a.

Lorenzo Lotto’s Husband and Wife, c1523—4

Analysis of this painting shows how an understanding of geometrical optics and perspective provides a powerful way to show certain images were created with the aid of lenses, as well as to extract quantitative information from those images. Figure 1 is a detail from Lorenzo Lotto’s Husband and Wife of c1523—4, showing an octagonal pattern on a table covering that appears to go out of focus at some depth into the painting. Overlaid on the painting are three segments of a perspective-corrected octagon whose details we calculate below. Although there are several features of interest in Lotto’s Husband and Wife, here we are concerned only with the way the image in this portion of the painting seems to go out of focus as it recedes into the distance.

Fig. 1.

Detail of Husband and Wife. The blue, green, and red overlays are perspective-corrected sections of an octagonal pattern fit to the painting. If this portion of the painting is based on a projected image of an actual table covering, Lorenzo Lotto would have painted the front (blue) section, then moved his lens to refocus further back when the depth of field had been exceeded and continued to paint the center (green) section, then refocused one more time to finish the final (red) portion. As can be seen, the details of the painting are in excellent qualitative agreement with the three-segment, perspective-corrected octagon predicted by the laws of geometrical optics for such a projected image. As described in the text, the details are in excellent quantitative agreement as well.

While a simple lens can be focused at only one specific distance at a time, the brain causes the muscles of a human eye to quickly and automatically alter the shape to refocus to different depths as the eye traverses a scene. Because of this, we do not simultaneously see part of a scene in focus and part out of focus. In contrast with the eye, no matter what the distance of focus of a simple lens, only a certain field on either side of that distance will remain acceptably sharp, resulting in a depth of field that depends on the focal length and diameter of the lens. To change that distance of focus requires physically altering the position of the lens with respect to the subject and the image plane. Refocusing a lens to a depth further into a scene from its original plane of focus requires moving the lens closer to the image plane (and vice versa). Moving the lens closer to the image plane results in a small decrease in the magnification of the projected scene, as well as in a slight change in the vanishing point, since the lens is now at a slightly different position. While both of these effects are quite small for magnifications M<<1, which is the magnification range for most ordinary photographs, they increase in magnitude as M increases (note that the image of a 1.6 m woman will be ~2 cm tall when projected on a piece of film or a CCD sensor, so M≈0.012, whereas the woman in the Lotto painting discussed below is at M≈0.56, which is nearly 50× greater magnification). Such effects are fundamental characteristics of images projected by lenses, and are extremely unlikely to occur in a painting if an artist had instead laid out patterns following geometrical rules first articulated in the fifteenth century [9].

Although we discussed several aspects of this painting elsewhere [2,7], here we provide additional details of our analysis. As we showed in Ref. 7, the width of the wife’s shoulders in the painting provides an internal length scale that lets us determine the magnification to be M≈00.56, and from this we determine the depth of field to be 16±1.5 cm. From geometrical optics the focal length (FL) and magnification (M) are given by:

and

As indicated by the colored segments on Figure 1, there are three regions of this octagonal pattern resulting from Lotto refocusing twice as he exceeded the depth of field of his lens. We label these Regions 1, 2, and 3, with Region 1 the closest to the front of the painting (blue, in Fig. 1). Thus, for the first two Regions,

and

However, as we showed in Ref. 7, the measured depth of field is 16 cm, so for Region 2

and

Because Region 2 is further into the scene, and therefore at slightly lower magnification, than is Region 1, its depth of field will be somewhat larger than 16 cm. However, for the purposes of this paper we will omit this complexity. Thus, Region 3 of the pattern starts at a depth of approximately 2*16 cm = 32 cm into the scene, so

and

The fit of the three segments of the perspective-corrected octagon overlaid on Figure 1 allows us to determine the relative magnifications M of the three regions as 0.56, 0.48, and 0.42, respectively, from which we obtain:

Thus, we have six equations (3, 6, 8, 9, 10 and 11), but only five unknowns: FL, d_{lens subjectl}, and d_{lens-image1,2,3}. Solving these equations uniquely determines:

Note that there are no adjustable parameters in our analysis of this image. In going from Region 1 to Region 2 the magnification, as measured from our fit of a perspective-corrected octagon, decreases by 14.5% from the original 0.56 of the painting, in excellent agreement with the 14.3% we calculate from the above equations. Similarly, the measured magnification decreases by a further 12.3% when going to Region 3, again in excellent agreement with the calculated value of 12.5%.

