The preceding examples show how to find shadows when cast upon a vertical plane; shadows thrown upon curved surfaces are ascertained in a similar manner. (Fig. 434.)
By an inspection of Fig. 434, it will be seen that the only difference between this and the last examples is that the rays in the plan die against the circle ab, instead of a straight line.
Cast the ray ab (Fig. 435) from the end of the shelf to the face of the wall, and from b draw b c parallel to the shelf; cast the ray de from the end of the shelf; then the lines de and ec will define the shadow.
These examples might be multiplied, but enough has been given to illustrate the general principle by which shadows in all instances are found. Let us attend now to the application of this principle to such familiar objects as are likely to occur in practice.
From the points a, a, etc. (Fig. 436), cast rays upon the wall; the intersections e, e, e of those rays with the perpendiculars drawn from the plan will define the shadow. If the beam be inclined, either on the plan or elevation, at any angle other than a right angle, the difference in the manner of proceeding can be seen by reference to the preceding examples of inclined shelves, etc.
From the point a (Fig. 437) in the plan, and b in the elevation, draw the rays ac and be; from c erect the perpendicular ce, and from e draw the horizontal line ed; then the lines ce and ed will show the extent of the shadow. This applies only where the back of the recess is parallel with the face of the wall.
In Fig. 438, A shows the section and B the elevation of a recess of this kind. From b, and from any other point in the line b a, as a, draw the rays bc and ae; from c, a, and e draw the horizontal lines eg, af, and e h; from d and f cast the rays di and fh; from i, through h, draw is; then s i and ig will define the shadow.
577. - To Find the Shadow in a Fireplace, - From a and b (Fig. 439) cast the rays a c and b e, and from c erect the perpendicular ce; from e draw the horizontal line eo, and join o and d; then ce, eo and od will give the extent of the shadow.
579 - To Find the Shadow of a Moulded Window-Lintel. - Cast rays from the projections a, o, etc., in the plan (Fig. 440), and d, e, etc., in the elevation, and draw the usual perpendiculars intersecting the rays at i, i, and i; these intersections connected, and horizontal lines drawn from them, will define the shadow. The shadow on the face of the lintel is found by casting a ray back from i to s, and drawing the horizontal line s n.
From a (Fig. 441) and its corresponding point c, cast the rays a b and cd, and from b erect the perpendicular bd; tan-gical to the curve at e cast the ray ef, and from c drop the perpendicular e o, meeting the mitre-line ag in o; cast a ray from o to i, and from i erect the perpendicular if; from h draw the ray h k; from f to d and from d to k trace the curve as shown in the figure; from k and h draw the horizontal lines kn and hs; then the limit of the shadow will be completed.