CONSTRUCTION OF SHADOWS IN AXONOMETRIC PROJECTIONS USING THE LS-2 UNIVEROGRAPH

L. S. SKRIPOV

(Presented by the scientific seminar of the Department of Descriptive Geometry and Graphics)

Introduction

The present work is a continuation of the previous one, entitled “The LS-2 Univerograph,” and investigates the possibility of constructing, in axonometry, proper and cast shadows by a mechanical method using the “LS-2 Univerograph” instrument.

The purpose of the work is to continue research into the creation of designs for such instruments, which could perform various drafting tasks more simply, more quickly, and more cheaply than the manual method, using generally known drafting instruments.

The paper shows various examples of the application of the LS-2 Univerograph for constructing shadows in axonometry, providing yet another property of the instrument in addition to those described earlier, namely:

1) construction of perspective images from two orthogonal ones,
2) construction of proper and cast shadows on the orthogonal projections of an object,
3) construction of axonometric images of any kind from two orthogonal projections of an object,
4) use of the instrument as an ordinary drafting machine.

Construction of Shadows of a Right Prism

The use of the LS-2 Univerograph for constructing shadows in axonometric projections can at first be considered most simply on the image of a right prism (Fig. 1).

The drawing shows the construction of shadows with ordinary drafting instruments according to the laws of descriptive geometry. The prism is given in rectangular dimetric projection, and the rays of light are specified by the direction \(S(s)\).

The construction of the shadow, for example, from the edge \(AA_1\), is carried out by drawing the line \(Aa_0\) parallel to the ray \(S\) and the line \(A_1a_0\) parallel to the projection of the ray \(s\). The line \(A_1a_0\) will be the shadow of the prism edge \(AA_1\). In the same way the remaining points of intersection of the light rays with the planes \(H\) and \(V\) are found, namely \(b_0\), \(b_1\), \(c_1\). The contour of the cast shadow is hatched.

To construct the shadow on the drawing it is necessary to draw auxiliary lines parallel to the ray \(S\) and to its projection \(s\) on the plane \(H\), for example, \(B_1b_x\), \(Bb\), \(b_xb_1\), etc., which are then to be removed.

The same construction of the shadow is performed with the aid of the LS-2 Univerograph in Fig. 2. Figure 3 shows the scheme for setting up the instrument in this case. Rod 5 is moved along frame 1 to the right; rods 8 are set on it in the most convenient way for constructing the shadow and in such—

Fig. 1.

Fig. 2.

Fig. 3.

positions, are rigidly fastened to rod 5. Along rods 8, rulers 24 can move freely; their directions are parallel to the light ray and to its projection \(S\). The intersection of the two rulers gives the required shadow of some point, for example, \(a_0\) from the vertex of the prism \(A\).

By gradually moving rulers 24, all the necessary shadows of points are found, for example, \(b_1, c_1, b_0\), which are then connected, already with the aid of an ordinary ruler, by straight lines in order to construct the outline of the cast shadow. The drawing is freed from superfluous construction lines, such as, for example, the lines \(Bb_1, Cc_1\), which, of course, speeds up the work.

In the example given, the shadow from the edge of the prism \(AA_1\) can be constructed at once as the line \(A_1a_0\) along ruler 24 of the instrument.

The instrument is usually placed on a box with a glass cover and with a grid of lines on it, as was described in the previous paper “Univerograph LS-2,” which makes it unnecessary to draw vertical lines on the drawing, such as \(b_x b_1, c_x c_1\).

Only the rods or rulers necessary for constructing shadows are left on the instrument; all the rest are removed.

Construction of the Shadows of an Arch

Figure 4 shows the construction of the shadow of an arch for a given direction of the light rays \(S(s)\), different from the direction in the preceding example, in order to show the possibility of using the LS-2 Univerograph for any cases of illumination.

Fig. 4

In Fig. 4, the shadows of points 1, 2, 3, 4, 5, 6, 7, 8, taken on the arc of the arch, as well as the points \(A, B, C\), are constructed by the generally accepted method.