To summarize, using the measured size of the woman in this painting as the only input parameter, geometrical optics dictates that if Lotto used a lens to project the octagonal pattern there must be three regions corresponding to the depths into the scene where he would have been forced to refocus, with three different magnifications, and hence three sets of vanishing points. All of these complex features are found in the painting, and all are in excellent quantitative agreement with the predictions of geometrical optics, providing extremely strong evidence indeed that a lens was used. Further, the focal length of the lens as well as the distances from the lens to the table covering as well as to Lotto’s canvas are all quite reasonable, allowing significant insights into the actual layout of the artist’s studio.

b.

Robert Campin’s Merode Triptych, c.1425–28

The earliest optical evidence we have found to date is in the center and right panels of Robert Campin’s Merode Triptych of c.1425–28, a detail of which is reproduced in Figure 2. This detail exhibits the same complex change in perspective resulting from having refocused a lens as discussed above for Lorenzo Lotto’s Husband and Wife. Overlaid on this painting are three segments of an ideal perspective-corrected lattice that fit the painting to an accuracy of better than 1 mm. Not shown is that a single-segment ideal lattice cannot be made to fit. As can be seen, the latticework in the seat back contains unmistakable evidence of two small changes in perspective that occur at depths into the scene where a lens would have had to have been refocused. A similar change in the perspective at roughly the same depth into the scene also is in the latticework of the seat back in the central panel of this triptych. The complex perspective exhibited by the latticework is an inevitable outcome from the depth of field of a lens, but would be very difficult (and extremely unlikely) to reproduce by any geometrical technique.

Fig. 2.

Detail of right panel of the Merode Triptych. The blue, green and red overlays are perspective-corrected sections of the same ideal latticework, revealing where Campin had to refocus his lens when he exceeded its depth of field.

c.

Jan van Eyck’s Arnolfini Marriage, 1434

The perspective, texture, and detail of the complex chandelier in this painting are all quite remarkable, leading us to examine it for evidence that it might be based on an optical projection. The elementary rules of perspective (see, for example, Fig. 24 of Reference 9) seemingly dictate that lines drawn through common elements of an optical projection should meet at well defined foci, all of which must be on a single horizon line. However, any real object, such as a chandelier, inevitably will deviate from absolutely perfect hexagonal symmetry. While this should be obvious, the consequence of even very small variations is that vanishing points will not obey the simplified laws of perspective as taught in most textbooks [10]. This is shown on the bottom of Figure 3, where lines accurately drawn through corresponding features on a photograph of a light fixture show marked deviation from optical perspective as it is described in elementary textbooks. Interestingly, the top of this figure shows that lines carefully drawn through corresponding features on van Eyck’s chandelier (the tops of the six candle holders) show similar deviations. Although this might seem an obvious test to conduct on an image, as can be seen from these examples, drawing lines on the image of this chandelier is too simplistic an approach to reveal anything useful.

Fig. 3.

(top) Chandelier in the Arnolfini Marriage. Lines accurately drawn through the tops of the corresponding six candle holders do not meet at foci on a horizon line as seemingly dictated by the simplified laws of perspective. (bottom) Photograph of a light fixture. Lines accurately drawn through the corresponding six bulb holders show deviations that are remarkably similar to those of the chandelier. In this case there is no doubt whatever the image is based on an optical projection, showing that even slight deviations from defect-free, perfect hexagonal symmetry are sufficient to make inapplicable the simplified laws of perspective.

Another important point is that any painting of a three-dimensional object has reduced that object to a two dimensional space, so some spatial information inevitably has been lost. This is unlike the cases discussed earlier of the Lotto and the Campin paintings, where the carpet and seat back both were two-dimensional. Finally, we note that the decorative elements on the top and bottom of each of the six chandelier arms on this painting obviously would have been attached individually (by soldering or riveting) in fabricating such a complex piece from metal. This easily can be seen by examining a detail of the chandelier arms, as shown in Figure 4. Because of this, the placement of these decorative elements will vary from arm-to-arm, thus making them completely unsuitable for any kind of perspective analysis. For all of these reasons, it is important to analyze appropriate aspects of the chandelier in the painting in order to determine whether or not it was based on an optical projection.

Fig. 4.

Chandelier in the Arnolfini Marriage (detail). It can be seen from this image that the decorative features above and below each of the six main arcs would have been attached individually (by soldering or riveting), making their locations completely unsuitable for any analysis of the perspective of this painting. This is confirmed by the findings of Fig. 6.

We first need to establish the approximate magnification of the chandelier, since the depth of field of a projected image depends on this parameter, and in turn establishes whether or not a lens could have captured the entire image without the magnification-change effects of refocusing described above for the Lotto painting. Comparing the measured height of an actual candle flame, 3.9 cm, to the one van Eyck included in the painting, provides a size scale. We note, however, that none of the conclusions from our calculations are sensitive to the precise height of the flame. With this value, the overall width of the chandelier between opposite candle holders is 96.7 cm, and the magnification is 0.16. This magnification is small enough that the depth of field for a lens falling within any reasonable range of focal lengths would be over 1 m, and hence the entire image could have been captured without refocusing. This would have made the task simpler for the artists and, indeed, makes analyzing the image easier than if it had been composed of several segments.

