Soft Shadow Images

Abstract

In traditional optics education, shadows are often regarded as a mere triviality, namely as silhouettes of obstacles to the propagation of light. However, by examining a series of shadow phenomena from an embedded perspective, we challenge this view and demonstrate how in general both the shape of object and light source have significant impact on the resulting soft shadow images. Through experimental and mathematical analysis of the imaging properties of inverse objects, we develop a generalized concept of shadow images as complementary phenomena. Shadow images are instructive examples of optical convolution and provide an opportunity to learn about the power of embedded perspective for the study of optical phenomena in the classroom. Additionally, we introduce the less known phenomenon of the bright shadow.

Full text

Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da
Soft Shadow Images
Johannes Grebe-Ellis and Thomas Quick
University of Wuppertal, Faculty of Mathematics and Natural Sciences, Gaußstraße
20, 42119 Wuppertal, Germany
E-mail: grebe-ellis@uni-wuppertal.de, quick@uni-wuppertal.de
Abstract. In traditional optics education, shadows are often regarded as a mere
triviality, namely as silhouettes of obstacles to the propagation of light. However, by
examining a series of shadow phenomena from an embedded perspective, we challenge
this view and demonstrate how in general both the shape of object and light source
have significant impact on the resulting soft shadow images. Through experimental
and mathematical analysis of the imaging properties of inverse objects, we develop a
generalized concept of shadow images as complementary phenomena. Shadow images
are instructive examples of optical convolution and provide an opportunity to learn
about the power of embedded perspective for the study of optical phenomena in
the classroom. Additionally, we introduce the less known phenomenon of the bright
shadow.
Keywords: soft shadow image, embedded perspective, convolution, complementarity,
bright shadow
1. Introduction
In optics teaching it is common to view shadows as trivial phenomena, simply as
silhouettes of opaque obstacles which prevent the propagation of light [1]. However,
a simple experiment can cause confusion (figure 1, upper row): a small round cardboard
disc is positioned in front of an annular light source, so that its shadow falls on a
projection screen. Moving the disc from the screen towards the source, we observe
a remarkable change in the shadow image. The shadow of the disk takes the form
of a ring – a challenge to the conventional concept of shadow formation [2], which
raises the question: How the shadow of the cardboard disc and the dark image of the
annular light source are related? A similar, more familiar situation can be found in
pinhole imaging. When the obstacle is replaced by a geometrically equivalent aperture
(i.e., swap the transmissive and occluding region), the geometrical imaging properties
remain unchanged. We get a bright ring on the screen: the pinhole image of the light
source (figure 1, bottom row). The dark image, sometimes called the anti-pinhole or
pinspeck image [3,4] and the bright pinhole image of the light source are complementary
phenomena.

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Figure 1. Inverse objects: a small round cardboard and a panel opening with the
same size (Ø = 5 cm) are illuminated with an annular light source (Ø = 25 cm,
middle). The photographs in the upper and the bottom row show the transformation
of the shadow images on the projection screen, while the inverse objects are moving
towards the light source.
There is a long-standing tradition in teaching optics to study special cases of shadow
formation where either the extent of the light source or the extent of the aperture is
neglected. Hard shadows as produced by point-like light sources are easy to model,
because they represent projections of the illuminated objects. On the other hand, the
pinhole camera is about producing bright images of the light source. In both cases, the
geometric influence of the imaging element is suppressed. As special cases, however,
they are only poles of a wide range of possible imaging geometries (figure 1and 2),
which generally both image and are imaged, and further produce intriguingly appealing
shadow images. One aim of this paper is to expand and generalize the concept of shadow
to include this variety of shadow phenomena for the context of optics lessons.
In computer graphics technology, the question of how to render realistic soft
shadows for complex lighting scenes has sparked vast research effort which led to an
advanced understanding of shadow formation on the base of what we call the embedded
perspective [5,6,7]. Shadow is defined here as it is experienced standing in the shadow
oneself: as an occlusion phenomenon. The approach can provide a useful explanation
that makes complex irradiance distributions more accessible. For example, the puzzle of
the dark ring from the beginning can be quickly solved by looking at the spatial layout of
light source and obstacle from the location of the shadow: Viewed from the illuminated

