Vermeer and the Art of Linear Perspective: Perspective Diagrams of Vermeer's Paintings

In order to ensure the highest degree of geometrical precision and visual clarity, the two sets of perspective diagrams provided above were devised using high-resolution images of Vermeer's paintings and a scalable vector-based graphics (SVG) application (Inkscape). This format allows greater precision than raster-based applications such as Painter, Photoshop or GIMP. In order to speed download time, the SVG images were sized down and transformed to PNG, and then in the internet-friendly JPEG. The SVG images can be obtained upon request, free of charge.

The diagrams were created with a common procedure loosely referred to as reverse perspective (not to be confused with a primitive form of perspective), or reverse geometry, in which the visual evidence embedded in the painted surface is utilized to retrieve hypothetical vanishing lines, vanishing points and horizon lines rather than, as an artist himself would have done, to create the positions and foreshortening of the objects by means of these elements. The reconstructions are, naturally, based on the assumption that Vermeer's pictures obey the rules of linear perspective.

By examining the evidence presented in the painted surface of Vermeer's pictures, what can we deduce about his knowledge of perspective and the method(s) he may used to achieve his results? First of all, not all Vermeer's perspectives are equal in degree of sophistication or precision. There are three or four paintings whose perspective crudeness suggest that linear perspective was not employed. There are 10 or so that can be created with one-point perspective alone, and another 10 that demand knowledge of both one-and two-point perspective.

One-Point Perspective
The fact that the orthogonals of Vermeer's one-point perspectives do not intersect precisely at the central vanishing point (fig. 1) but are dispersed over a relatively discreet area—whose range varies from painting to painting—does not necessarily indicate a lack of systematic understanding of the laws of linear perspective on the part of the artist. Dispersion occurs even in those paintings that present a pin-point hole in the canvas. Since as far as we know the sole function of the pin hole was to act as a fixed vanishing point by which orthogonals could be mechanically and consistently measured with a sting or straight-edge tool, it is likely in these cases that the lack of geometrical coherence can be traced to variables pertaining to the painting process or methodology.

Although not generally highlighted by those who aspire to reconstruct the perspective armature of centuries-old paintings, the retrieval and subsequent interpretation of perspective elements from a painted surface is fraught with hazard. Absolute precision is not attainable, if for nothing else, the difficulties of interpreting the often ambiguous visual information of the paintings. Even the slightest misinterpretation of the angle of a painted segment will cause a fully projected vanishing line to stray from its theoretical vanishing point, sometimes dramatically so.

In various paintings by Vermeer the straight edges of architectural features or movable objects are sometimes lengthy enough to provide a relatively secure basis for projection, such as the edge where a tiled floor meets a side wall (see The Music Lesson). On the other hand, many segments are so short that mismeasurement is practically guaranteed, such as those of the braces of chairs or the edges of a foot stool, or, even smaller, those of a foot warmer (see The Milkmaid ). This is why diagrams of Vermeer's perspectives, in which subjectivity cannot be avoided, vary from author to author. Many authors choose simply to ignore any discrepancies in favor of a theoretically correct rendering, assuming that Vermeer's perspectives are by definition geometrically sound and highly accurate. Hopefully, the diagram (fig. 2) of Vermeer's Allegory of Faith, a relatively carefully painted work, illustrates the point set out above.

Points "a," "b" and "c" are located precisely at the points where the upper and lower corners of a receding row of white floor tiles touch each other on the painted surface. These points should fall on a single line. However, it becomes evident that when point "a" is connected to point "b" it produces a line (yellow) that passes to the right of the vanishing point, which is described by a small red dot inside a larger red circle at area "d." The vanishing point in this case can be securely located by the presence of a pin hole that was presumably made by the painter himself. Similarly, when a line is projected from point "a" through point "c" it passes slightly to the left of the pin hole. At the height of the central vanishing point the distance between the two orthogonals can be easily noted.

Two-Point Perspective
Two-point, or distance-point, construction is applied to objects rotated 45 degrees to the viewer's line of sight, be they flat, such as diagonally set floor tiles, or three-dimensional (e.g., foreground chairs of the Lady Writing a Letter with her Maid or Allegory of Faith). In Vermeer's perspectives both the locations of the distance points and their relative vanishing lines are generally less precisely contrived than those of the orthogonals notwithstanding the fact that it would have been possible to attain a similar level of accuracy by means of the same workshop procedure (pin-and-string) used for one-point construction, had, of course, perspectival accuracy been of overriding importance. In fact, since the slight, localized changes in the angle that inevitably occur during the painting process do not alter even minimally the reading of spatial depth by the observer, and become noticeable only when they are projected, the painter would have had little incentive pursue precision for its own sake. In fact, there are few instances in Vermeer's paintings in which the consequences of minor geometric imprecision lead to the awareness that something has gone awry. One example is the near isometric projection of lower braces of the chair on which the male lute player is seated in The Concert (fig. 3).

