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Use of Planes of Light Perpendicular to the Co-Ordinate Planes 79

line, shadow, plane, edge, intersection, profile, mouldings, parallel, sphere and projection


79. Another method often necessary and convenient in casting the shadows of double curved surfaces is the use of planes of lightperpendiettlar to the co-ordinate planes.

These auxiliary planes of light are passed through the given object. They will cut out lines of intersection with the object and to these lines of inter section can be applied the projections of the rays of light which lie in the auxiliary planes of light. The points of contact or tangency, as the case may be; of the projections of the rays and the line of intersection are points in the required shadow.

80. The use of this method will be illustrated by finding the shadow of a sphere in the following problem. The shadow of the sphere serves to illustrate this method well, but a more accur ate and convenient method is given later in Problem XXIX for determtning the shade line of the sphere and its shadow.

81. Problem XV. To find the shade line of a sphere.

In Fig. 33 is shown the plan and elevation of a sphere.

Through the sphere in plan, pass the auxiliary plane of light P, perpendicular to H. This cuts out of the sphere the "line of in tersection," shown in the elevation. This "line of intersection" is determined by using the auxiliary planes A, B, C, D, etc., each plane giving two points in the line. To this line of intersection made by the plane of light P, with the sphere, we apply the projections of the ray and obtain two points, xvyv, in the required shade line. Other points can be determined by using a number of these planes of light, as shown in Fig. 34, P, Q, R and S.

The points xv and yv can be projected to the plan to deter mine the shade line there. The ends of the major axis of the ellipse ce and bv are de termined by applying directly to the sphere the of the ray. The same is true of the plan.

82. *Problem XVI. To find the shadow of pediment Mouldings.

Fig. 35 shows a series of pediment mouldings in elevation, the mouldings being supposed to extend to the left and right indefinitely. At the left is a "Right Sec tion," showing. the profile of each moulding forming the pediment.

The shadow of such an object can be most conveniently found by the use of a plane of light perpendicular to the V plane and intersecting the mouldings.

If such a " Plane of Light" (45° line) as that shown in Fig. 35 is passed through the mouldings, it will be evident that this plane will cut the mouldings along a line of intersection which can be made use of in determining the shadow of each moulding upon the others". If we find the profile projection of this line of intersec tion using the right section,. we can apply the profile projections of rays of light to the line of intersection. It will then be evident what faces the light strikes directly and. to what edges the rays are tangent.

The line of intersection in Fig. 35 made by the Plane of Light is shown in vertical projection by the 45° line etc. The profile aPbPeP,c/P, etc., is the profile projection of this line of intersec tion; the point LP is evidently on a horizontal line to the left of the point a° at a distance from the line (profile projection of V) equal to a'b', obtained from the Right Section,. In the

same way the point c° is on a horizontal line to the left of cv and at a distance from the line ir" equal to the distance of' also ob tained from the Right Section. In a similar manner the other points in the profile projection are found. The vertical line bP is the profile projection of the line of intersection which the Plane of Light makes with the fillet, this line in direct elevation is bvc:.

If we now apply to this profile projection of the line of inter section the profile projections of the ray (45° lines) we see that the fillet bPcP is in the light, and that the ray is tangent to its lower edge ci'. We also see that this tangent ray strikes the face D at the point /P; this means that the shadow of the edge c falls upon the face D. Since the mouldings of the'pediment are all parallel to each other, the edge c is parallel to the face D, therefore, (30) the shadow of c on D will be parallel to C itself. This shadow is found in the elevation by drawing a horizontal line from the point 1P back to the Plane of Light. This operation gives the point 1° and we draw through the point 1° a line parallel to the edge C, as a part of the required shadow. Evidently that portion of the ele vation between the edge C and its shadow will be in shadow.

In a like manner the edge dP is found to cast its shadow on the plane V, below the pediment mouldings proper, and its shadow is of course a line drawn through 2° parallel to the lines of the mouldings.

To return to the shadow of the edge C on the face D. It will be noticed that, if this is extended far enough, it will cross the pediment mouldings on the right-hand slope; as these are not parallel to the edge C, the shadow on them will not be a parallel line and we must use a separate, though similar, method for deter mining this portion of the shadow.

If auxilialy planes 0, Q and .R parallel to V are passed through the crowning moulding, they will cut out of it lines of intersection which will be parallel to the other lines of the pedi ment. (See the enlarged diagram at A showing the line of inter section of the auxiliary plane 0.) If we cast the shadow of the edge C on this plane 0, by draw ing the 45° line from cP to the line PO (the profile projection of 0) and from the point 4P draw a horizontal line back to the Plane df _Livia, we shall obtain the line 0 (see "shadow on PO" in dia gram A). This shadow will cross the Line of intersection of PO at the point 5°. The point 5° will be one point in the shadow of the edge C (indefinitely extended) on the slope of the pediment. Other points, 8° and 9°, can be found in a like manner by use of the auxiliary planes Q and Rs Through a suffi cient number of these points the curve 5°9°8° is drawn. This curve is the required shadow. The shadow of the end of the edge C is found by drawing a 43° line from the point qn° (diagram A) to the curve. The point of intersection, 10y, is the shadow of the end of -the edge C. It is also the beginning of the shallow of the edge B on the right-hand slope, which shadow is parallel to B.

The remaining shadows of the pediment are found in the same manner, and may be understood, from the diagram, -without a detailed explanation.