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Figure 2. For terminology: an object can be either an aperture or an obstacle, always
referring to the totality of the transmissive and occluding region. Hard shadows are
projections of the objects. The pinhole or pinspeck image is an image of the light
source. Soft shadows are always mixtures of the geometry of object and light source.
center of the dark ring, the annular light source is fully visible. – Although the approach
of the embedded view is reasonable and powerful, it has been uncommon in optics
education so far. Thus another aim of this paper is to demonstrate the explanatory
power of the embedded perspective as a possible approach to shadow imaging in optics
teaching.
The paper is organized as follows: In section 2we present shadow phenomena from
nature, everyday life, and the laboratory, which exemplify various cases of the totality
shown schematically in figure 2. They provide the starting point for an exploration of the
relevant conditions in section 3. On the basis of the embedded perspective we describe
the irradiance distribution in the shadow image and show that this corresponds to a
convolution of light source and object [8]. We investigate the transformation properties
of shadow images and we show the invariance of the imaging conditions under inversion
of the object [9]. We arrive at the little known fact that shadows are complementary
phenomena. In this context we describe another complementary shadow phenomenon
that has not yet received proper attention. The bright shadow occurs by inverting the
light source [10]. In section 4we give a simple mathematical model for the irradiance
distribution within shadow images which can reproduce all the phenomena described in
this paper.
2. Shadows in the laboratory, nature and everyday life
To say that the edges of shadows appear smooth because the light source is extended, is
not wrong, but it downplays the fact that there is a characteristic imprint of the source
geometry that can be discovered in the shadow image. Figure 3shows shadow images
of a pliers illuminated with various shapes of light sources. The distances between light

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Figure 3. Shadow images of a pliers illuminated with different light source geometries.
Figure 4. Shadow of a fly during its journey on a lampshade. That it is the fly’s
shadow, can only be recognized by the movement. It has no resemblance to the fly,
but shows the dark image of the illuminating filament.
source, pliers and screen remain unchanged, only the geometry of the light source is
varied (shown at the bottom left of each picture). The shadow image obviously refers
not only to the pliers as an obstacle, but also to the particular shape of the light source,
which gives it a characteristic, aesthetic appearance. It seems as if the sharp outline of
the pliers’ shadow has been softened by the shape and orientation of the light source –
the shadow image could be thought to have been created by drawing with a pen whose
tip has the shape of the light source geometry [9,11].
While in figure 3the geometry of the light source mainly influences the penumbra
of the pliers’ shadow, the next example from everyday life shows that the shadow image
can adopt the source geometry completely [12]. In figure 4a fly wanders along the edge
of a lampshade and creates a shadow on the wall. The shadow image takes on the very
different but distinct shape of a filament, which is a dark image of the incandescent
filament of the bulb. Similar transformations of shadow images can be discovered in
everyday life [13,14]. Figure 5shows a bouncing ping-pong ball illuminated from above
by a rod light. Whenever the ball’s distance from the ground increases, its shadow
stretches in a direction determined by the orientation of the rod light: the shadow
becomes the dark image of the latter.
Though somewhat more subtle, the influence of source geometry is often noticeable

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da 5
Figure 5. Shadow of a bouncing ping-pong-ball illuminated by a rod light.
Figure 6. Shadow image of crossed fingers on a sunlit wall. The larger the distance
to the wall, the more dominant the image of the solar disk: ”sun coins”. Fotos: Laila
Ellis.
at shadow edges. For example, where the trunk of a tree or a street light meet the ground
the shadows always appear much sharper, richer in contrast and darker than the shadow
areas from higher up. As the distance from the obstacle increases, the shadows become
softer and lighter, i.e., the geometry of the circular Sun makes a stronger influence.
When we look at our own shadow in sunlight, the penumbra always carry inscribed
images of the Sun, since the Sun has an extension of about 0.5 degrees (figure 6).
More familiar than imaging with obstacles is imaging with apertures [15,16,17].
Figure 6and 7show ”sun coins” (in German called ”Sonnentaler”), i.e., pinhole images
of the Sun on the ground (figure 7a) under the high canopy of a tree, which are described
frequently in the literature [18,19,20]. They are the complementary counterpart to the
dark ”sun coins” [21] in figure 7b, where the imaging elements are the leaves themselves.
The extent to which the appearance of shadows in nature is influenced by the size and
shape of the solar disk is particularly evident in the transformations of shadow images
during the covering phases of a solar eclipse (figure 8). The symmetry of the light sources
in the previous examples hid the fact that the pinhole image is point-symmetrically
mirrored. Figure 9shows the transition of complementary shadow images for an L-
shaped geometry of the light source.
Little known so far is the bright shadow [10]. It appears, for example, in the interplay
of two obstacles standing offset to each other, which are illuminated by an extended light
source. It can be discovered easily in the diffuse shadow of a bar placed on a table in

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Figure 7. The leaves of a tree creates both pinhole images (a) and pinspeck images
(b) of the sun (reprinted with kind permission of H.-J. Schlichting).
Figure 8. Images of the partially eclipsed sun during a solar eclipse. Foto: Bill
Gozansky.
Figure 9. Complementary shadow images created with an L-shaped light source (9
x 9 cm) and a square geometry of the object (2.25 x 2.25 cm) (setup shown in figure
13). The distance between the light source and the projection screen was 2.4 m. The
ratio of the distances to the object is given in the upper row.

