For the advanced landscape painter, reflections are one of the most popular components in a showpiece painting, and one that many viewers value highly. While they can be painted successfully by following the realist rule of ‘paint what you see, not what you think you see’, understanding the optical principles involved is important for avoiding error. Unlike with cast shadows, errors of commission or omission are readily noticed by the viewer.

Basic optics

The fundamental optical principles underlying all reflections on water surfaces are:

1. light travels in a straight line when passing through a medium of constant refractive index;
2. when light reaches the interface between two media of different refractive indexes, some will be reflected from the boundary, and some will be refracted through the boundary;
3. the angles between a ray of light incident on, and that being reflected from a boundary (between media of different refractive indexes), with that boundary surface, are equal;
4. a water surface with air above it presents a suitable boundary at which reflection and refraction occur;
5. still water surfaces are horizontal planes which therefore act as horizontal planar mirrors as far as incident light rays are concerned.

Here a post (with marks A and B), on the left of the diagram, is on the bank of a completely flat lake, with the observer on the opposite side of the lake at C, looking over the lake towards the post. A ray of light scattered from point A on the post passes over the lake to point A’. There, in passing from air to water, it reaches a boundary between media of different refractive indexes, and most of that light will then be reflected towards the observer at C.

Measured at the point A’ on the lake, the angle of incidence (between the ray and the horizontal water surface) equals the angle of reflection (between the ray and the water surface). The same process of reflection occurs to light scattered from point B on the post, with respect to its reflection at point B’ and the observer’s eye at C, although the angles at point A’ are less than those at point B’.

The relationship between heights above the water plane and the distances from the reflection on the water are a matter of simple geometry. Taking first the right-angled triangle formed by the points B, B’, and the base of the post, the perpendicular height of B above the water (H) and the distance from the base of the post to the point of reflection B’ (D) is the tangent of the angle of incidence of the light ray at B’, as H/D.

As the angle of reflection at B’ must be the same as the angle of incidence (point 3 above), the ratios of the heights at B and C to the distances along the water surface from the base of the post to the point of reflection, and the point of reflection to the observer, must be equal. Hence if three of the four variables are known, the fourth can be calculated, e.g. H = H’.D/D’.

Putting it together

Repeating this simple tracing of light rays enables visualisation of more complex examples of reflection, such as that shown above. It quickly becomes tedious for humans to trace individual light rays in a three-dimensional model, but computers excel at the task. The resulting image makes it clear that corresponding points in a reflection on water are vertically below their originals, although the slight waves seen here can result in limited lateral shift in the reflected points.

As is well known from real life, the reflected image in a horizontal mirror is effectively reflected on a local horizontal plane, so that the left of the original remains on the left of the reflection, but the reflected image is inverted (the top of the reflection shows the lowest part of the original); that is unlike reflections in a mirror positioned vertically, such as those in which we inspect our face, shave, or apply make-up, which (fortuitously) remain uninverted. Whether a mirror is vertical or horizontal, the left of the reflected image shows the left of the real image, and similarly for the right.

The patterning and slant of the pole illustrates the differences between such a reflected image and one that has simply been rotated through 180˚ (as might be achieved by a painter rotating the canvas on an easel). In the latter case the chirality (handedness) of the transformed image is opposite, the post would lean the opposite way, and the red spiral pattern on it would differ too.

This is a composite image: alongside the original and its untransformed reflection, I have reflected the image of the reflection, and moved it up, alongside the original.

Complex 3D scenes

Reflections in a more complex 3D scene, above, show slight alteration in vertical dimensions, but remain strictly aligned with the original. Although the slight waves produce very small lateral displacements in the reflection, every point in the reflection corresponding to a point in the original remains in alignment across the width of the image.

Those objects at the front of the original remain at the front of the reflection: for example, the tree with red leaves retains its position relative to the green conifer tree behind it. The further back from the water’s edge that an object is situated, the more truncated is its reflection; truncation appears as if it occurred at the base of the original.

One useful rule of thumb that is sometimes given to gauge the effect of truncation is to imagine that the water extends right back to the base of the original, to construct the reflection on that imaginary water surface, then to erase the reflection on the water where it does not actually exist.

There are complex differences between the original and its reflection with respect to the position of details, tones and highlights. Although highlights are seen on the red leaves, their pattern is not the same, because the original leaves are seen direct, but those in the reflection are viewed as if from below and in front of the tree (from their points of reflection on the water).

This makes it impossible to create a completely accurate representation of a reflection unless you are able to see that reflection; even using modern ray-tracing software on a computer it is extremely difficult to construct or reconstruct a reflection from real life. Any painter who paints the original en plein air and later attempts to paint its reflection in the studio is only going to be able to guess its form and appearance, and will be unable to get tones and highlights correct with respect to those seen in nature.

Difficulties encountered in extensive reflections on water in the real world are illustrated in the composite image above. Note how the distant mountains almost disappear from the reflection with the change in vertical dimensions, but rigorous vertical alignment is maintained across the entire image. Painting reflections in images such as this is an incredibly difficult task without the complete view in front of the painter at the time.

