECTIVE CIRCLE INTO ANY NUMBER OF EQUAL PARTS

This is simply a repet.i.tion of the previous figure as far as its construction is concerned, only in this case we have divided the semicircle into twelve parts and the perspective into twenty-four.

[Ill.u.s.tration: Fig. 176.]

[Ill.u.s.tration: Fig. 177.] We have raised perpendiculars from the divisions on the semicircle, and proceeded as before to draw lines to the point of sight, and have thus by their intersections with the circ.u.mference already drawn in perspective divided it into the required number of equal parts, to which from the centre we have drawn the radii.

This will show us how to draw traceries in Gothic windows, columns in a circle, cart-wheels, &c.

The geometrical figure (177) will explain the construction of the perspective one by showing how the divisions are obtained on the line _AB_, which represents base of square, from the divisions on the semicircle _AKB_.

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