If the overall image of this chandelier is based on an optical pro9ection, it is reasonable to expect that, after correcting for perspective, the positions of the tops of each of the six candle holders should exhibit something very close to perfect hexagonal symmetry. If, on the other hand, van Eyck painted this very complex three-dimensional obj ect only by observation, large deviations should occur in the positions of these candle holders. The blue dots in Figure 5 are the positions of the tops of each of the candle holders in the painting after correcting for perspective (i.e. to convert it to a plan view as viewed from the beneath the chandelier. with the candle holders nearest to the viewer at the top of this figure). As can be seen, the agreement with the points of a perfect hexagon is remarkable. The hexagon is rotated 6° counterclockwise, which is due to the chandelier being viewed from that angle. The maximum deviation of any of the candle holders from a perfect hexagon is only 7°, which in turn corresponds to the end of this arm being only 6.6 cm away from its “ideal” hexagonal position. The root-mean-square deviation of all six candle holders from perfect hexagonal symmetry is only 4.2°, or 4.1 cm. While there is no reason to expect a real 15th century chandelier to have been made to an accuracy any greater than this in the first place, there is also the possibility some or all of the individual deviations could have resulted from slight bends during transportation, hanging, or cleaning. As Figure 7 shows, the arms of a 21st century light fixture show imperfections comparable to those on van Eyck’s chandelier.

Fig. 5.

The blue ovals mark the perspective corrected positions of the tops of the six candle holders in the Arnolfini Marriage. In green is a hexagon, rotated by 6°. The rms deviation of the angular positions of the candle holders from this hexagon is only 4.3°, with a maximum deviation of 7°. Assuming a magnification of 0.16 as determined from the measured height of a candle flame, the yellow circle through the candle holders has a diameter of 96.7 cm. The maximum radial deviation of any of the six candle holders from the radius of this circle is only 7.4 mm (1.5%).

While it would have been relatively easy to have accidentally bent the angular positions of such chandelier arms with respect to each other, or to have attached them with slight errors to the central core, it would be much less likely that the lengths of the arms would vary significantly. Hence, if this painting is based on an optical projection, we would expect the radial positions of the candle holders to deviate much less than their angular positions. Shown in yellow on Figure 5 is a circle through the perspective-corrected positions of the candle holders. As can be seen, the radial positions are identical to within a worst case of only 7.4 mm; i.e., only 1.5% of the radius of the chandelier.

Figure 6 shows the shapes of the perspective-corrected arms of the chandelier, each outlined in a different color, and with the three arms to the viewer’s right flopped horizontally so all six overlap. Where the painting doesn’t show a complete arm because it is partially hidden by the arms in front of it, only the visible portions are outlined. As can be seen, the main arcs of all six arms are identical to within 5% in width and 1.5% in length. The latter figure is completely consistent with the above, independent, analysis of the radial positions of the candle holders. However, as would be expected for such a complex, hand-fabricated object, there are large variations in the positions of the decorative features attached to those arcs, confirming their uselessness for any sort of perspective analysis.

Fig. 6.

The outlines of all six arms on the Arnolfini chandelier in different colors, corrected for perspective and with the arms to the viewer’s right flopped horizontally to overlay on the arms to the left. As can be seen (highlighted in grey), the main arc of all six arms are the same to within 1.5% in length and 5% in width. However, as would be expected, there are significant variations in the locations where the decorative elements are attached to each of the arms.

Fig. 7.

(left) The Arnolfini chandelier analysis of Fig. 5. (right) The same analysis done to the light fixture at the bottom of Fig. 3. Note that the imperfections in this 21st century light fixture are comparable to those of the 15th century chandelier.

Given that the arcs themselves are nearly identical, the lowest point on the position of each arc provides an independent set of data to analyze in the same way as we did with the candle holders. With one exception, these lowest points, when corrected for perspective, are in excellent agreement with the same 6°-rotated hexagon as the candle holders. That one exception is the arm extending directly to the viewer's right, which is significantly below the position it would have if it had been faithfully traced from an optical projection (it is nearly the full width of the arm lower than the predicted position).

To summarize the evidence that van Eyck’s painting of this chandelier is based on an optical projection:

From the above evidence, with a high degree of certainty we can conclude that van Eyck’s chandelier is based on an optical projection of a real chandelier. Although there are some small differences (e.g. the height of one arm) that provide insights into artistic choices van Eyck made to deviate from tracing the proj ection, possibly to accentuate some features, overall the remarkably realistic perspective of this complex object is a result of the optical projection technique used as an aid by the artist.