The Theory and Practice of Perspective Part 21

I have here placed the perspective drawing under the ground plan to show the relation between the two, and how the perspective is worked out, but the general practice is to find the required measurements as here shown, to mark them on a straight edge of card or paper, and transfer them to the paper on which the drawing is to be made.

This of course is the simplest form of a plan and elevation. It is easy to see, however, that we could set out an elaborate building in the same way as this figure, but in that case we should not place the drawing underneath the ground-plan, but transfer the measurements to another sheet of paper as mentioned above.

CIX

THE OCTAGON

To draw the geometrical figure of an octagon contained in a square, take half of the diagonal of that square as radius, and from each corner describe a quarter circle. At the eight points where they touch the sides of the square, draw the eight sides of the octagon.

[Ill.u.s.tration: Fig. 198.]

[Ill.u.s.tration: Fig. 199.]

To put this into perspective take the base of the square _AB_ and thereon form the perspective square _ABCD_. From either extremity of that base (say _B_) drop perpendicular _BF_, draw diagonal _AF_, and then from _B_ with radius _BO_, half that diagonal, describe arc _EOE_.

This will give us the measurement _AE_. Make _GB_ equal to _AE_. Then draw lines from _G_ and _E_ towards _S_, and by means of the diagonals find the transverse lines _KK_, _hh_, which will give us the eight points through which to draw the octagon.

CX

HOW TO DRAW THE OCTAGON IN ANGULAR PERSPECTIVE

Form square _ABCD_ (new method), produce sides _BC_ and _AD_ to the horizon at _V_, and produce _VA_ to _a_ on base. Drop perpendicular from _B_ to _F_ the same length as _aB_, and proceed as in the previous figure to find the eight points on the oblique square through which to draw the octagon.

[Ill.u.s.tration: Fig. 200.]

It will be seen that this operation is very much the same as in parallel perspective, only we make our measurements on the base line _aB_ as we cannot measure the vanis.h.i.+ng line _BA_ otherwise.

CXI

HOW TO DRAW AN OCTAGONAL FIGURE IN ANGULAR PERSPECTIVE

In this figure in angular perspective we do precisely the same thing as in the previous problem, taking our measurements on the base line _EB_ instead of on the vanis.h.i.+ng line _BA_. If we wish to raise a figure on this octagon the height of _EG_ we form the vanis.h.i.+ng scale _EGO_, and from the eight points on the ground draw horizontals to _EO_ and thus find all the points that give us the perspective height of each angle of the octagonal figure.

[Ill.u.s.tration: Fig. 201.]

CXII

HOW TO DRAW CONCENTRIC OCTAGONS, WITH ILl.u.s.tRATION OF A WELL

The geometrical figure 202 A shows how by means of diagonals _AC_ and _BD_ and the radii 1 2 3, &c., we can obtain smaller octagons inside the larger ones. Note how these are carried out in the second figure (202 B), and their application to this drawing of an octagonal well on an octagonal base.

[Ill.u.s.tration: Fig. 202 A.]

[Ill.u.s.tration: Fig. 202 B.]

[Ill.u.s.tration: Fig. 203.]

CXIII

A PAVEMENT COMPOSED OF OCTAGONS AND SMALL SQUARES

To draw a pavement with octagonal tiles we will begin with an octagon contained in a square _abcd_. Produce diagonal _ac_ to _V_. This will be the vanis.h.i.+ng point for the sides of the small squares directed towards it. The other sides are directed to an inaccessible point out of the picture, but their directions are determined by the lines drawn from divisions on base to V2 (see back, Fig. 133).

[Ill.u.s.tration: Fig. 204.]

[Ill.u.s.tration: Fig. 205.]

I have drawn the lower figure to show how the squares which contain the octagons are obtained by means of the diagonals, _BD_, _AC_, and the central line OV2. Given the square _ABCD_. From _D_ draw diagonal to _G_, then from _C_ through centre _o_ draw _CE_, and so on all the way up the floor until sufficient are obtained. It is easy to see how other squares on each side of these can be produced.

CXIV

THE HEXAGON

The hexagon is a six-sided figure which, if inscribed in a circle, will have each of its sides equal to the radius of that circle (Fig. 206). If inscribed in a rectangle _ABCD_, that rectangle will be equal in length to two sides of the hexagon or two radii of the circle, as _EF_, and its width will be twice the height of an equilateral triangle _mon_.

[Ill.u.s.tration: Fig. 206.]

To put the hexagon into perspective, draw base of quadrilateral _AD_, divide it into four equal parts, and from each division draw lines to point of sight. From _h_ drop perpendicular _ho_, and form equilateral triangle _mno_. Take the height _ho_ and measure it twice along the base from _A_ to 2. From 2 draw line to point of distance, or from 1 to distance, and so find length of side _AB_ equal to A2. Draw _BC_, and _EF_ through centre _o_, and thus we have the six points through which to draw the hexagon.

[Ill.u.s.tration: Fig. 207.]

CXV

A PAVEMENT COMPOSED OF HEXAGONAL TILES