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Figure 10. The ordinary shadow of a bar in diffuse lighting from a window to the
bright sky (top left) is maximally blurred. Where the darkening by a second bar is not
effective because of the occlusion by the first, it is brighter than in the surrounding
area.
Figure 11. A square frame of paper, suspended by a nylon thread, is illuminated by a
horizontal rod lamp (a). If the lamp is covered at one point, a bright shadow appears
in the diffuse shadow of the frame (b and c).
front of a window in the glare of the bright sky, if you move a second bar in between.
Then a bright and relatively sharp stripe appears in the diffuse shadow of the first bar
(figure 10). Its trace on the table refers to the position of the moving bar, which so to
speak acts as an “inverse light source”. Similarly, the vertical bars of a window front
become visible as bright stripes in the diffuse shadow of an upright note-taking pencil.
Figure 11 shows what matters for the formation of bright shadows. If an extended light
source (here a rod lamp) is darkened at a certain place (figure 11, below), a relatively
sharply contoured but now bright shadow image of the object appears in the diffuse
shadow of the object.
The showcases mentioned exemplify how diverse and impressively beautiful the
world of shadow images is. What can we learn from these examples? In general, both

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Figure 12. View from the embedded perspective on light source and object while
moving from the left into the shadow. At the border of penumbra, the light source
is just unobscured (left). In the penumbra, the light source is partially obscured: the
smaller the unobscured part, the darker the penumbra (center). At the boundary from
penumbra to umbra, the light source is just fully obscured (right).
the shape of the light source and the shape of the object interact in shadow images. They
are involved both as imaging and as being imaged. Seemingly complicated shadows are
often created by the intertwining of unusual geometries of light sources and object. At
first glance, which of the effective geometries predominates in the shadow image depends
on the relative distances between the light source, object and projection screen. Closer
examination in the next section shows that what matters is the relative size of the light
source and the object – as seen from the projection screen.
3. Exploring soft shadow images
3.1. Investigating shadow images from the embedded perspective
As it is known from the observation of solar eclipses, the formation of shadows can be
explained by relating the irradiance of a given point in the shadow to the degree of
occlusion of the light source by the object – as seen from that point on the projection
screen (where it can be a human eye or a technical eye, e.g. a detector) [22]. In contrast
to the detached perspective typically discussed in the side view, here the eye is part of
the setup. We therefore refer to this view from the screen as embedded perspective. It
is always used when considering what a detector ”sees”, i.e. when thinking through the
conditions of an optical imaging system not only from the light source but also in the
opposite direction, i.e. from the detector.
In the case of shadow images, these conditions are given by very simple perspective
and parallactic properties of the spatial interplay between light source and object with
respect to the projection screen (figure 12): No shadow is where the light source is
unobscured from; penumbra is where only part of the light source can be seen from;
umbra is where the light source is fully obscured from. To illustrate the approach with
an experiment we drilled a small hole in the projection screen of the setup shown in figure
13. It projects a pinhole image of the lighting situation onto a second semitransparent
screen. The pinhole image then shows what is visible from the respective pinhole position
while moving the first screen through the shadow image (figure 14, 1-6 on the right): a
projection of the object into the plane of the light source.
In optics education, however, the embedded perspective has been rather uncommon,

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da 9
Figure 13. Experiment to illustrate the embedded perspective. While the projection
screen with the pinhole is moved through the shadow image, the respective pinhole
image shows what can be seen from the location of the pinhole: a projection of the
object (here an obstacle) into the plane of the light source (figure 14, right).
Figure 14. Left: Photo of a shadow image at the projection screen from the setup
shown in figure 13 with six pinhole positions. Right: The corresponding pinhole images
show the transformation of the occlusion situation as seen from the respective pinhole
position in the shadow image: While the ”pinhole eye” moves from position 1 to 6, the
square obstacle passes in front of the light source due to parallax from right to left.
although it has didactic potential. It links the respective optical conditions
to the student’s own experiences, thereby encouraging a more phenomenon-based
understanding of optical concepts. In the German literature there is a phenomenological
tradition in optics teaching where the change between embedded and detached
perspective as a powerful exploration tool has long been established [10,23,24]. The
idea of teaching optics close to the phenomena (”the world through my eyes”), especially
in the early grades, has been increasingly taken up in recent years and investigated in
empirical studies of phenomenological optics teaching [25,26,27,28,29].
3.2. Adjusting the conditions of shadow images
How bright it is at a given point on the screen depends on the degree of occlusion of the
apparent sizes of light source and object in the field of view of the embedded observer.

