Applying Minnaert
Min Liu
Professor Marc Levoy
The Science of Art
February 20, 2003
All photos were taken
with an Olympus D-520 Zoom
digital camera
Leaves
“When the leaf is illuminated from the front
(relative to the observer), a bluish hue
mingles with the green, and when from the
back, a yellow hue.”
(Minnaert, 356)
One can see that the more yellow leaves are
illuminated from the back side and the
deeper green-colored leaves are directly
illuminated.
Minnaert comments that leaves take on complex
forms of illumination due to variables that
include double-sided illumination, leaf
surface texture, chlorophyll presence, and
infrastructural intensity that reflects light.
-This photo was taken on a sunny late morning
in late August, 2002.
Leaves, continued
“… in a leaf, though much less than 1mm thick, all the
processes of reflection, absorption, and scattering
take place in the same way as in an ocean tens or
hundreds of meters deep. Absorption is caused here
by the chlorophyll grains; scattering is probably
brought about by innumerable grains of all kinds in
which the contents of cells are so rich, or perhaps by
the unevenness of the leaf’s surface.” (Minnaert,
356)
The intricate infrastructure of leaves is
illuminated by sunlight. One can see the play
of light on the leaves, emphasizing small dark
spots on the clover-like petals (right) and the
red hues on the leaves in the photo above.
Photos taken late morning, Feb 18, 2003.
Leaves, continued
“From an optical point of view, a leaf is much
more complex than a lake or a sea; … [I]t is
illuminated not on one, but on two sides;
moreover, one side is matt and the other
shiny, while the intensity and color of the
incident light are usually different at the two
sides. The possible combinations of optical
phenomena are astronomical!”
(Minnaert, 356)
This image of a large clove-like plant has a
“matt” or fuzzy underside as the sunlight
shows. The shiny front side is absorbing
light as well, causing certain areas of the
leaf to look yellow-green. Light on both sides
create complex green hues on this leaf.
Photo taken late morning, Feb 18, 2003.
Images of the Sun
“In the shade of a group of trees the ground is dappled
with spots of light, some small, some large, but all
regularly elliptical … they are… sunlight that falls
through some opening in the foliage: all we see
here and there between the leaves is a blinding
ray of light.”
(Minnaert, 1-3)
The “sunspots” are caused by myriad beams of light
that together flood through tiny openings in the
foliage. The collective light that floods through the
openings and land on the ground is conical
because the sun is not a point source; light shines
through the foliage opening at different angles,
thus filling in an ellipse on the ground.
The sunspot is an ellipse because the ground cuts the
light cone at a non-right angle. The elliptical
sunspots are especially apparent during the
morning and afternoon, when the sun is not
directly overhead.
-This photo was taken during the late morning of Feb,
18, 2003.
On the other hand, when the sun is overhead, there is a bigger chance that the sunspots on the
ground form circles. Similarly, when light shines through foliage openings and hit a wall in the morning,
the sunspots formed tend to be more like circles than ellipses. This is due to the angle at which the
sun shines through the foliage opening and the upright angle of the wall.
Minnaert states that “we see the sun’s disk at an angle of 1/108 radian.”
With this formula, we can calculate the height of the tree and the length of the sunspot from the foliage opening:
H = (k/b)L = 108k(k/b) where H = height of tree; L = distance of sunspot to foliage opening;
k = minor axis of sunspot; b = major axis of sunspot.
I measured the elliptical sunspot of an oak tree. k = 5”; b = 20” ; H = 135” or 11’ 3”; L = 540” or 45’. The results turned out to be
erroneous. A problem I encountered was that the sun moved rapidly through the foliage openings, hence changing the ellipse.
(Notice the changing shape of the ellipse as I measure the major and minor axes. The time interval between these images was
about a minute).
Shadows
“When you look at your shadow on the ground,
you will notice that the shadow of your feet
is clearly defined, whereas that of your head
is not. The shadow of the bottom part of a
tree or post is sharp, while that of the higher
part becomes increasingly unclear toward
the top.”
(Minnaert, 4)
Although this is a perspective view of the tree
trunk shadow, Minneart’s observations still
hold.
-This photo was taken in the late afternoon of
January 2003.
Window glass & Plate glass
“The reflections from windows indicate whether they are normal window glass or plate glass; if the latter, the images are
fairly clear; if the former, they are so irregular that the unevenesses of the glass can be seen clearly” (Minnaert, 22)
Window Glass - “warped” reflection
A Stanford engineering building
Noon, Feb 18, 2003
Plate Glass - a clearer reflection
Elliot Program Center
Late morning, Feb 18, 2003
Closed Coils of Light
“Remarkable is the appearance of closed coils of
light seen when the water surges gently, the
waves have short crests, and the light
source is high … When you look at the
water at a sufficiently large angle, you will
see the light source reflected by two
separate spots of light [S1 and S2] on each
wavelet, for instance, one at the crest and
the other at the trough of the wavelet … the
associated reflections [S1 and S2] are
always in the same plane of the wavelet …
When you look slightly to the left or right you
will see the reflections getting closer and
closer together until they fuse into one
closed coil whereby annulus is formed. After
all, the wavelets not only have a given
wavelength, but also a certain crest length;
when two crests merge, the tangent is
horizontal. But before that, a point must
have been reached where the slope was still
steep enough to reflect the light source to
our eyes: at that point S1 and S2 coincide.”