Finding the points \(a_0, b_0, 1_0, 2_0, 3_0, 4_0, 5_0, 6_0, 7_0, 8_0\) is clear from the drawing.

In Fig. 5 the construction of the shadows from the same arch is repeated, but with the aid of the LS-2 Univerograph. The arrangement diagram for it is shown in Fig. 6,

Fig. 5

where both rods 8 are joined at one point \(P\); however, this arrangement is not unique, and another may be selected depending on the convenience of the placement of the rods and the direction of the light rays.

In Fig. 5 the finding of the shadow from point \(C\) is shown. Rulers 24 are set parallel to the direction of the light ray \(S\) and its projection \(s\).

To construct the shadows of points \(1, 2, 3, 4, 5, 6, 7, 8\), it is necessary first to draw in the drawing the line \(DC\), on which the projections of the above-listed points are located.

This drawing also shows the absence of auxiliary lines as compared with Fig. 4.

Fig. 6

Construction of the Shadows of a Cylinder

In Fig. 7 the construction of the shadow of a hollow cylinder is presented in isometric projection, with a quarter cut away in order to make it possible to construct the internal shadow.

The direction of the light rays \(S(s)\) is given.

Fig. 7

Fig. 8

The construction of the cast shadow has been carried out through the points \(A, B, C\) and the intermediate points \(1, 2, 3, 4, 5\), according to the general rules for such constructions. The points \(C\) and \(C_1\) lie on a generator of the cylinder on lines tangent to the upper and lower bases of the cylinder, drawn parallel to the projection \(s\) of the ray \(S\). The inner cast shadow \(6_0 7_0 8_0 9_0 10_0\) is constructed through the points \(6, 7, 8, 9, 10\), taken on the inner cylinder, and through their projections onto the horizontal plane, located at a distance equal to the height of the cylinder from the selected points.

Fig. 9

Figure 8 shows the construction of the shadows of such a cylinder with the aid of the LS-2 Univerograph, the arrangement of whose rods and rulers is shown in Fig. 9. The direction of the ray is taken from left to right in order to demonstrate the possibility of using the instrument in this case as well.

To construct the shadow, for example, of point \(B\), rulers 24 are placed parallel to the ray \(S\) and \(s\). The intersection of rulers 8 gives the required point \(b_0\).

To construct the cast shadow from the cylinder, it is only necessary first to find on the drawing the horizontal projections of the selected points \(1, 2, D, 3, 4, 5\), and likewise of the points \(6, 7, 8, 9, 10\) on the inner cylinder.

In Fig. 8 the auxiliary constructions that are unnecessary when the instrument is used are also absent.

Construction of the Shadows of a Bracket

Figure 10 gives the construction of the shadows of a complex spatial figure—a bracket—in order to check the possibility of using the LS-2 Univerograph also in such cases of axonometric projections.

The construction of all points was carried out according to the rules of descriptive geometry for a given direction of the ray \(S(s)\). In Fig. 10 one can see the construction of each point of the figure by drawing lines parallel to the given direction of the rays.

Figure 11 gives a diagram of the positions of the rods and rulers of the instrument as one of the possible installation variants.

Rod 5 can move along frame 1 to the right for constructing the shadows of points located on the right-hand side of the drawing.

Figure 12 shows the same construction of shadows with the aid of the LS-2 Univerograph; here there are no longer any auxiliary lines, and the drawing is cleaner than in the usual construction (Fig. 10), not to mention the shorter time required for constructing the shadows of such a complex part.

Figure 13 shows the arrangement of the rods and rulers of the instrument for constructing the shadow of point 1.

The examples given are already sufficient to become convinced of the advantage of using the described mechanism instead of a T-square and set square for the speed and cleanliness of executing the drawing, especially for complex constructions.

Fig. 10

Fig. 11

Fig. 12

Fig. 13