Revolutions (1961)

The principle of revolution consists of the rotation of a point or points about an axis. In revolution, the observer maintains a fixed position while the object changes position. This is opposite to the auxiliary view procedure. Based upon this principle of axis controlled rotation, we find in moving bodies many applications. These uses range from helping us to see true values in ordinary experience to such considerations as the transmission of power and balance and the complex problems involving outer space. The rotation of a point about an axis is basic to all problems of revolution. When the axis of revolution appears as a point, the path of revolution shows as a circle. Where the axis appears in true length, the path of revolution shows as a line. As the point revolves about its axis, it maintains a constant distance from the axis and remains in a plane perpendicular to the axis. When the axis is tipped, the top and front views no longer show these relationships. However, when a view three is added, showing the axis in true length and when a view 4 is added, showing the axis as a point, these fundamental relationships between the revolving point and its axis again appear. The behavior of a line revolved about an axis is observed by assigning points to the line. When the line intersects the axis, each of the end points traces its own circular path as shown in the top view. And these planar paths maintain the 90° relationship to the axis as shown in the front view. Note the pattern formed when the line does not intersect the axis. This revolution no longer forms a cone of two naps but a hyperbooid of revolution. All points on this revolving line also generate circular paths as we see here in the orthographic views. The rotation of a plain surface follows a similar procedure. For example, starting with a vertical triangle, revolve it to a horizontal position about a horizontal axis on its surface. The axis here chosen through one corner of the triangle shows as a point when viewed from direction three. In the revolving procedure, this corner of the triangle on the axis is fixed and will not move. The other two corners of the triangle will travel in circular arcs. Observe in picture how the triangle rotates about the axis. As this movement occurs, the corner points move at right angles to the axis in the top and front views and the arcs generated show in true shape in view three where the axis appears as a point. When the movement is completed, the top view shows the triangle in true shape with its front view as an edge coinciding with the horizontal axis. Here is another application for rotation of a plain surface. Using an inclined triangle with a vertical axis located as shown, revolve the triangle about the axis until it appears as an edge in the front view. This procedure begins by assuming a horizontal line ax on the triangle. Revolve this line about the axis until the line shows as a point in the front view. Then the triangle also will show as an edge in the front view. You will note in this example that the axis is vertical and the line on the triangle is horizontal. If desired to see the triangle as an edge in the top view, use a horizontal axis and a frontal line on the plane. We may also apply the principles of revolution to measure the true size of the dihedral angle between two planes. The measuring plane passed perpendicular to the line of intersection of the two planes cuts the lines XY and XZ from the given planes. When the measuring plane is revolved about an axis containing points Y and Z. The true size of the angle between XY and XZ will be seen in the top view. This is the dihedral angle between the two planes. Orthographically, the axis of revolution may be seen as a point if viewed from direction three. In this view, the points Y and Z located on the axis remain fixed and the point X revolves about the axis into the horizontal plane. When the measuring plane is in this revolve position, its true shape will appear in the top view. Revolution may also be used in a part of the solution to measure the angle a line makes with a plane. To see the required angle, the plane must show as an edge in the same view that the given line shows in true length. To affect this condition without changing the given relationship of line with plane, first choose a normal to the plane to be used as an axis of revolution. When the given line is revolved about this axis, it will generate a right cone whose base is parallel to the plane. All elements of the cone maintain a constant angle with its base and also with the given plane. A view three will show the plane as an edge and the right cone and plane relationship. View 4 shows the axis as a point and the position which the revolved line must take to show it in true length in view 3. Here the angle can be measured because the true length of the line and the edge view of the plane appear in the same view. Given the horizontal and vertical lines, show the location of a connecting line which makes a specified angle with the horizontal line and a specified angle with the vertical line. Two right cones having elements of equal length are used to determine the direction of a connecting line. Every element of the horizontal cone makes 45° with the horizontal line. Likewise, every element of the vertical cone makes the 60° angle with the vertical line. There are four possible answers to this problem. to affect the solution for the direction of one of these connectors. The horizontal line is moved to a parallel position intersecting the vertical line as shown. This makes a common vertex for the two cones. The elements generating these two cones will coincide in two positions. One of which is emphasized as line AO. Since AO is common to both cones, it makes the specified angle with each of the given lines. This direction line is shown in top and front views. Note that point A is one of two points where the base circles intersect. It is important to remember that both of these right cones have elements of equal length. To locate the connecting line XY, make it parallel to these views of the directed line AO. This completes the given projections and locates the X and Y points on the given lines. Pictorially X is located on the vertical line at an elevation H taken from the front projection view. The line XY is placed parallel to the intersection line AO of the cones. Then the position of Y on the horizontal line should check with its location in the projection top view. In summation, the basic principle in all revolution problems is the control of a point as it rotates about an axis. A line revolves about an axis with each point of the line moving in a circular path. The plane of this path always maintains a right angle relationship to the axis of revolution. A plane revolves about a line on its surface as an axis. Each point of the plane follows the path of a circle. When the axis of revolution is outside the plane, a suitable line is chosen on the plane and this line is revolved to its new position. The plane is then reconstructed on the chosen line. The dihedral angle between two planes is measured by revolving the measuring plane to a true shape view position. The measuring plane is chosen perpendicular to the line of intersection of the given planes. To see the true angle a line makes with a plane, revolve the line about an axis perpendicular to the plane until the given plane shows as an edge and the line shows in its true length. Right cones are used in determining the position of a connecting line making specified oblique angles with two given lines. Revolution not only provides a convenient method for the solution of problems but also develops a broader sense of feeling for the manipulation of moving lines, surfaces and objects in space.