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Figure 15. To the observer in P, the light source ¯
Alappears at solid angle ¯
Ωl,
whose projection ¯
Ωlp =¯
Ωlcos ε2is a measure of the irradiance in P. For small angles
ε2,¯
Ωlp ≈¯
Ωl.
As known from astronomy the apparent size can be quantified using the concept of solid
angle, which is measured with respect to the observer’s eye. In the case of an obstacle,
the solid angle describes the opaque part of the object, while in the case of an aperture,
it describes the transmissive part of the object. In the following, we will focus on the
case of an obstacle and later return to the general case of objects, which includes the
inverse case of the aperture (subsection 3.5).
We assume that the shadow images under consideration are created in the vicinity
around the optical axis. Then we can simplify the solid angles of light source Ωland
obstacle Ωoto appear approximately constant for the observer. That means for different
positions Pon the screen the seen sizes of light source and obstacle only shift to each
other due to parallax without perspective foreshortening. Then we can approximate
Ωl≈Al/(d1+d2)2and Ωo≈Ao/d2
1for the solid angles of the light source and the
obstacle, where Aland Aoare the absolute areas of the light source and obstacle,
d1+d2is the distance from the light source to the screen and d1is the distance from
the obstacle to the screen. For small angles ε1,¯
Ωl≈¯
Al/(d1+d2)2should then be the
solid angle of the seen, i.e. unoccluded, area of the light source for a certain occlusion
of the light source by the obstacle.
On the other hand, the irradiance on a given surface is also affected by its orientation
ε2relative to the light source, which is determined by the projected solid angle ¯
Ωlp (figure
15). This is especially apparent in the case of Sun position-dependent illumination,
where the amount of light received by a surface is influenced by the position of the Sun
in the sky. Although the seen solid angle of the Sun remains approximately constant,
the setting Sun near the horizon delivers much less light than a Sun in the zenith. For
small angles ε2the projected solid angle can be approximated by the solid angle itself,
i.e. ¯
Ωlp ≈¯
Ωl. Thus, the solid angle can be equated to the unoccluded portion of the

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Figure 16. To illustrate the convolution of light source and projected obstacle they
are expressed as functions in the source plane. Right: If the observer Pmoves on the
projection screen, the obstacle shifts in the opposite direction in the source plane. The
visible part of the source then corresponds to the overlap of the two functions of source
and obstacle.
light source that is visible to the observer.
If we further assume that the light source is Lambertian with constant radiance,
then the irradiance E(P) at a given point Pon the screen is approximately proportional
to the solid angle ¯
Ωlfor a certain occlusion of the light source by the obstacle. It holds
that E(P)∝¯
Ωl(more in section 4).
Although the idea of the embedded perspective applies much more generally,
shadow images can be easily modeled under these assumptions and it is approximately
satisfied in most teaching experiments.
3.3. The shadow image as convolution
To determine the solid angle ¯
Ωlof the unobscured light source, one must appropriately
weight the solid angle Ωlof the complete area light source with the solid angle Ωoof the
obstacle for each location Pon the screen. Mathematically, this process corresponds
to a convolution operation between the geometries of light source and obstacle, thus
shadow imaging is a beautiful example of convolution [8,30].
Qualitatively speaking, we represent light source and obstacle as functions in the
source plane. We therefore project the obstacle into the plane of the light source so that
Ωo=Ao/d2
1=A′
o/(d1+d2)2applies, with A′
othe projection of the obstacle’s area Aointo
the source plane. For simplicity, we consider the one-dimensional case here; the general
case will be examined in more detail in section 4. Let l(xl) and o(xl) be representative
functions of the light source and the projected obstacle in the source plane as shown in
figure 16. The integration Rl(xl)dxland |Ro(xl)dxl|respectively give the ”areas” of the
light source Aland the projected obstacle A′
o. If the observer moves to the left on the

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screen, the obstacle moves to the right in his field of view, i.e. the projected obstacle
shifts with some xi, so o(xi+xl). The ”area” of the unoccluded light source ¯
Alis then
given by Rl(xl)o(xi+xl)dxl[31]. Because of E(P)∝¯
Ωlfor the irradiance E(xi), the
following applies
E(xi)∝Z∞
−∞
l(xl)o(xi+xl)dxl.
With the substitution xl=−x′
lwe get the definition of the convolution product
E(xi)∝Z∞
−∞
l(−x′
l)o(xi−x′
l)dx′
l=l(−xi)∗o(xi).
3.4. Transformation conditions of soft shadow images
So far, we related the transformation of soft shadow images to the change in the relative
distances of the light source, obstacle, and projection screen (figure 1): As the obstacle
moves closer to the screen, the shadow image almost assumes its shape. In the other
limiting case the obstacle moves toward the source and the pattern converges with the
shape of the source. Evidently not every arbitrary obstacle produces the image of the
light source when approaching it. And with respect to the sun, an indication of the
change of distance is meaningless. The relevant conditions are easily understood from
the embedded perspective: the darkness of the shadow is determined by the ratio of the
solid angles of the obstacle Ωoand the light source Ωl, as seen from the screen. As long
as the distance between the light source and the projection screen remains unchanged,
Ωlis constant. For the ratio to Ωo, this leads to the following characteristic cases: If
Ωo>Ωlholds, the light source is fully occluded for an extended area at the screen. The
shadow image is dominated by the umbra and thus by the geometry of the obstacle. In
the reverse case Ωo<Ωl, i.e. the seen size of the obstacle is smaller than the light source.
The obstacle can only cover a small part of the light source. It thus changes from being
the imaged element to being the imaging element. The shadow image is dominated by
the penumbra and thus by the geometry of the light source. In between is a position
where the magnitude of Ωoand Ωlare in the same range: Ωo≈Ωl. The visible size
decreases with the square of its distance from the embedded observer. Thus the image
shows a balance between the two shapes when Ao/Al=d2
1/(d1+d2)2. In summary,
referring back to the scheme in figure 2, the following three cases can be distinguished
(figure 17, top row) [9]:
(i) Ωo>Ωl: If Ωois larger than Ωl, the image of Aopredominates, →hard shadow
(ii) Ωo≈Ωl: The image shows a balanced mixture of shapes Aoand Al→soft shadow
(iii) Ωo<Ωl: If Ωois smaller than Ωl, the image of Alpredominates →pinspeck image
If the light source is very small (Ωo≫Ωl) we obtain very sharply contoured shadow
images of the obstacle, which belong to the common notion of shadows.

