P. A. Maslennikov

Projection onto Three Planes:

The method of changing the plane of projection for determining the true dimensions of distorted forms and for obtaining additional views of projected bodies, in the form in which it is set forth in the existing literature, gives auxiliary views by rotation until the planes of projection coincide, about the axis of projection obtained as a result of the intersection of the new plane of projection with the remaining old plane of projection.

Such a method, while fully solving the problems indicated above, is not entirely convenient when comparing the new projections obtained with the remaining old ones, in order to reconstruct in the imagination, from all the projections, the projected body. Thus, for example, in the case of changing the vertical plane of projection, the new vertical projection of the projected body may turn out below the remaining horizontal one, i.e., as if upside down; consequently, in order to reconstruct the projected body in the imagination by comparing the old vertical projection of the body with its new vertical projection, one has to rotate the drawing, which distracts the attention, since in most cases we are accustomed to considering the horizontal projection of the projected body beneath its vertical projection. Among the inconveniences of using this method one may also include the dimensions of the drawing, which sometimes do not allow this method to be used, since the new projections given by this method extend beyond the limits of the drawing.

The above-mentioned inconveniences will be eliminated if, in order to obtain an auxiliary view when solving the above problems, the introduced plane of projection is rotated not about the axis formed by the intersection of the new plane of projection with the remaining old one, but about the axis formed by the intersection of the new plane of projection with the plane of projection being changed; that is, if \(V\) and \(H\) (see Drawing 1-a) are the principal vertical and horizon-

Drawing 1-a.              Drawing 1-b.

projection plane, and \(V_1\) is the newly introduced vertical projection plane, and if \(OX\) is the axis of projections of the system of planes \(HV\), \(OY_1\) the axis of the system \(HV_1\), and \(OZ\) the axis of the system \(VV_1\), then, in order to obtain the diagram expressing the change of the vertical projection plane, we cut the trihedral angle formed by the planes \(V\), \(V_1\), and \(H\) along the line \(OY_1\), and about the lines \(OX\) and \(OZ\) rotate the planes until they coincide with one of them.

The proposed method resembles the existing method of projection (see Drawing 2-a) onto three mutually perpendicular planes \(H\), \(V\), and \(P\), where \(H\) is the horizontal, \(V\) the vertical, and \(P\) the profile projection plane, and the lines \(OX\), \(OY\), and

Drawing 2-a.

Drawing 2-b.

\(OZ\) are the projection axes. Obviously, projection onto “three mutually perpendicular” projection planes \(V\), \(H\), and \(P\) will be a special case of the proposed method of projection onto the three projection planes \(V\), \(H\), and \(V_1\).

If it were necessary to change the horizontal projection plane \(H\), i.e. to introduce (see Drawing 3-a) a new plane \(W\), inclined to \(H\), then in order to

Drawing 3-a.

Drawing 3-b.

To obtain the epure we proceed as follows: the trihedral angle formed by the planes \(V\), \(H\), and \(W\) is cut along the line \(OZ^1\) and along the lines \(OY\) and \(OX\); we rotate the projection planes until they coincide with one of them.

In the appended drawings (1, 2, and 3) the construction of the epure of a point is shown for the given cases.

In a number of problems of descriptive geometry there are cases in which it is necessary to obtain such a projection as results from changing both the vertical and the horizontal projection planes.

The method of changing the projection planes solves this problem by a successive change of the projection planes (an indirect solution), i.e., first one projection plane is changed—for example, the horizontal plane \(H\) to the plane \(W\)—and then the vertical plane \(V\) to the plane \(V_1\); in this case the result of solving the problem is obtained on the new plane \(V_1\), while the plane \(W\) is only a means for obtaining this result.

Thus, when solving these problems by the method of changing projection planes, one must first carry out constructions on the auxiliary projection plane \(W\).

Since the position of the plane giving the final solution of the problem when two projection planes are changed, relative to the basic projection planes \(V\) and \(H\), is usually known, one can again make use of special projection in a system of three projection planes, without resorting to intermediate constructions, i.e., obtain a direct solution of the problems.

