Geometric perspective, often referred to as simply perspective, may be defined as the natural law which says the further something is from our eyes, the smaller its image will be. Since there are two types of perspective, the other being atmospheric perspective, it is necessary to distinguish between them by referring to what is normally called perspective as geometric perspective, in reference to the fact that it is a geometric breakdown of a natural optical phenomenon. It is sometimes referred to as linear perspective, as it involves the use of lines in its construction. As simple as it sounds, the problem of how to convincingly render this visual phenomenon had baffled painters for centuries, until a mathematical approach was discovered in the early Renaissance. The painters Masaccio and Uccello, as well as the architect Bruneleschi have each been credited with its discovery. The Roman architect Vitruvius may actually have preceded all of the others named, and may in turn have been influenced in it by someone still earlier, but the discovery does not seem to have reached painters until the early Renaissance. Whichever of these attributions is correct is less important than the fact that the discovery was made, and that it constituted a major breakthrough in illusionistic painting. Once artists learned the system, they could indicate spatial recession more realistically than had previously been possible. The system involves the use of vanishing points; points at which lines intended to depict parallel lines converge. These vanishing points are on the horizon if the lines are level. It is important to note that the horizon is always at the viewer's eye level. When two or more vanishing points are necessary, as in all but the simplest perspective problems, their placement may be worked out following the mathematical system, or, if we are working from life, by simply copying the angles we see and extending them to the horizon. The points at which the extended lines cross the horizon are the Vanishing Points. All lines parallel to the one used to establish the Vanishing Point will converge at that Vanishing Point. If there is any question as to their accuracy, the mathematical system may then be employed to double-check. The mathematical approach must be learned first, and practiced until a point is reached whereby the artist is able to visualize the scene in correct perspective automatically, without the need of actually drawing in the vanishing points and guide lines. The subject is taught to students of architecture, but is not part of the curriculum of fine arts programs in most universities at the time of this writing. It may be that a Fine Arts major could take it as an elective. There are several books on the subject, the best of which are listed in the bibliography. However one chooses to study, whether alone, in an institution, or with a private instructor, the importance of mastering geometric perspective cannot be stressed too highly. It is imperative that any serious aspiring artist absorb this fundamental principle completely, if he or she is to ever create genuinely Great Art. It must become second nature, so thoroughly assimilated that virtually no effort is required to visualize it correctly. As the subject is so completely covered in Rex Vicat Cole's book, Perspective for Artists, there is little point in addressing it in full detail in this book.

Errors in perspective are far too common in modern times. Such an error immediately destroys the illusion of spatial recession, and prevents the viewer from receiving the artist's message.

In the simplest exercises in perspective, one Vanishing Point only is used, and may be placed arbitrarily, on the Horizon. A straight road on absolutely level ground may be indicated by drawing lines from points on each side of the road to the Vanishing Point on the horizon, where they converge. Suppose we want to add a line of telephone poles, or fence posts, running parallel to the road and placed at regular intervals. The spaces between them must diminish as greater distance from the viewer's eye is indicated. The interval between the nearest pole and the second pole is established arbitrarily by the artist. The placement of the base of the third pole may be determined by drawing a guideline from the top of the first pole through the center of the second pole, and extending it until it intersects the line running from the base of the first pole to the Vanishing Point.

A vertical line drawn from the point thus established becomes the third pole. Its height is found by drawing a line from the top of the first pole to the Vanishing Point. The fourth pole is located by drawing a line from the top of the second pole through the center of the third pole and extending it to the line connecting the base of the first pole with the Vanishing Point, and so on. This example is quite simple, and should serve only as an introduction to the geometrical system of indicating three-dimensional depth on a two-dimensional surface. Refer to the insets and accompanying illustrations for solutions to some of the more complex perspective problems.

The system is not quite perfect, as it fails to take into account the curvature of the Earth. It works well because the Earth is so large that in most cases the curvature is not apparent. Its limitations are that it can become quite complicated, and artists are generally not mathematicians, nor are they likely to be interested in approaching the scene from such an analytical, as opposed to intuitive, standpoint. For this reason, many artists, or would-be artists, are weak in their understanding of this fundamental principle. It is imperative that the student, the aspiring artist, apply the discipline necessary to learn the mathematics of the system so well that all awkwardness with its application disappears and ceases to interfere with the creative, intuitive processes so essential to art. Once it is committed to second nature, it becomes a help rather than a hindrance. The artist should then be able to "eyeball" the scene accurately, without having to actually draw the vanishing points and guide lines. Its parallel in music would be the learning of music theory; perhaps no fun at first, but Great Music cannot be created without it.

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