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3.5. Complementarity of shadow images
Principles of complementarity are introduced in optics mainly to order color spaces, i.e.,
to define the complement of a given color [32]. Somewhat generally, the complement
is understood to be something like the completing counterpart of a whole, where the
whole remains invariant under certain conditions. Following on from this, we call two
irradiances E1and E2complementary to each other if they satisfy the relation
E1(P) + E2(P) = E0.(1)
For a given irradiance E0(the ‘whole’), the irradiance E2=EC
1is then complementary
to E1, where the ‘C’ stands for ‘complementary’ [33,34]. As we will now see, inverting
the object (obstacle ⇀
↽aperture) of a given irradiance distribution produces a second
one for which this criterion is satisfied for all locations on the screen.
To verify the complementarity relation for shadow imaging we consider the
following. Inverting the object means swapping the opaque and the transmissive parts
of the object while keeping the geometrical properties invariant (figure 17 top row and
bottom row). From the embedded perspective, however, this also means that the visible
and non-visible parts of the light source are swapped. If Ωlis the total solid angle
of the light source, and ¯
Ωlthe unoccluded, i.e. visible part of the source for a given
position Pon the screen then we get ¯
ΩC
l= Ωl−¯
Ωlas the visible part of the source
in the inverted case. The previously invisible parts of the source are now visible. As
discussed the irradiance E(P) is under the specified simplifications proportional to the
magnitudes of the respective solid angles of the visible part of the area light source, i.e.
E1(P)∝¯
Ωl,E2(P)∝¯
ΩC
land E0∝Ωl, where E0is the constant irradiance at Pdue to
the unoccluded light source with the fixed solid angle Ωl. Because this holds for every
point on the screen, it results in E2(P) = E0−E1(P) and the complementarity relation
(1) is fulfilled. Therefore E2=EC
1. In this sense, all pairs of images in the top and the
bottom row from figure 1respectively figure 9are complementary to each other.
The above mentioned transformations (i-iii) are invariant under inversion of the
object, i.e. if we replace an obstacle by a geometrically equivalent aperture, where Ωo
now describes the solid angle of the transmissive part of the aperture (see figure 17).
The first case (i) with Ωo>Ωlleads to the image of the aperture where the geometry
of the source approximately being suppressed. The third case (iii) with Ωo<Ωlleads
to the pinhole image where the solid angle of the aperture is small in comparison to the
solid angle of the area light source.
3.6. Conditions of the bright shadow
Inversion was previously limited to the plane of the object. What happens when the
light source is inverted? The interchange of luminous and nonluminous regions in the
plane of the light source yields an inversion of the ordinary shadow, called bright shadow
(figures 10 and 11) [10]. An inversion of the radiance can of course only be realized
approximately, e.g. by covering an extended luminous field of homogeneous radiance

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Figure 17. Shadow images are determined by the ratio of the solid angle Ωlof light
source (L-shape, white) and object Ωo(square obstacle above, corresponding aperture
below, both black; background: grey) for cases (i)-(iii), seen from the embedded
perspective. The case Ωo<Ωlin the top row on the right corresponds to the pinhole
image no. 4 in figure 14.
Figure 18. Schematic of the bright shadow (top) and the conditional occlusion as seen
from the embedded perspective of the projection screen (bottom). By inserting G2, it
becomes darker everywhere on the projection screen except for the area from where
G2is masked by G1(view from P2). Therefore the bright shadow appears not because
this region becomes brighter, but because the surrounding region becomes darker.
in a limited area. The covered area in the bright environment then corresponds to the
“inverse light source“.
Initially, we aim to elucidate how the occurrence of brightening in shadows can be
comprehended from the embedded perspective. In the setup of figure 10 we used a large
window to the bright sky which was partially covered by a bar. Since the important
thing here is to cover an area within an extended field of homogeneous radiance, for

