From the point as the shadow of the line repro duces the profile ABCD and we obtain as nsosb5, the required shadow.
94. Problem XXVII. To construct the shad ow on the intrados of a circular arch in section, the plane of the arch being in profile projec tion. .
Let AL (Fig. 48) be the "springing line" of the arch. Let CD be the radius of the curve, The point F is determined by the Construction used in finding the shade element of a cylinder. Problem XXIII. At the point F draw the 11110 with an inclination to the "horizontal" of 1 in 2. Through the point D draw the 45° line DB. The curve of the line of shadow will be tangent to these two lines at the points F and B. The required shadow is that portion of the curve be. tween the lines DC and MN.
A similar construction is used in the case of a hollow semi. cylinder when its axis is vertical, except, that the line GH has then an inclination to "the horizontal" of 2 in 1. Fig. 49.
95. Problem XXVIII. To construct the shadow of a spheri cal hollow with the plane of its face parallel to either of the co-ordinate planes.
The line of shadow is a semi-ellipse.' The projections of the rays of light tangent to the circle determine the major axis. The semi minor axis is equal to A the radius of the circle.
Given the vertical projection of a spherical hollow, the plane of its face parallel to V. Fig. 50.
. Determine the ends of the major axis by drawing the pro jections of the rays of light tangent to the hollow. The semi minor axis, oa, equals A the radius ob. On be and oa.construct the semi-ellipse, the required shadow.
96. Problem XXIX. To con struct the shade line and shadow of a sphere. Fig. 51.
Let the circle whose center is o be the vertical projection of a sphere whose center is at a dis. • tance a from the V plane.
The shade line will be an ellipse.
The major axis of this ellipse is de termined by the projections of the rays of light tangent to the circle. The semi minor axis and two other points can be deter mined as follows : .
Through the points, A, o, and B, draw vertical and horizon tal lines, intersecting in the points E and D.
The points E and D are two points in the required shade,line. Through the point E draw the 45° line EF. Through the point F, where this line intersects the circle, and the point B, draw the line FB. The point C,.where this line FB intersects the 45'
line through the center of the sphere, o, is the end of the semi minor axis. The shadow of the sphere on the co-ordinate plane will also be an ellipse. The center of this ellipse, 0, will be the shadow of the center of the sphere. It will be determined Problem XVII. The ends of the major axis MN, will be on the projection of the ray of light drawn through the center of the sphere. The minor axis PR will be a line at right angles to this through the point 0. Its length will be determined by the projections of the rays of light BR and AP tangent to the circle, and is equal to the diameter of the sphere. The points Di and N, which determine the ends of the major axis, are the apexes of equilateral triangles PMR and PNR, constructed on the minor axis as a base.
97. Problem XXX. To construct the shade line of a torus. Fig. 52, in elevation: The points 1 and 5 can be determined by drawing the projections of the rays of light tangent to the vation. Since the shade line is symmetrical on either side of the line MN in plan, the points 3 and 7 can be found from 1 and 5, by drawing horizontal to the axis. The points 4 and 8 are determined by the construction used in finding the shade elements of a cylinder. Problem XXIII.
The above points can be determined without the use of plan.
The highest and lowest points in the shade line, 2 and 6, can be found only by use of plan. It is not necessary, as a rule, to determine accurately points 2 and 6. The shade line in plan will be, approximately, an ellipse whose center is o. The ends of the major axis R and S, are determined by the projections of the rays of light tangent to the circle. Other points can be determined without the use of the elevation as follows: With center o, con struct the plan of a sphere whose diameter equals that of the circle which generated the torus. Determine the shade line by Problem XXIX. Draw any number of radii OE, OF, OG, etc.
On these radii, from the points where they intersect the shade line of the sphere, lay off the distance RT, giving the points f and g. These are points on the required shade line.