Line And Plane Relationships Pt1 (1961)

In the combinations of lines and planes, we have innumerable examples of pure space relationships. The study of these problems presents opportunities for judging how to view these arrangements to see true values. Here we can exercise mentally and visually the basic controlling principles always present in the analysis and synthesis of these elements of form. It is intended that this presentation will promote a sense of feeling for the graphic manipulation which is important in the consideration of space problems. From the given points, numerous lines appear on the plane surface. Notice that the orange lines are the only parallel lines of the group. All the lines lie flat on the plain surface and thus show here in their true length. That they lie flat on the surface is evident when the surface is tipped to see it as an edge. Here the forcehortening now becomes apparent. Now when the lines are flipped normal to the plain surface, obviously all lines become parallel. This parallelism shows from all points of view although the observed lengths may vary between their maximum limit or true lengths and their minimum limit or as points. If we return to the two parallel lines lying on the plane and tip them upward, they look shorter than their true length but still appear parallel. But a view three taken normal to the lines indicates that they are no longer parallel and that their true lengths will show in this view and also true slopes since this is an elevation view. In summation, as illustrated with the edges of the book, parallel lines always show parallel when viewed from any and all directions, although their lengths may appear to range from their maximum or true lengths to their minimum or point views. Unlike the relationships for parallel lines, lines which are perpendicular to each other do not show their right angle relationship when viewed from all directions. When a line is chosen perpendicular to the book in both picture and in projection then all lines lying in the plane of the book or parallel to it are perpendicular to the given line. And also a line need not necessarily intersect another line in order to be considered at right angles to it. In other words, lines may be at right angles to each other without intersecting. Note that the line in front and the line behind are parallel to the plane of the book but do not lie in this plane. Only the white line and the given line intersect at 90°. The yellow lines have a perpendicular relationship but do not intersect the given line. The right angle relationship between two lines becomes apparent when either one or both of the lines show in their true lengths. When a right angle triangle assumes the position indicated, the top view shows only the longer side in true length and the right angle in true size. As the triangle is rotated to the horizontal, both sides show in true length in the top view, and the right angle still appears in true size. The top and front views no longer show the perpendicular relationship. However, in a side view where the longer side shows in true length, the right angle again appears. As the triangle is rotated to a vertical position, the triangle appears as an edge in both the top and front views while the side view shows the two edges in true length. Thus, the right angle appears in true size. So far we have employed only principal views to display the right angle relationship. The general case however may require the use of auxiliary views or rotation to show the plane as an edge and finally in true shape. A horizontal line on the triangle when viewed from the end will show the triangular plane as an edge. A view taken perpendicular to the plane will show the true length of the bounding lines and the angular relationship between them. We have here changed the position of the object by rotating it so that you may see the true shape, true lengths of the sides and the true size of the included angle. The same result can be achieved by changing the viewpoint instead of the position of the object. Then this top view would be identical with an auxiliary view 5. The shortest distance between a point and a line in space shown here with their projections on the base is the true length of the perpendicular from the point to the line. Any number of connecting lines may be drawn from the point to the given line. But the 90° relationship is evident only when either the shortest connecting line or the given line appears in true length. From the previous treatment to see the 90° relationship for determining the connecting line, we repeat either the given line or the connecting line must show in true length. To see the given line in true length, we may look perpendicular at a vertical plane which contains the line so as to see the plane in true shape. View three. In view three, a line from the point drawn perpendicular to the given line which shows in true length locates the point on the line. Although the given line is now in true length, the connecting line is not but shows shorter. However, when the given line is shown as a point, the connecting distance will appear true and is the shortest distance between the point and the line. Another solution is to see both the given line and the connecting line in true length in the same view. To do this, form a plane containing the point and the line. View the plane to see it in true shape and draw from the given point perpendicular to the given line. The perpendicular from the point to the line locates the point X on the line. Now let us consider the shortest connection between two oblique lines. The location and length of the shortest distance between two lines is based on the concept that this shortest distance must be the common perpendicular between the two lines. An infinite number of lines may be constructed perpendicular to the blue line. Likewise, lines perpendicular to the orange line may be constructed. The line common to both groups of perpendiculars will be the shortest connection between the orange and blue lines. The procedure to see this common line in true length follows. All the numerous lines perpendicular to the blue line will show at right angles to it when this line appears in true length. A view of this same blue line as a point will show all the perpendiculars in their true lengths. One of these perpendiculars will make the required 90° connection with the orange line. Although the orange line is not shown in true length in this view, one of the innumerable lines perpendicular to it will coincide with one of the lines perpendicular to the blue line. Because this shortest connection appears in true length, the right angle relationship between it and the orange line shows in this particular view of the two original lines. The point Y is now fixed and the point X may be located in the view where the blue line appears in true length. The shortest connection between two oblique lines can also be found by a plane method. This method in principle also applies to connectors having specified slopes other than the right angle connection. An auxiliary line is taken through one of the given lines parallel to the other given line. A plane is thus formed containing the one line parallel to the other line. When perpendiculars are dropped from the line to the plane, in other words, when the line is projected on the plane, the projection intersects the lower given line at a fixed point. From this fixed point, a perpendicular to the plane is erected and this perpendicular will intersect the upper line. This connecting line is the common perpendicular between the given lines and thus is the shortest connection between them. To sum up, the concepts expressed are related principally to the parallel and right angle relationships between lines as they are visualized in space and projection. Planes are involved in these solutions to aid in assimilating these two concepts. Lines which are parallel in space will show parallel in all views. The exception is when they show as points. The perpendicular relationship between two lines show the 90° angle when either one or both of the lines appear in true length. One application of this perpendicular relationship is presented in obtaining the shortest distance from a point to a line. The shortest connection between two oblique lines by the line method and the plane method presents another application of the principles of perpendicular relationship. The consideration of these numerous line positions should help to develop a more realistic understanding of the basic concepts of spatial relationships.