To draw the Development, Fig. 3.—Make the straight line a-c equal in length to the circumference of the helix m plan ; draw a -b porpendicuhir and equal in length to the given rise a-A, Fig. 2; join b -c, then b -c is the true inclination of the helix. A wreathed handrail to a circular stair is an example of a helical curve, and in Fig. 5 a projection of one is given.
To drawthe Projection of a Wreathed Handrail .—Let Fig. 4 be the plan of the rail and the radiating lines may be considered the risers of the steps beneath. For convenience of explanation we will consider first the outside of the rail in plan, and the under arris in elevation. Draw the perpendicular 1-8, Fig. 5, as a projection of point 1 in the plan; set off upon it eight divisions equal to the distance between the points in plan ; this, of course, would not be so in practice. The " rise " of a step is always less than its " going," but the above arrangement facilitates explanations.
Draw horizontal projectors from each division and intersect them by perpendiculars from the correspondingly numbered points in the plan. Draw, either freehand or by aid of the French curve, a continuous line through the points so found, lhis will produce the lower external arris as shown. Now if we consider the rail to have vertical sides, as it actually does in practice, before it is moulded, it will be obvious that the top external arris will be directly over the line just drawn, and is represented in the plan by the same circle. Therefore all that is necessary to obtain the. projection of the top edge is to utilise the same projectors by producing them upwards as shown 11 full line, and making them equal in length to the thickness of rail; a second series of points will thus be obtained through which to draw this curve. The two inner arrises are obtained similarly, but in this case, though we use the same heights as before, we project from the inside circle in plan, as, for example, on radiator 3, point III taken into the elevation becomes point III' upon the horizontal projector 3. As the rail winds around the cylinder the view of the inner edge is
176 THE RISING SPIRAL
intercepted by the outside of the rail, therefore that portion is shown by a dotted line.
The Spiral is a curved lhie whose consecutive ¡joints continuously and uniformly approach or recede from a certain fixed point called the pule. A line drawn from the pole to any point in the curve is called its radius at that point. Herein lies the essential difference between a true spiral and a helix ; the radii of a helix are all of one length ; the radii of a spiral are ;dl of different lengths. The spiral ascends upon a cone, the helix upon a cylindric. or ellipsoidial prism.
To draw a Spiral Curve.—Draw' the generating cone and its plan as Figs. 6 and 7. Divide the plan into any number of segments by radial lines through its centre. Project their extremities into the elevation as indicated by the figures. Join the points upon the base to the apex, thus producing a series of lines upon the surface of the cone of which the first drawn radial lines are the plans. Next erect a perpendicular to the base, and upon it set off the same number of equal divisions as are shown in the plan. These may bear any proportion to the " going " in the plan, and, as in the helix, they indicate the pitch. Draw horizontal projectors from these divisions 2, 3, 4, etc., to the correspondingly numbered sections upon the surface of the cone, and trace the curve through the points so found ; this gives us the " rising spiral."
To find the plane or spiral "scroll," Fig. 6, drop perpendiculars from the points found 011 the surface of the cone upon the similarly numbered radii in the plan, as, per example, I III to 4a, and trace the curve through these points.
The Scroll, Fig. 9, is a plane spiral curve, also known as the volute. There are several methods of drawing these by diminishing arcs of circles; some more applicable to handrails are given in the author's "Modern Practical Joinery." The one described here is more suitable for architectural draughtsmen.
We have first to obtain proportional radii, sec Fig. 9
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