## Shadows Of Points And Lines

To cast the shadow of a point upon a given surface draw a conventional ray through the point and to the surface. Where this ray strikes the surface is the required shadow of the point. This is the fundamental operation in all shadow casting, but its application is not always easy, therefore the following detailed helps are given for various cases.

Where the shadow of a straight line is to be cast on a plane surface, Fig. 66, the shadow of each end of the line is located and these points connected to give the shadow of the line. If the given line or the receiving surface is curved, a number of these shadow points are determined and the shadow of the line drawn through them.

In Figs. 66 and 67 of Plate 12 are shown both pictorially and in orthographic projection, a vertical line casting a shadow on horizontal and vertical planes. It will be readily seen that the shadow of but one point a in Fig. 66 is necessary to determine the shadow of the line, while in Fig. 67 two points b and c must be used.

When a vertical line casts a shadow on a horizontal moulding (or a horizontal line on a vertical moulding), as in Fig. 68 on Plate 12, the front view of that shadow is the same as the profile of the moulding. This fact, if remembered, will make short work of many problems which would otherwise be quite tedious. A similar labor saver is shown in Fig. 69 on Plate 12. Here it is seen that the shadow of a vertical line on a sloping surface, when viewed from the front, takes the same angle as the slope of the surface. This is useful when working with the shadows of chimneys and dormers on sloping roofs.

## Cylinders, Cones And Spheres

If a circle is parallel to a plane surface, its shadow on that surface is circular, and if it is oblique to the surface, its shadow is an ellipse. This is true of circle shadows on any plane surface and is illustrated in the drawing of the three objects on Plate 13. A practical way to draw the shadow when it is elliptical is to first draw a square enclosing the circle which casts the shadow, then locate the shadow of the square (as in Fig. 70 and 70a) and in it sketch the ellipse, being careful that it touches and is tangent to the sides of the parallelogram at their center points, a, b, c and d.

## Cylinder

After the circular or elliptical shadows of the ends of the right circular cylinder have been drawn, the straight tangent lines are drawn completing the outline of the shadow. The shade lines of the cylinder are found by drawing the 45-degree tangent lines as noted in the top view, Fig. 70a, and locating the tangent points at e and f. Projecting down to the front view from these points will locate the front view of the shade line.

## Cone

The shadow of a conical object is determined by first locating the shadows of the apex and the base and then connecting the former with the latter by tangent lines as in Figs. 71 and 71a. The shade lines of the cone are found by projecting from the tangent points g and h in the top view, back (at 45 degrees) to the base at j and k, then connecting these points with the apex.

## Sphere

The shadow of a sphere is determined by the use of a cylinder of tangent light rays as in Fig. 72. Since this is a rather tedious process, a shorter method is given; see Fig. 72a. In the top view draw the 45-degree tangent lines through m-m„ and l-l, and a center line through n-os lightly. Draw line m-l through p (at 45 degrees). Draw m-o at an angle of 30 degrees with m-l, locating 0. Lay off p-n equal to p-o. This gives the major and minor axes of the elliptical shade line in the top view. Draw the shade line by the method of Fig. 38, Plate 5. Next on the front view draw the 45-degree line r'-s' through center p'. Locate t' by drawing s'-t' at an angle of 30 degrees with r'-s', then lay off p'-u' equal to p'-f and draw the ellipse as in the top view. The shadows of points m, l, 0, and n are found by drawing conventional light rays through the points in the top and front views, locating ms, ls,os, and ns and drawing the elliptical shadow line in the top view by the trammel method. If the shadow is cast on an irregular or a curving surface, the shadows of a number of points of the shade line may be found and the shadow line drawn through them.

The methods thus far explained are known as oblique projection methods. They are difficult of application where double curved surfaces are to be dealt with. For the solution of a problem involving such surfaces, the slicing method is found useful. This method, while not difficult, necessitates a great deal of very careful work for an accurate shadow.

PLATE 14.

## Slicing Method

Through the column base on Plate 14 a series of imaginary vertical cutting planes (see also upper left-hand corner of plate) is passed and the lines which these planes cut on the surface of the object are drawn. The cutting planes are passed at 45 degrees in the plan view and consequently contain a number of the conventional rays. Then the lines of intersection are drawn on the elevation (see also upper right-hand corner of plate). These lines of intersection are found by drawing the projections of a series of circular lines on the surface of the base, finding where these circles go through the cutting planes as at a, b, c, d, e, etc., in the plan and projecting up to the corresponding lines at a', b', c', d', e', etc., on the elevation. Locate enough points to accurately fix the lines of intersection. Where the cutting planes pass through a moulding of circular section, as the upper two members of this base, the line of intersection with that moulding will be an ellipse, and may be drawn by the method of Fig. 38, Plate 5, after the two axes or half-axes have been located, as m'-n' and k'-l' on the elevation. The cutting planes may be passed through wherever lines of intersection are thought necessary.

Now in the elevation draw the 45-degree tangent lines as shown. These light rays will locate points of tangency and intersection through which the shade and shadow lines may be drawn.