Minneart (30-33)
The crests of these closed coils of light on the
water reflect the concrete rim of the fountain.
-Midday, fountain outside Memorial Auditorium
Feb. 18, 2003
Irregularities on a Water Surface
“[On a water surface,] tiny mounds of water show up light or dark depending on the direction in
which you are looking … [for example,] inside each eddy [in a river] the tension is a little less and its
surface is slightly hollowed out to a depth of a few mm. In the vicinity of the boundary between light
and dark reflections, you can see clearly even the tiniest eddies. This is an application of natural
‘schlieren’.” (Minnaert, 21-22)
Schlieren are “regions of varying refraction in a transparent medium.”* Schlieren in these images
are the eddies found between the elliptical water crests. They almost seem like compressed lines of
the sun’s reflection, the blue of the tile bottom, and the sky.
*http://www.m-w.com/
Irregular
water
surface
from the
fountain
outside
Memorial
Auditorium
Noon
2/18/2003
Refraction by an Undulating Water Surface
“When a water surface is not perfectly smooth,
this is revealed by a change in direction of
broken rays of light and an uneven
brightness at the bottom.”
Minnaert (46-47)
The broken rays and uneven brightness are
caused by “rays of light [that] spread at the
center and then close up concentrically” on
the bottom of the pool. (Minnaert, 47)
In this image of the fountain in front of Memorial
Auditorium, the tiles at the bottom are lit
unevenly due to the light rays that spread
out concentrically. The light on the crest of
the water above the tiles combined with the
refracted rays at the fountain bottom are
causing a symphony of reflecting and
refracting rays.
Differences between an object and its reflection
“The closer the objects are to us, the lower their
images with respect to that of the
background”
“Although the reflection is identical to the object,
it looks different in perspective because the
two are shifted with respect to each other.
We see the landscape as if we were looking
at it from a point beneath the water’s surface
where the image of our eye is. The
difference becomes smaller the closer we
bring our eyes to the water and the farther
away the objects are.”
Minnaert (11-13)
Although it is difficult to distinguish in this image,
the palm tree and the conical tree on the hill
appear to be the same height. In the water’s
reflection, however, one can see (up close to
the image) that the conical tree is lower in
the water. This means that the conical tree is
closer to us than the palm tree, even though
they appear to be of the same height when
viewed directly. This optical effect occurs
because a reflected image does not
represent the true image.
Photo taken Feb 18, 2003; Lake Lagunita in the
late morning.
Reflections in Puddles
“The reflection of trees and shrubs in small
ponds and puddles at the roadside
sometimes have a more pronounced clairty,
sharpness, and warmth of color than the
objects themselves … The cause of these
differences is more psychological than
physical … The reduced brightness of the
mirror image is in itself beneficial for looking
at the sky and clouds, which otherwise are
somewhat too bright for our eyes.
Furthermore, the reflection is polarized, so
that it may attenuate the luster of certain
objects and saturate colors.”
Minnaert (13)
In this image, the puddle is in the shadow of a
tree, thus significantly reducing the
brightness of the sky. The sky in the puddle
has a blue hue opposed to the white sky on
that particular day.
-Trail alongside Lake Lagunita; late morning.
Feb 18, 2003
Freak Reflections
Freak reflections are caused by windows.
“The spots of light caused by standard window
glass are irregular, whereas those caused
by plate glass are far more uniform.”
Minnaert (15)
The uniformity and clarity of this reflection shows
that the reflecting window was made of plate
glass.
-The wall of a stairway in Adams dorm.
Late afternoon, Feb 18, 2003
The Rainbow in Artificial Clouds
“The way in which a rainbow arises in a mass of
drops of water is immediately visible to us
when we see the sun shining on the fine
spray floating above fountains and waterfalls
… Always look for rainbows in a direction 42
degrees away from the anti-solar point, and
preferably against a dark background.”
Minnaert (191-192)
-Rainbow formed by waterfall spray;
Vernal Falls, Yosemite. August, 2002.
Parhelic Circle
“After the small ring, the mock suns or sun dogs
are the most frequently encountered halo
phenomenon. These mock suns are two
concentrations of light on the small halo at
the same altitude as the sun … The intensity
of these mock suns is usually very great;
they are distinctly red on the inside, then
yellow, before changing into a bluish white.”
Minnaert (214)
Minnaert cites Greenler in explaining parhelic
circles; he states that “parhelia are formed
when the air contains enough crystals
floating horizontally like leaves. Through
such prisms, the rays of light no longer
travel along the path of minimum deviation,
because they do not lie in a plane
perpendicular to the axis.”
Q: From Greenler, it seems that parhelic circles
occur on cold days with floating crystals.
This photo, however, was taken on a sunny
warm August day in Southern California. Is
this really an image of a parhelic circle?