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da15
simplicity we consider in the following a ”one-dimensional” extended rod light. Figure
18 schematically shows how an upright cylinder G1is illuminated with a horizontally
aligned, stretched rod light. The inverse light source is realized by the covering part of
the cylinder G2. As in the ordinary case the geometry of G2determines the size of the
penumbra, the one of G1the size of the umbra. The view from the embedded perspective
(figure 18, below) makes immediately clear what matters: Because the unobscured part
of the rod light seen from P2is larger than from P1and P3, it is also brightest there.
To clarify the bright shadows relation to the obstacle, we utilize the transformation
conditions outlined in subsection 3.4. We define Ωoagain as the solid angle of the
obstructing region of the obstacle, and Ωlas the solid angle of the ”inverse light source,”
which arises from the obstructed area in the extended illuminated field. In figure 11,
for example, the condition Ωo>Ωlis met. When Ωois larger than Ωl, the image of Ao
becomes dominant, resulting in a hard but bright shadow.
One might think the bright shadow is a mere trick. In fact, it is nothing more
and nothing less than the inverse of the ordinary shadow. The latter arises where the
brightening of the dark space by a light source does not become effective because of an
obstacle. The bright shadow arises where the darkening of the illuminated space by an
inverse light source does not become effective because of an obstacle.
Although the conditions may seem somewhat artificial at first glance, it is easy to
discover the bright shadow in nature and in everyday life, you just have to learn to see
it [10,35]. As a more detailed analysis of the inversions presented here shows, these are
only partial inversions. A generalization of the issue is beyond the scope of the present
article. As has been shown by Rang, a strict generalized complementarity condition
requires the inversion of the entire space, i.e., the swapping of the flux density at each
point in space, and thus leads to the theory of bright space [36].
4. Modeling soft shadow images
4.1. Assumptions of the model
In the following we present a simple model for the computation of soft shadow images,
which is still elaborate enough to capture all phenomena discussed in this paper (see
also [8,37]). The model is based on the following assumptions:
(i) Object and light source are treated as shapes that lie in planes parallel to the screen.
(ii) The area light source is treated as a Lambertian emitter, i.e. the radiance L[W
m−2sr−1] is constant over the whole surface and in all directions. For a uniform
diffuse emitter, the radiance is given by the radiosity (radiant exitance) M[W m−2]
radiated over the hemisphere, i.e. L=M/π = const. for all points on the source
[38].
(iii) The shadow image is generated in close proximity to the optical axis on the screen,
thereby satisfying the paraxial approximation. As a consequence, we assume that
the projected solid angle of ¯
Ωl, which corresponds to the visible and unobscured

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da16
light source, remains constant and can be approximated to the solid angle itself.
Moreover, we ignore the effect of perspective shortening on the area of the light
source and object when they are viewed at oblique angles.
(iv) The screen is a Lambertian surface, which is a perfect diffusing reflector that
exhibits uniform radiance in all directions, and the incoming irradiance is
proportional to the radiosity of the surface. Consequently, the shadow image that
is observed on the screen can be characterized by the irradiance distribution.
(v) The object is large with respect to the wavelength of light, which is why effects
due to the wave nature of light (e.g. diffraction or interference) are neglected [39].
Physiological and psychological effects, for example the Machband effect or Koffka’s
effect are also neglected [40,41].
4.2. Visible part of the light source
We define a cartesian coordinate system in which the xy-plane is the plane of the screen
and the z-axis is in the direction of the illuminating source with area Aland the object
with area Aoof the occluding part. Parallel to the imaging plane Σi(the screen) the
object plane Σois positioned at the distance d1. Furthermore the plane Σl, where the
light source is located (figure 19), is positioned at distance d2. We can describe an object
as the binary function o:R3→Rin the following way:
o(ro) = 


0 if rois on the occluding regions of the object,
1,otherwise, (2)
where 1 means transmission and 0 opacity. We get the inverted object by defining
the function 1 −o(ro). Similarly we can define a binary function of the light source
l:R3→Rwith
l(rl) = 


1 if rlis on the source,
0,otherwise, (3)
where 1 means luminous and 0 means not luminous. We will now consider a general
representation of the projection of the object Aothat is produced by an arbitrary viewing
point Pon the screen. For this riis an arbitrary but fixed vector in Σipointing towards
P. For a vector ropointing towards Aoand the corresponding vector rltowards Σlwe
can formulate the relationship
ro−rl=λ·(ri−rl),(4)
which requires that ro−rland ri−rlbe linearly related with the vectors
ro=



xo
yo
d1



,rl=



xl
yl
d1+d2



,ri=



xi
yi
0



.(5)