Let the solution of such a problem lie on a certain plane \(M\), for which the line of intersection \(OY_1^1\) with the horizontal projection plane \(H\) (see Drawing 4) makes, in the horizontal plane, an angle \(\alpha\) with the line \(OX\), while the line of intersection \(OZ_2^1\) with the vertical projection plane \(V\) makes, in the vertical plane, an angle \(\beta\) with the same line \(OX\), which is the line of intersection of the projection planes \(V\) and \(H\).

For obtaining the epure of such planes, we cut the trihedral angle formed by them along the lines \(OZ_2^1\) and \(OH_1^1\) and bring the planes \(H\) and \(M\) into coincidence with the plane \(V\), which lies in the plane of the drawing.

To bring the plane \(H\) into coincidence with \(V\), we rotate \(H\) by \(90^\circ\) about the axis \(OX\); thus on the plane of the drawing we obtain the horizontal line \(OX\), with the plane \(V\) lying above it and the plane \(H\) below it.

Drawing 4-a.

To bring the plane \(M\) into coincidence with the plane of the drawing, by rotating the plane \(M\) about the vertical axis \(OZ\) through the angle \(\alpha\), we bring it into the position-

position \(M_1\), corresponding to the plane passing through the axis \(OX \equiv OX^1\), and by rotation about the axis \(OY^1\) bring it into the position \(M_0\), corresponding to the position coincident with the plane of the drawing.

When the plane \(M\) is rotated about the vertical axis \(OZ\) through an angle \(\alpha\), the point \(Z^1_2\), taken on the vertical trace of the plane \(M\) (Fig. 4-a), moves in space along a circle whose plane is parallel to the horizontal plane \(H\), into the position \(Z^1_1\); it is obvious that the horizontal projection of this point \(Z^1_2\), which was previously on the axis \(OX\) at the point \(Z^0_2\), will likewise move along a circle through the same angle \(\alpha\), into the position \(Z^0_1\). The point \(Y^1_1\), taken on the horizontal trace of the plane \(M\), will occupy, as is not difficult to see from Fig. 4-a, the position of the point \(Y^1\), lying on the extension of the axis \(XO\).

Fig. 4-b.

When the plane \(M\) is rotated about the axis \(OY^1\) from the position \(M_1\) to \(M_0\), for complete coincidence with the plane \(V\), the selected point \(Y^1_1\) on the horizontal trace of the plane \(M\), as a point lying on the axis of rotation, will remain in the position \(Y^1_1\), while the point \(Z^1_1\) on the vertical trace will move from the position \(Z^1_1\) along a circle whose plane is perpendicular to the principal planes \(H\) and \(V\), into the position of the point \(Z^1\) in the vertical plane \(V\), at a distance from the point \(O\) equal to \(OZ^1_1\), and lying on the same perpendicular to the axis \(OX\) as the horizontal projection, which during this rotation passes along the perpendicular to the axis \(OX\) from the position \(Z^0_1\) to the \(X\)-axis.

On the epure, the superposition of these planes must be carried out in the following way. Having constructed the epure of the principal planes \(V\) and \(H\), according to the rules set forth in descriptive geometry (Fig. 4-b), we draw, at an arbitrarily chosen point \(O\) on the line \(OX\), in the vertical plane \(V\) the line \(OZ^1_2\) at an angle \(\beta\) to the line \(OX\), and in the horizontal plane of projections \(H\) the line \(OY^1_1\) at an angle \(\alpha\) to the line \(XO\); the point \(Y^1_1\) and the horizontal projection \(Z^0_2\) of the point \(Z^1_2\), taken on the vertical trace of the plane \(M\), we rotate through an angle \(\alpha\); the point \(Y^1_1\) will occupy the position of the point \(Y^1\), and the point \(Z^0_2\)—the position of the point \(Z^0_1\); from the point \(Z^0_1\) we draw a perpendicular to the line \(OX\) until it intersects, in the plane \(V\), the arc of a circle with center \(O\) and radius \(OZ^1_2\) at the point \(Z^1\); by joining the points \(Y^1\) and \(Z^1\) with \(O\), we obtain the line of the plane \(M\) that bounds the position coincident with the plane of the drawing.