The Concert (detail)
Johannes Vermeer
c. 1663–1666
Oil on canvas, 72.5 x 64.7 cm.
Isabella Stewart Gardner Museum, Boston

In brief, this is how it can be done. Once the central vanishing point has been determined, the drawing or canvas is attached to a wall or laid face up on a large table. The horizon line, determined by the height of the central vanishing point, is developed by extending a horizontal line through the vanishing point to the left and right, lengthy enough so that the two distance points can be properly accommodated, equally spaced from the central vanishing point. Then, a pin is inserted into the locations of the distance points. Finally, a string is attached to the pin and pulled taught thereby allowing the artist verify or construct the "distance" lines with precision and without, if necessary, disturbing the surface of the canvas. Obviously, the pin-and-string technique is more elaborate and time consuming when used in conjunction with distance points, especially once the actual painting phase has commenced. In part, this may explain why Vermeer's distant-point constructions are somewhat less accurately contrived that the orthogonals. However, it must also be considered that the effects of misinterpretation in two-point perspective construction are greatly exacerbated given that the distance between the original segment and its appropriate distance point is very long: even the slightest measurement may give disastrous results.

Another factor crucial in gauging the accuracy of Vermeer's perspectives involves neither theoretical understanding nor interpretative ambiguities, but the painting process. It is not only possible but virtually inevitable that even when the perspective lines of the underdrawing are geometrically perfect, they will become lost when painted upon, and at times repainted, in order to give color and tone to the objects to which they belong. We find faulty vanishing lines not only in the earlier works done in a painterly style, but in the later works in which the painter went to great lengths to render the straight edges of objects as straight as possible. It is also difficult to rule out that occasionally the actual objects that Vermeer painted from were not geometrically regular (see the braces of the Spanish chairs).

In Vermeer's perspective there are a number objects which are set neither perpendicularly nor at 45 degrees to the observer's line of sight, but rotated at intermediate angles, such as chairs, window frames and stools. In the relatively few cases in which the perspectives of such oriented objects are contemplated in perspective literature of the past, the vanishing points associated with their construction were referred to specifically as "accidental" points (poinct accidental), rather than "tier" points, as distance points were often referred to. Today, there exists no equivalent for this term. It is likely that a need for a specific term came about—"accidental" may suggest an element of unusualness—because those preoccupied with high degrees of perspective accuracy found that objects set perpendicularly or at 45 degree angles to the viewer's line of sight were far easier to develop than those set intermediate angles, in both conceptually and in practice. In Vermeer's paintings there are more than a dozen of such objects, which include chairs, stools, windows frames and an easel (The Art of Painting). It must be said that degree of accuracy of the perspective construction in these cases is generally significantly inferior with respect to those of objects developed with one- or distance-point constructions. In some cases the vanishing lines appear so inconsistent with perspective geometry that one has the impression that they were, as artists are want to say, eye-balled (e.g., the foreground chair of Officer and Laughing Girl).

Given the variability of the perspectival inaccuracies encountered in Vermeer's paintings, it may be arbitrary to relate them to any particular process he might have used to develop perceptive.

Errors Vermeer's Perspectives

Since there is no objective method for determining the exact angle of a hand-painted segment, it is difficult to identify the nature of errors that are encountered. Notwithstanding, it could be said that there are essentially three types of errors in perspective: systematic errors, accidental errors, and ad hoc errors.

A systematic error is an error that results from a misunderstanding of one or another rule of perceptive. For example, in Woman with a Pearl Necklace, the different bundles of orthogonals of the mirror, window muntins and table intersect at very different points rather than meeting at a single vanishing point.

An accidental error is usually a casual mismeasurement in the initial projection, a slip of the artist's hand or repeated repainting during the painting phase. Such errors may have been so small as to be considered as negligent by the painter, or simply too small to be noticed by him. For example, the orthogonal lines caused by uppermost leadings of the window in A Lady Standing at a Virginal do not converge at the central vanishing point. Moreover, it is impossible to know if the leadings were incorrectly drawn by Vermeer or if, in the case they represent real objects in Vermeer's studio, physical irregularities in the objects themselves. This may explain also why the vanishing lines associated with the braces of the Spanish are generally less conforming to geometric accuracy than the orthogonals.

An ad hoc error is an intentional error meant to purposely alter some element of a correct perspective scheme in order to modify an aesthetic outcome. Ad hoc errors are the most difficult to identify in that it is largely a matter of interpretation as to what is intentional and what is not. If the perspective of the jug of The Milkmaid were correct, the viewer would have been able to see the flat disk of milk inside it. SInce this passage is so crucial to the painting, it is doubtful that Vermeer would not have noticed it.

Perspective Diagrams

There are two groups of diagrams below. The first group (left-hand column) is the fruit of a common-sense approach. It is based on the reasonable assumption that Vermeer was familiar with linear perspective and therefore knew that all orthogonal lines must converge at a single, vanishing point, referred to in the present study as the central vanishing point. This assumption is supported by physical evidence found in 13 paintings, which present a detectable hole in the canvas in which, presumably, a pin was inserted. This trick was used by a number of other Dutch painters. For this reason and for the sake of illustrating the geometric principles behind Vermeer's perspectives, only those lines which deviate clearly from what would be expected are not connected to their appropriate vanishing points.

In three paintings, Christ in the House of Martha and Mary, A Maid Asleep and The Little Street, it was not possible to derive a vanishing point, and hence, a horizon line due to the combined difficulties of interpreting the physical evidence presented by the pictures themselves as well as the uncertainties of the perspective construction of these three works.

The second group of diagrams is based on the most careful consideration of visual evidence embedded in the painted surface, and nothing else. Therefore, there are no vanishing points or horizon lines; only raw perspective lines which have been retrieved but not mediated in any way.