The natural way to place an object to be shown by projections would be in the simplest position; that is, with an edge or face parallel to either the horizontal or vertical plane of projection. Sometimes it is necessary, however, to draw the views of an object in a position at an angle to the planes. In such case it is usually advisable to draw the object parallel to one of the planes, and then rotate it to the required position about an axis perpendicular to a plane of projection.
When an object is rotated in this way, about an axis perpendicular to a plane, its projection on that plane will remain unchanged in size and shape, and the dimensions parallel to this axis on the other planes will remain the same.
Fig. 125. Plan, Front, and Side Views of a Square Pyramid.
In Fig. 125, the plan, front, and side views of a pyramid are shown, and in Fig. 126 is shown the same pyramid after it has been rotated through 30 degrees about an axis perpendicular to the horizontal plane. The height of the pyramid has not been altered by this rotation and, therefore, the front and side views are the same height as in the original front view.
Now, if the pyramid in Fig. 125 is rotated about an axis perpendicular to the vertical plane, the front view will not be altered, and may be copied in the new position at an angle of 30 degrees, as shown in Fig. 127. The distance above the ground line to any point in the top view are not altered, and the distances of the various points can be taken on the lines projected up from the points of the front view with a pair of dividers, or the points can be obtained by projecting across from the original top view to meet the projection lines drawn up from the front view. The side view dimensions are not altered, and this view can therefore be obtained in the usual way, by projecting across from the front view, and revolving down from the plane at right angles to the horizontal and vertical planes the points projected across from the top view.
Fig. 126. Flan, Front and Bide Views after Revolving Pyramid in Fig. 125. through 30 Degrees with Vertical Plane.
As shown in Fig. 128, first draw the plan, a circle, at A. Then draw the rectangle at B, representing the front view. Now, draw the rectangle at C, representing the front view at the desired angle. This rectangle C is the same size as the view at B, since the cylinder has simply been inclined to the horizontal plane, but kept parallel to the vertical plane. The point D, the center of the circle forming the base of the cylinder, is projected up to the point E, and with this point as a center, a circle representing the plan view of the base is drawn. Then from F project up to G, and with this point as a center draw the circle representing the plan view of the top of the cylinder. Connecting these two circles with horizontal lines HI and JK, representing the sides of the cylinder, completes the plan view, and the problem is finished.
Fig. 127. Plan, Front, and Side Views after Revolving Pyramid in Fig. 125 through 30 Degrees with Horizontal Plane.
Fig. 128. Projections of Cylinder Inclined to Horizontal Plane,.
Fig. 129. Method of Finding the Projection, in the Form of an Ellipse, of the Top of a Cylinder Greatly Inclined to a Plane.
As the cylinder is at an angle with the horizontal plane, it will be seen that the top and bottom of the cylinder in the plan view are not circles, but ellipses. It is, however, customary to draw them with the compass, as circles, when the angle of the cylinder with the plane is not great.
In Fig. 129 the plan and front elevation of the top of the cylinder are drawn at the desired angle with the horizontal plane at A and B, respectively. The plan view at A is then transferred to C. In each of these plan views divide the lower semicircle into a number of equal parts, eight in this case. From the view of A, project the points 0 - 8, parallel to the center line, down to E F, and then project across to the projection lines drawn vertically down from the points 0 - 8 in C. The points of intersection of projection lines, correspondingly numbered, form the shape of the ellipse representing the top of the side view of the inclined cylinder D, and the ellipse drawn through these points completes this view. The side lines of the cylinder may now be drawn, and the curve representing the bottom of the side view may easily be copied from the lower half of the ellipse representing the top view. When the points have been located, the ellipses may be drawn through them with the aid of an irregular curve.