Points And Lines (1961?)

A point is a unit of form. When points are aimlessly scattered, they hold no particular interest. It is only when we recognize and understand a pattern in their arrangement or consider their movement that these combinations of points challenge our minds. The changing patterns of such points as stars in the sky have always fired man's imagination. Although this film is not an attempt to search for the why of the movement of points, we will consider how the lines formed by such movement should be observed for us to see their various characteristics. A point may move through space in a completely random manner forming no pattern which would enable us to predict its future behavior. A point may also move through space in a controlled manner, generating line paths with precise mathematical characteristics. For example, the point generating this line path for an ellipse has precise predictable positions which may be computed from previous positions. Another line with precise mathematical characteristics is formed by controlling the generating point at a constant distance or radius from the origin. If we increase the radius of this circle an infinite amount, a short segment of this infinitely enlarged circle may be called a straight line. There are other theoretical lines which have mathematical equations. All these lines have predictable conformity and are generated by the mathematically controlled movement of the generating point. The development of lines with mathematical predictability is not limited to the movement of a generating point. Controlled lines may also be cut from theoretical surfaces by slicing methods. Lines lacking in precise predictability of new positions may be cut from surfaces not conforming to theory. For example, if this natural surface is cut by a series of planes, irregular lines will result. These lines are not mathematically predictable, although they may be plotted by coordinate methods. These irregular lines as well as numerous other theoretically controlled lines such as the parabola, hyperola and sine wave are called planer lines. In addition to planer lines, we will include in our consideration one example of a space line, namely the helix. To construct a helix, we may begin by drawing a straight line diagonally across a rectangular plane. When this plane is viewed from the top, the line appears shorter than its true length. When the plane containing the line is wrapped to form a cylinder, the line now appears as a circle. By rotating the cylinder to a vertical position, the line which was originally a planer line has by controlled bending become a space line called the helix. Several turns of such a helix when made of spring steel may be found in many applications as illustrated by the helical spring of an automobile wheel suspension. This baseline is a three-dimensional form where the locus of the point defining the line revolves uniformly around an axis and at the same time moves uniformly parallel to the axis. With this general introduction of line forming, we are now ready to learn the names and properties of straight lines and how they may be viewed to see true lengths, true slopes, and when they appear as points. You are already familiar with this two-dimensional system for planer points and lines. In a three-dimensional system, the axes may be replaced by reference planes and the images of the points and lines are then used. Consider points A and B and their image or projection on the horizontal image plane, front image plane and profile image plane. The names horizontal image plane, front image plane and profile image plane are assigned to the principal planes. To illustrate this naming and viewing of lines, we will consider the edge lines formed by the intersection of planes of a portion of this house. AB is a horizontal line and will be projected onto the horizontal image plane in true length. When viewed from the top, AB appears in true length. If AB is viewed from the front, it will again appear in true length. This is so because line AB is parallel to the front image plane. AB is called a frontal horizontal line. AB is projected onto the profile image plane as a point viewed from either side. We see AB as a point. BC is a vertical line. It is seen from the top or plan view as a point. When we view BC from the front, it appears vertical and in true length. If BC is viewed from the right side, it will again appear vertical and in true length. BC is a vertical line. B D is a sloping line. Viewed from the top, BD appears shorter. From the front, line BD appears in true length because it is perpendicular to our lines of sight and parallel to the front image plane. The angle BD makes with the horizontal is called the true slope. Observed from the right side, line BD appears shorter and vertical. If we assemble these three lines, you will observe that they all lie in the same plane. That is, the lines are parallel to a front image plane and appear as an edge when viewed from the side and the top. Any of these lines projected on the front image plane appear in true length and true slope. These are all frontal lines and all lie in a plane called a frontal plane. Be is another horizontal line which appears in true length from the top as a point when viewed from the front and in true length when viewed from the right side. AE is an imaginary horizontal line and like be or any other horizontal line will appear in true length when viewed from the top. EF is a sloping line. When viewed from the top or front, it appears shorter than true length. It is only when we view this line from either side that it appears in true length. True slope may also be measured from either side if the line appears in true length. Now if we assemble lines BE, EF and BC, you can see that they all lie in the same plane. This plane appears as an edge from the top and front views. This is called a profile plane. The lines on the plane or parallel to it are called profile lines. Such lines reveal their true length and true slope when viewed from either the right or left side. Another type of line which must be considered is the oblique line. The oblique line is not horizontal, frontal or profile since it is not in or parallel to any of the principal image planes. The line EG does not appear in true length in top, front or side view. Thus, true slope will not be seen in any of these views. If we can view this line from a position where all points on line eg are the same distance from the image plane, you can then see the true length. If this image plane is a vertical plane, you will also see the true slope of line eg. Further consideration of this line will reveal it as a point when the line is projected onto an image plane to which it is perpendicular. Let us review these lines while you try to imagine the correct viewpoints from which to see the true lengths, true slopes and point views. Will you also visualize the proper image planes and their relationship to each other? First consider the horizontal lines. Second, the frontal lines. Then the profile lines. In conclusion, consider the many lines, all of them generated by points, that you see around you every day. To represent these lines so as to evaluate their location, position and magnitude, you must remember how lines are formed. Remember also the names of straight lines and how to see them in true length and true slope and as points.