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da17
Figure 19. Simple geometric model for the generation of shadow images.
Substituting the vectors in (5) into (4) gives the modification factor
λ=d2
d1+d2
.(6)
From (6) and (4) we obtain
ro=d1·rl+d2·ri
d1+d2
.(7)
For a given point rion the screen the function o(ro) in (2) is transformed by (7) into the
projection o(rl,ri) on the plane of the light source. We can now define a new function
¯
l(rl,ri) = l(rl)o(rl,ri) which is 1 when a point of the light source is visible from P, i.e. it
is not obscured by the projected object. Therefore we get the value of the unobscured,
i.e. the visible luminous area ¯
Alin Σl
¯
Al=Z¯
Al
dAl=ZΣl
¯
l(rl,ri)dAl=ZΣl
l(rl)o(rl,ri)dAl.(8)
4.3. Irradiance distribution in soft shadow images
The differential solid angle dω of the luminous area element dAl, as seen from Pat the
angle ε1and at the distance |rl−ri|is given by dω =dAlcos ε1· |rl−ri|−2(figure 20).
For the radiant power d2Φ [W] transmitted from the surface element dAlto dAiwe then
obtain:
d2Φ = LdAicos ε2dω =LdAicos ε2·dAlcos ε1
|rl−ri|2.(9)
This relationship (9), also known as the photometric fundamental law, states that
the radiant power transmitted from dAlto dAiis proportional to the apparent areas

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da18
Figure 20. Radiant power on dAi.
dAlcos ε1and dAicos ε2and inversely proportional to the distance squared |rl−ri|2,
where Lis the radiance from dAlto dAi. The radiated power towards the element dAi
from all elements dAlof the unobscured area light source ¯
Alvisible from Pgives the
irradiance E(ri) [W m−2] at P:
E(ri) = dΦ/dAi=Z¯
Al
Lcos ε2·cos ε1
|rl−ri|2dAl.(10)
The expression dω = cos ε1· |rl−ri|−2·dAltakes into account the perspective
shortening of the area light source and dωp=dω·cos ε2is the projected solid angle, which
describes the orientation of the illuminated surface relative to the light source. From the
geometric setting of the model, it follows from assumption (i) that ε1=ε2. According
to assumption (ii) we can replace the radiance L=M/π. Considering assumption
(iii) of the paraxial approximation, we can set cosε1= cos ε2≈1 for small angles and
|rl−ri| ≈ d1+d2for the distance. With these simplifications and with (8) the irradiance
distribution (10) finally results in
E(ri) = M
π(d1+d2)2Z¯
Al
dAl=M
π(d1+d2)2ZΣl
l(rl)o(rl,ri)dAl.(11)
The expression ¯
Al/(d1+d2)2=¯
Ωlcorresponds to the solid angle of the visible area
light source. Therefore, E(ri)∝¯
Ωlis approximately valid under the given assumptions.
If we use the substitution rl=−(d2/d1)r′
l, (11) can be transformed with (7) into a
convolution integral [8]. We obtain
E(ri) = M
π(d1+d2)2 d2
d1!2
ZΣl
l −d2
d1
r′
l!o d2(ri−r′
l)
d1+d2!dA′
l(12)
=Md2
2
πd2
1(d1+d2)2l −d2
d1
ri!∗o d2
d1+d2
ri!.(13)

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da19
We obtain from (11) the irradiance distribution EC
1for inverted objects or EC
2for
inverted light sources (bright shadow situation) by replacing o(rl,ri) by 1 −o(rl,ri)
in the first case and l(rl) by 1 −l(rl) in the second case. To confirm the relation for
complementarity (1) in each case, we add either EC
1or EC
2to E(ri) from (11). In the
first case we get
E(ri) + EC
1(ri) = M
π(d1+d2)2ZΣl
l(rl)dAl=MAl
π(d1+d2)2=E0,(14)
which is a constant for fixed distance d1+d2. Inverse objects therefore lead to
complementary shadow images. For the second case we obtain
E(ri) + EC
2(ri) = M
π(d1+d2)2ZΣl
o(rl,ri)dAl.(15)
Due to the definition of o(rl,ri) the integral in (15) diverges. In practice, the non-
luminous geometry of the inverted light source is embedded in a luminous area of finite
size. Therefore the integral is constant for every location P. It follows, that the bright
shadow is also a complementary shadow phenomenon.
4.4. Test of the model
An analytical closed-form solution is not possible even for simple geometries, so all
calculations are numerical, performed with Mathematica. For comparability of model
and experimental data, we used the relative irradiance E/E0=¯
Al/Al. The range of
values of (8) was mapped to a linear gray scale between white and black (8 bit), where
we set the endpoints to black for 0 ( ¯
Al= 0, the light source is not visible) and to white
for 1 ( ¯
Al= 1, the light source is completely visible).
We will apply the model to the L-shaped light source and the square object from
figure 13 with the ratio of their edge lengths of 4 : 1. For (3) of the source we let
l(rl) = 


1 if (−4.5≤xl, yl≤4.5) ∨(−4.5≤xl≤0∧0≤yl≤4.5)
0,otherwise (16)
and with (2) for the obstacle we let
o(ro) = 


0 if (−2.25 ≤xo, yo≤2.25)
1,otherwise. (17)
We chose the distance to be d1+d2= 240. Figure 21 shows for different stages d1/d2
the computed transformation of the shadow images for an obstacle (below) and the
inverse aperture (above). The values d1/d2were chosen so that the model reproduces
the transformation characteristics of the shadow images from figure 9.
For experimental testing of the predicted irradiance distribution (11) we used a
light source which allows to realize extended fields of homogeneous radiosity [42]. It
consists of a hemisphere (Ø = 30 cm), the inside of which is coated with a high-matte
white and is illuminated with 4x500W halogen lamps. In front of the opening can be