The construction, on the epure of the planes \(V\), \(H\), and \(M\) thus obtained, of the third projection of a point from two given projections is carried out, as is evident from the drawing, with some difference from the usual rule for obtaining the third projection on three mutually perpendicular planes \(V\), \(H\), and \(P\); this difference consists in a double rotation of the projecting directions to the corresponding axes of projection, since the epure of the projection planes of the form \(V\), \(H\), and \(M\) has two breaks.

Since the plane \(M\) is introduced chiefly for the solution of problems

for determining the true magnitude of distorted forms; instead of changing the projection planes twice, then, obviously, in most cases this plane will have to be placed either parallel or perpendicular to the principal lines of the projected figure (horizontal, line of greatest slope, etc.); and therefore, unlike the other projection planes, we shall call the plane \(M\) the principal projection plane of the figure.

To illustrate the convenience of applying projection onto three projection planes of the kind \(V\), \(H\), and \(M\), let us solve the following problems, often encountered in courses of descriptive geometry.

1st problem. A line \(AB\) is given (see Fig. 5) by its projections on the principal projection planes \(V\) and \(H\). It is required to introduce such a projection plane \(M\) that the line \(AB\) is projected onto it as a point.

Since a line is projected as a point only onto a plane perpendicular to it, the traces of this plane on the principal planes of projection must therefore be perpendicular to the like-named projections of this line.

Fig. 5.

projection planes must be perpendicular to the like-named projections of this line.

Having drawn from an arbitrary point \(O\) on the axis of the principal projection planes \(OX\) the lines \(OY_1\) and \(OZ_1\), perpendicular to the projections \((ab)\) and \((a^1b^1)\) of the line, we move them, according to the rule stated above, into the position \(OY\) and \(OZ'\); after this we draw the continuations of the projections of the segment of the line until they intersect on the epure with the lines \(OZ_1\) and \(OY_1\); the intersection points thus obtained are transferred through the break and by arcs of circles of the corresponding radii onto the lines \(OZ'\) and \(OY\); from the latter to the lines \(OZ^1\) and \(OY^1\) we restore perpendiculars up to their mutual intersection, which gives the required point \(a^0 b^0\).

2nd problem. Determine the natural magnitude of the area of the quadrilateral \(ABCD\) (Fig. 6), given by its projections on the principal projection planes \(V\) and \(H\).

To solve this problem, the principal plane of projection of the figure must be arranged so that its plane is parallel to the plane of the projected quadrilateral, and for this purpose, in the plane of the quadrilateral we draw, according to the general rules of descriptive geometry, the horizontal line \(k^1 l^1,\ kl\); from point \(a\) we drop to the line \(kl\) the perpendicular \(an\), and transfer it as \(a^1 n^1\) onto the vertical projection plane \(V\). The horizontal line \((k^1 l^1,\ kl)\), lying in the plane of the quadrilateral, and the line of greatest slope of the quadrilateral to \(H\) perpendicular to it \((a^1 n^1,\ an)\), determine the position of the traces of the principal plane \(Oy_1\) and \(OZ_1\) on the principal projection planes, namely: \(Oy_1\) must be parallel to the line \(kl\), and \(OZ_1\) parallel to the line \(a^1 n^1\).

Drawing 6.

Having transferred the lines \(Oy_1\) and \(OZ_1\) into the position \(OY^1\) and \(OZ^1\), we obtain on the epure the angle \(Z^1 OY^1\), the plane of which will represent the principal plane of projection of the figure, and, finally, according to the rule set out above, we project all points of the quadrilateral onto the plane \(M\). The figure \(a^0 b^0 c^0 d^0\) obtained on the plane \(M\) will represent the true size of the quadrilateral \(ABCD\).

As can be seen from the examples given, the method of introducing the principal projection plane of a figure, while simple in construction, completely replaces the method of double replacement of projection planes, i.e. it eliminates cumbersome constructions, gives a direct solution to a large number of problems, reduces the time spent on solving them, and possesses greater accuracy of the results of graphical constructions in comparison with the successive replacement of projection planes.