Bevels and Angles in Oblique Planes. Simple Angles. Com pound Angles—rotation of inclined plane. Cuts for Purlins against Hips. Oblique Cuts in Angle Braces -various positions of brace, development oi inclined surfaces. Bevels in Splayed Liaittgg- setting out the soffit. Properties oi and methods of drawing Ellipses—definition of ellipse and terms connected with 'arnc. Sections of Cylinders. Describing Ellipse by intersecting lines, ditto by T rammflling. To find the Foci, Normal and Tangent of an Ellipse. False Ellipses. The Cone and its Properties- -definitions, projections. The Conic Sections -how to produce them. The Covering of Cones. Development of Frustum of Cone. The Covering of Domes and Vaults—types of Domes. To obtain projection of boarding. To obtain Shape of boards laid vertically; ditto laid horizontally. A Gothic Dome—to project the ribs. An Elliptic Dome—to obtain the covering of. Setting out Arches in Brick and Stone—the gauged camber, round, elliptic, lancet, equilateral Tudor, horseshoe, stilted, basket handle, ogee and squinch arches. c arpentry Arches—Gothic, method of finding centres for. The Wave, Ogee and Bell Arches. Complex Curves. The Ilelix--methods of drawing. " Pitch " explained. Development of Helical Curve. Projecting a Wreathed Handrail. The Spiral—definition. To draw the rising Spiral, the Plane Spiral. Drawing Scrolls
The Determination of Revels and Angles in Oblique Planes-—This is a class of problem which in one form or another is constantly occurring in the workshop and drawing office. The carpenter meets with it in variety in traming up a roof ; the joiner when fitting the mitres of splayed linings or joints ot curved and splayed fascias, etc.; the bricklayer and the mason in preparing templets for splayed and skew arches or in the angles of window and door openings; the plumber when obtaining shapes of lacing 141
sheets for turret roofs, etc. The reader who has followed the chapter oil orthographic projection will scarcely need telling that a drawing of an angular or any other shaped plane surface gives the true size and shape of that surface only when projected upon a plane parallel with itself.
In most technical drawings in which oblique surfaces occur, however, these will be found projected upon a plane not parallel with the oblique surface, and to obtain cither the real shape and dimensions or the t rue angles at the intersections special projections must be made, and it is with these we now deal. To make the matter quite clear as to when a special drawing or development is required, a few examples of intersecting surfaces at various angles which do not require special treatment are first given (see page 143)-
Simple Angles.—Figs, i and 2 show two pieces of wood at right angles to each other, and it is obvious that the square applied as shown will give the correct cut for the joint. Again, in Figs. 3 and 4 we have the piece B fitted against the piece A, and as it lies horizontal in one direction its true shape is shown in the plan and a " square " gives the joint. The piece B is inclined in the other direction, conse-quentlj' the view at right angles to tiie plan will show its true inclination, and the edge bevel is obtained as shown in Fig. 4. If piece B is level and piece /I vertical, but making any angle with B in plan, then the true bevel is shown in the plan as at Fig. 6, and obviously the depth of the cut is square. Other simple angles are shown on page 146, on the left side of Figs. 1 and 2, and on Fig. 4.
Compound Angles.—These arise when one or both of the interseding surfaces are at angles other than right angles—that is, square to each other-—and one or both of the surfaces are inclined in transverse direction. Such cases occur in splayed linings or jambs, in topped roofs, in angle braces, hoppers, etc., and the method of development now to be described will disclose the bevels or " cuts " required in all such cases.
COMPOUND ANGLES 143
Figs. 7, 8 and 9 show a horizontal board inclined at 300 across its width, fitting against a vertical board,
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