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da20
Figure 21. Above: The function of the light source (16) and the obstacle (17). Below:
Computed transformation of soft shadow images produced by aperture (above) and
obstacle (below) for a L-shaped light source and different stages d1/d2. – Compare to
the respective setup in figure 13 and the photographs in figure 9.
placed apertures cut from sheet steel with different geometries. The distances were set
to d1/d2= 1. The photos of the shadow images were taken with a photo camera (Nikon
D60, Sigma DC 1008072). A camera usually has a nonlinear grayscale value output. To
avoid this problem, we used the RAW files (NEF) for the analysis, where the camera
writes the data to the storage medium after digitization largely without processing.
For lossless image editing, the RAW files were converted compression-free into an 8-bit
grayscale TIFF-image. To ensure the linear gray scale, all further image editing was
performed without any gamma-correction (γ= 1). Using tone correction, we mapped
the photo to a scale from black to white. Furthermore, the image analysis shows noise
superimposed on the irradiance distribution due to the graininess of the screen. We
averaged out the fluctuations by local blurring. The photographs edited in this way
were analyzed with the image processing program Image Analyzer.
Not only do figures 22 and 23 show remarkable agreement between the
photographed and computed shadow images in the visual inspection, comparing E/E0
for the selected profile lines allows a quantitative reproduction of the characteristic
irradiance distributions by the model.

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da21
Figure 22. Left: The photo of the shadow image was taken with an exposure time
of 0.125s, an aperture of f/10 and a sensitivity of ISO400. Right: The corresponding
computed shadow image. Below: The measured curves of E/E0together with the
theoretical curves as predicted by the model for the two lines 1 and 2.
5. Summary
Our primary goal in this paper was to provide the reader with an overview of multiple
shadow phenomena that can be motivationally accessed with the embedded perspective.
For practical teaching we can only give some hints at this point. The material for soft
shadow images presented in this paper provides several starting points for teaching
optics at both the high school level and the undergraduate level.
The idea of the embedded view makes difficult irradiances on the projection screen
understandable even to students at the undergraduate level. ”Bright is where bright
can be seen from” sounds almost trivial, but operated systematically, leads to basic
imaging principles in shadow imaging. Even without the theoretical model, students
can understand the transformation behavior or the irradiance distribution of shadow
images [43]. In addition, the connection to their own visual experience can motivate
and increase the feeling of self-efficacy. The approach of the embedded perspective can
also be applied to other optical contexts such as the mirror, the lens or diffraction and
can therefore also be used in optics lessons [26,44,45,46,47,48].
Shadows have long been part of the canon of optics education, usually focusing
on special cases. Either one considers hard shadows caused by point light sources or

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da22
Figure 23. Left: The photo of the complementary shadow image was taken with an
exposure time of 1.3s, an aperture of f/14 and a sensitivity of ISO400. Right: The
computed shadow image. Below: The measured curves of E/E0together with the
theoretical curves.
pinhole camera images caused by expansionless apertures. As was shown, the shadow
image generally contains geometric information of both the object and the light source,
i.e. both have an imaging effect at the same time. As we have also seen, soft shadow
images can be generalized by the principles of complementarity and inversion. In the
classroom, the example of shadow images can illustrate how the same imaging laws
are at work in the variety of shadow phenomena and provides an example of the idea
of unification – an important principle in physics. From a more technical perspective,
students may be interested in how to reconstruct the geometries of the light source and
object from a given shadow image [49,50].
As shown in (12), the integral in (11) can be written as a convolution product. This
is an important result, because it means that the irradiance distribution is finally the
consequence from a convolution operation between the geometries of object and light
source. Thus, a fundamental concept as convolution, whose first important application
is usually the lens, can already be introduced in shadow images. The same applies to
the ideas of complementarity and inverse objects, which proved fruitful here, and which
reappear in the context of complementary colors and spectra [33]. Thus, the shadow
image not only offers access to some high-level optical ideas, but it is also a nice example

Soft Shadow Images – Grebe-Ellis & Quick 2023 Eur. J. Phys. DOI 10.1088/1361-6404/acc7da23
of what Berry called the arcane in the mundane, i.e. ”to bring out connections between
what can be seen with the unaided, or almost unaided, eye and general explanatory
concepts in optics and more widely in physics and mathematics” [51].
Shadows are ubiquitous and speak to our sense of beauty. They offer several starting
points for teaching optics, provide insights into modern imaging techniques and can still
lead to exciting discoveries like the bright shadow.
Acknowledgments
The authors thank all anonymous referees for improving the clarity and presentation of
this document. In particular we would like to thank the second reviewer for the idea to
illustrate the embedded perspective by the setup shown in Figure 13 and for a valuable
discussion.
Data availability statement
The data that support the findings of this study are available upon reasonable request
from the authors.
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