Those who become enamoured of the practice of the art, without having previously applied to the diligent study of the scientific part of it, may be compared to mariners, who put to sea in a ship without rudder or compass, and therefore cannot be certain of arriving at the wished-for port. Practice must always be founded on good theory; to this, Perspective is the guide and entrance, without which nothing can be well done.
Jan van Eyck
Leonardo da Vinci.
The Mathematics of Linear PerspectiveThe idea behind linear perspective is simple: First, we assume that we are observing a physical scene (a basket of fruit, a sunset in over the ocean...) with one eye only. Now put up an imaginary vertical screen (called the picture plane) between the eye and the scene. For each object in the physical scene (an apple, a flower, a grain of sand...), there is a collection of light rays which are reflected off the object and then enter our eye. (That is how we normally see.) Suppose that on the way to our eyeball, each of these light rays makes a mark (with the same color as the light) on the picture plane. The resulting colored picture plane has the property that: wherever we place it, and as long as we stand at the same distance from it as before, the picture plane presents to our eye an image which is indistinguishable from the original physical scene (our basket of fruits, the sunset). It apears as if the space we saw now lies inside the flat picture plane.
Now, let's flip the idea around. If we can figure out the laws which govern the placement of the marks by the light rays reflecting off the physical scene, then we could start with a blank canvas, place marks on it according to the same laws, and produce an illusion of a three-dimensional physical scene contained within the canvas (whether or not the scene thus illustrated actually exists in the real world).
What are these laws? We name a few here. In the following:
- We prefix any element in the physical scene with the word physical, and any element on the picture plane with the word pictorial.
- We call a physical line which is perpendicular to the picture plane an orthogonal line.
- We call a physical line which is parallel to picture plane a transversal line.
- We also assume the picture plane is infinitely large.
Laws of Linear Perspective►
In particular, parallel transversal lines project to parallel pictorial lines.
- Any set of parallel non-transversals share the same vanishing point.
- The vanishing points of horizonal non-transversals lie on the horizon of the picture plane. This is the horizontal line on the picture plane which is level with the viewer's eye.
The vanishing point every orthogonal line lies at the center of the horizon.
It is remarkable that with only these simple rules, and a little bit of ingenuity in their application, some great visual results could be obtained. For example:
Jan van Eyck
Besides employing the geometric rules outlined above to depict objects viewed in perspective, there are various mechanical or semi-mechanical means to produce the same effect. In the following illustration by Albrecht Dürer, the "imaginary screen" we used in devising linear perspective is realized physically as a gridded net (grid #1) placed between a draftsman and the physical scene (in this case a reclining nude). Before the draftsman lies a drawing surface with a similar grid system (grid #2) overlaid upon it. Viewing the physical scene with one eye placed at a fixed point, various points of the scene lies directly behind points on grid #1. If, say, the tip of the nose of the woman lies behind the grid point on grid #1 which is two from the top and one from the left, then the draftsman would draw the tip of the nose on the drawing surface at the corresponding point on grid #2. Repeating this process for each point in the physical scene, a perspectival drawing of the scene is obtained.
(Gallerie dell'Accademia, Venice, Italy)
¡Cada animal tiene su estrategia!
(Each animal has its strategy!)
Antonio Lopez Garcia (Spanish)
Metodología pictórica en la obra de Antonio López García (Pictorial Methodology in Antonio López García's work), David Serrano León, LABORATORIO DE ARTE 24 (2012), pp. 717-737.