Practical principles

Any faithful depiction of reflections on water will therefore show the following features:

1. a line joining any point on the original with its equivalent on the reflection will be strictly vertical, allowing for slight lateral shift resulting from the effects of small waves;
2. an object which is behind another object in the original will also remain behind that object in the reflection, as reflections preserve depth order;
3. the further back that an original object is from the water’s edge, the more its reflection will be cropped vertically;
4. vertical cropping loses the lower section of the original from the reflection, and the upper section remains in the reflection;
5. when the water surface is smooth, the position of reflections can be determined by simple geometry relating the height above the water surface to the distance between the point of reflection on the water and the perpendicular projection down onto the water plane (the tangent of the angle or incidence or reflection being equal to height/distance);
6. the view of each part of the original seen in the reflection will be that as seen from the points of reflection, those being lower than the observer and closer to the original;
7. what is seen on the (observer’s) left of the original appears on the left of the reflection, and what is seen on the right remains on the right of the reflection;
8. because the reflection is vertically inverted, what is seen at the top of the original appears at the bottom of the reflection;
9. the more the water surface departs from being a flat and smooth mirror, the more distortion will be introduced into the reflection, until eventually its form is lost in a series of vague areas of broken colour.

History

Although several early manuals on painting, such as that of Roger de Piles (1708), refer to the need to depict reflections accurately, detailed accounts first appeared in manuals on perspective, such as Brook Taylor’s in 1719.

Valenciennes’s Elements of Perspective (1820) gave sage advice to artists trying to tackle reflections, including those of the moon, and stress the vertical alignment of original and reflected images.

More recently, Rex Vicat Cole’s manual (1921) gives some good rules of thumb. However the rule of thumb which is often given — to extend the water surface to the base of the object, imagine the reflection as it would appear on that ‘virtual’ water, and then truncate that to the edge of the real water — isn’t strictly accurate.

Take a house on a hill some distance away from the edge of a perfectly-reflecting lake. To determine whether any of its reflection would appear on the lake, the hill under the house should be eroded to leave just a thin pillar directly under the house, so that the ground around that pillar is below the surface of the extended lake. You should then construct the reflection as it would appear on this extended water surface, before trimming the water back to its actual edge.

This may be both conceptually simpler and more accurate than working through geometric calculations using estimated heights and distances.

Cole also interestingly noted: “Though we have said that reflections are always vertically under the objects, I have noticed one is apt unconsciously to represent the reflection of vertical lines as very slightly inclined towards the spot we stand on, and I have seen them so painted, and I think rightly so, by artists of repute.”

Unfortunately some recent accounts of painting reflections also give bad or confusing advice. One such is the recommendation to work on the painting when it is upside down, which can (because of the preservation of chirality in reflections) make them much harder to paint correctly.

Examples

Look in the landscape behind Jan van Eyck’s Madonna of Chancellor Rolin (c 1435) and you will see one of the earliest examples of the meticulously accurate depiction of reflections on water. In the following three centuries, many Masters demonstrated their insights into scientific optics by tackling such reflections, including Lucas Cranach the Elder, Piero della Francesca, Raphael, Tintoretto, Titian, Caravaggio, Poussin, Rubens, Rembrandt, Canaletto, JMW Turner and his contemporaries such as John Sell Cotman and Thomas Girtin.

Claude Monet’s masterwork Autumn on the Seine, Argenteuil from 1873 is an excellent example of his waterscapes of this period, with its finely broken and rippled reflections.

Alder Trunks from 1893 is one of Laurits Andersen Ring’s finest landscapes, and has earned it place in the Danish Royal Collection. He shows these old coppiced alders mainly in reflection. Although their details are quite painterly, the overall effect is that of precise realism.

Eilert Adelsteen Normann’s views of Norway include many fine examples of reflections, including this unspecified and undated Norwegian Fjord, which may well be a view of Sognefjord with its slightly rippled mirror surface.

Secret-Reflection (1902) is a pair of paintings by Fernand Khnopff in pastel and crayon on paper. In the upper image, the artist’s sister Marguerite is swathed in clothes, while the lower painting shows a flooded town, probably the city of Bruges from Khnopff’s childhood, with its meticulous reflections.

One of Ferdinand Hodler’s finest landscapes from 1904 is Lake Thun with Symmetrical Reflection Before Sunrise. Unlike some of his later landscapes, he has here been largely true to optics.

Paul Signac’s Pointillist paintings such as Giudecca Anchorage from 1904 claimed no exemption from optics, although their tessellated patches of colour gave the artist more leeway.

JMW Turner was generally accurate in his painting of reflections, but as with other features, he sometimes bent the rules for pictorial effect, for example in his late oil painting War. The Exile and the Rock Limpet from 1842.

The left and more distant figure (of the guard to Napoleon) shouldn’t have appeared in reflection at all. However, strict adherence to optics would have significantly detracted from the effect of this painting. He has also omitted shadows which might have been cast by the figures, presumably for the same reason.

In the next article I’ll look at some painted reflections that aren’t optically correct, for whatever reason.

References

Brook Taylor (1719) New Principles of Linear Perspective, or the Art of Designing on a Plane the Representations of All Sorts of Objects, in a more General and Simple Method than has been done before, London.
Cole, Rex Vicat (1921) Perspective, Seeley, Service and Co, London. (Available in various reprints, and Archive.org.)
de Piles, Roger (1708) Cours de Peinture par Principes, Paris. (Available at Archive.org.)
de Valenciennes Pierre-Henri (1820) Élémens de Perspective Pratique à l’usage des artistes, 2nd edn., Paris.