The Theory and Practice of Perspective Part 23

CXXI

THE PYRAMID

Nothing can be more simple than to put a pyramid into perspective. Given the base (_abc_), raise from its centre a perpendicular (_OP_) of the required height, then draw lines from the corners of that base to a point _P_ on the vertical line, and the thing is done. These pyramids can be used in drawing roofs, steeples, &c. The cone is drawn in the same way, so also is any other figure, whether octagonal, hexangular, triangular, &c.

[Ill.u.s.tration: Fig. 221.]

[Ill.u.s.tration: Fig. 222.]

[Ill.u.s.tration: Fig. 223.]

[Ill.u.s.tration: Fig. 224.]

CXXII

THE GREAT PYRAMID

This enormous structure stands on a square base of over thirteen acres, each side of which measures, or did measure, 764 feet. Its original height was 480 feet, each side being an equilateral triangle. Let us see how we can draw this gigantic ma.s.s on our little sheet of paper.

In the first place, to take it all in at one view we must put it very far back, and in the second the horizon must be so low down that we cannot draw the square base of thirteen acres on the perspective plane, that is on the ground, so we must draw it in the air, and also to a very small scale.

Divide the base _AB_ into ten equal parts, and suppose each of these parts to measure 10 feet, _S_, the point of sight, is placed on the left of the picture near the side, in order that we may get a long line of distance, _S D_; but even this line is only half the distance we require. Let us therefore take the 16th distance, as shown in our previous ill.u.s.tration of the lighthouse (Fig. 92), which enables us to measure sixteen times the length of base _AB_, or 1,600 feet. The base _ef_ of the pyramid is 1,600 feet from the base line of the picture, and is, according to our 10-foot scale, 764 feet long.

The next thing to consider is the height of the pyramid. We make a scale to the right of the picture measuring 50 feet from _B_ to 50 at point where _BP_ intersects base of pyramid, raise perpendicular _CG_ and thereon measure 480 feet. As we cannot obtain a palpable square on the ground, let us draw one 480 feet above the ground. From _e_ and _f_ raise verticals _eM_ and _fN_, making them equal to perpendicular _G_, and draw line _MN_, which will be the same length as base, or 764 feet.

On this line form square _MNK_ parallel to the perspective plane, find its centre _O_ by means of diagonals, and _O_ will be the central height of the pyramid and exactly over the centre of the base. From this point _O_ draw sloping lines _Of_, _Oe_, _OY_, &c., and the figure is complete.

Note the way in which we find the measurements on base of pyramid and on line _MN_. By drawing _AS_ and _BS_ to point of sight we find _Te_, which measures 100 feet at a distance of 1,600 feet. We mark off seven of these lengths, and an additional 64 feet by the scale, and so obtain the required length. The position of the third corner of the base is found by dropping a perpendicular from _K_, till it meets the line _eS_.

Another thing to note is that the side of the pyramid that faces us, although an equilateral triangle, does not appear so, as its top angle is 382 feet farther off than its base owing to its leaning position.

CXXIII

THE PYRAMID IN ANGULAR PERSPECTIVE

In order to show the working of this proposition I have taken a much higher horizon, which immediately detracts from the impression of the bigness of the pyramid.

[Ill.u.s.tration: Fig. 225.]

We proceed to make our ground-plan _abcd_ high above the horizon instead of below it, drawing first the parallel square and then the oblique one.

From all the princ.i.p.al points drop perpendiculars to the ground and thus find the points through which to draw the base of the pyramid. Find centres _OO_ and decide upon the height _OP_. Draw the sloping lines from _P_ to the corners of the base, and the figure is complete.

CXXIV

TO DIVIDE THE SIDES OF THE PYRAMID HORIZONTALLY

Having raised the pyramid on a given oblique square, divide the vertical line OP into the required number of parts. From _A_ through _C_ draw _AG_ to horizon, which gives us _G_, the vanishing point of all the diagonals of squares parallel to and at the same angle as _ABCD_. From _G_ draw lines through the divisions 2, 3, &c., on _OP_ cutting the lines _PA_ and _PC_, thus dividing them into the required parts. Through the points thus found draw from _V_ all those sides of the squares that have _V_ for their vanishing point, as _ab_, _cd_, &c. Then join _bd_, _ac_, and the rest, and thus make the horizontal divisions required.

[Ill.u.s.tration: Fig. 226.]

[Ill.u.s.tration: Fig. 227.]

The same method will apply to drawing steps, square blocks, &c., as shown in Fig. 227, which is at the same angle as the above.

CXXV

OF ROOFS

The pyramidal roof (Fig. 228) is so simple that it explains itself. The chief thing to be noted is the way in which the diagonals are produced beyond the square of the walls, to give the width of the eaves, according to their position.

[Ill.u.s.tration: Fig. 228.]

Another form of the pyramidal roof is here given (Fig. 229). First draw the cube _edcba_ at the required height, and on the side facing us, _adcb_, draw triangle _K_, which represents the end of a gable roof.

Then draw similar triangles on the other sides of the cube (see Fig.

159, Lx.x.xIV). Join the opposite triangles at the apex, and thus form two gable roofs crossing each other at right angles. From _o_, centre of base of cube, raise vertical _OP_, and then from _P_ draw sloping lines to each corner of base _a_, _b_, &c., and by means of central lines drawn from _P_ to half base, find the points where the gable roofs intersect the central spire or pyramid. Any other proportions can be obtained by adding to or altering the cube.

[Ill.u.s.tration: Fig. 229.]

To draw a sloping or hip-roof which falls back at each end we must first draw its base, _CBDA_ (Fig. 230). Having found the centre _O_ and central line _SP_, and how far the roof is to fall back at each end, namely the distance _Pm_, draw horizontal line _RB_ through _m_. Then from _B_ through _O_ draw diagonal _BA_, and from _A_ draw horizontal _AD_, which gives us point _n_. From these two points _m_ and _n_ raise perpendiculars the height required for the roof, and from these draw sloping lines to the corners of the base. Join _ef_, that is, draw the top line of the roof, which completes it. Fig. 231 shows a plan or bird's-eye view of the roof and the diagonal _AB_ pa.s.sing through centre _O_. But there are so many varieties of roofs they would take almost a book to themselves to ill.u.s.trate them, especially the cottages and farm-buildings, barns, &c., besides churches, old mansions, and others.

There is also such irregularity about some of them that perspective rules, beyond those few here given, are of very little use. So that the best thing for an artist to do is to sketch them from the real whenever he has an opportunity.

[Ill.u.s.tration: Fig. 230.]

[Ill.u.s.tration: Fig. 231.]

CXXVI