Parhelic circle seen with a cloud; mountains
above Hetch Hetchy Dam; August 2002
Clouds
“Those not accustomed to studying the heavens will be
surprised to learn that clouds can often show the most
glorious and the purest colors, such as green, purple-red,
blue …They are distributed irregularly over the clouds in
the form of colored edges, spots, and bands; some
observers maintain that they have a ‘metallic’ luster; what
do that mean? Our feelings at the sight of such lovely
clouds are of intense delight, which is difficult to
describe, but which is certainly caused, to no small
extent, by the purity of the colors, their delicate
intermingling, and their radiant light. It is difficult to take
your eyes off their exquisite sight.”
Minnaert (250)
Scattering of Light by Clouds
“It is remarkable how certain kinds of cloud obscure the sharp outline of the sun until only a round
mass of light remains that grows fainter toward its periphery.” (Minnaert, 283)
“Every cloud has a silver lining”
Photo taken at 5PM, Feb. 19, 2003
Scattering of Light by Clouds, continued
“When the sun is hidden behind loose and heavy clouds, and the air is filled with a fine mist, groups of these
sunbeams can often be seen darting from the sun through the openings in the clouds, showing a path of light
through the mist, thanks to the scattering by the drops of which it is constituted” (Minnaert, 291)
These clouds were captured in the late afternoon of Feb. 19, 2003. The sky
was particularly clear that day due to morning rain. The rays of the sun
spearing through the openings in the clouds are radiant in this image. This
photo was taken from a balcony in Governor’s Corner.
Scattering of Light by Clouds, continued
Minnaert remarks that the best months to
study the colors of twilight are October and
November (293). During Autumn 2002, I took
photographs of various clouds at sunset.
Above: the set sun illuminates an impending
storm front.
Left: these clouds give the appearance of
smoke; the sun is hidden behind these thick
clouds; notice the beam of light(!).
Scattering of Light by Clouds, continued
(Above)
“The horizontal stripes (of sky) are
red only when the air contains
dust or water droplets”
(Minnaert,
304)
Photos taken at sunset
Autumn, 2002
Blue Lakes
“…the water of blue lakes is almost absolutely
pure and that the color is caused by
absorption by the water in the orange and
red parts of the spectrum. To account for
colors green, yellow-green, and yellowbrown, there is constantly increasing
proportion of iron-salt and humic acids and
also scattering by brown-colored particles in
these waters.”
Minnaert (345)
Second to Lake Tahoe and excluding the Pacific
Ocean, Hetch Hetchy Dam is one of the
bluest bodies of water I’ve seen.
Hetch Hetchy Dam; August 2002. Midday.
Grass
“The emerald green of grass in a bright light is
particularly lovely seen from a shady spot
against a dark background. It seems as if
each little blade were literally burning within
a green inward glow. The incident light
pouring on it laterally is scattered by the
millions of minute grains, so that each blade
casts a stream of light sideways towards
your eye.” (Minnaert, 357)
This picture was taken in the late morning of
February 18, 2003. Even then, the brilliance
of the meadow dominated the landscape at
Governor’s Corner. Some parts of the
meadow were shadowed by massive oak
trees, creating contrasting hues of green
across the landscape. The meadow,
covered with millions of dewdrops, created
an ever more ephemeral and heavenly
effect.
Dispersion of light by a dewy meadow
“When you … look over the heavily dewed fields
or meadows, … you will notice that they
disperse a remarkable amount of light into
the distance, toward the sun. The color of
the grass there can hardly be seen, it is
much whiter near you. It is, of course, the
dewdrops that reflect light; in the parts of the
field nearest you, only separate dots of light
can be seen here and there, but farther
away, there seem to be many more bright
spots” (Minnaert, 285)
Up front, the dew is scattered. At a
certain distance, however, one can see
the white blanket of the dew over the
grass.
Photo taken at Governor’s Corner,
February 18, 2003. Late morning.
“The grayish aspect of bedewed grass is caused
by the reflection of the rays of light in all the
tiny drops, inside as well as outside; a great
many of the rays do not even touch the
blade of grass. Large flattened drops have a
beautiful silver sheen when seen at fairly
large angles, because the rays are then
totally reflected at the back surface.”
(Minnaert, 56)
Vertical Reflection
“A chimney or a thin mast is reflected clearly, but the
strong lines of roofs have disappeared; only the
vertical lines are found back in the reflections.
Vertical trunks of trees are clearly recognizable,
but those that lean over are much less so, while
slanting branches have disappeared
completely. The slender neck of a swan is
reflected as a clear dab of light, but the body of
the bird is lost in the movement of the water …
In the case of an upright line, the columns are
neatly stacked together and magnify each
other; in the case of a horizontal line, they lie
side by side and broaden the line to a hazy
surface.”
Minnaert (24)
Minnaert explains that vertical shadows and reflections
are more prominent because their reflections
overlap each other at greater surface areas. On the
contrary, horizontal shapes lose their form because
their reflections and shadows do not overlap as
greatly, thus causing a hazy image.
In this image, the vertical shape of the trees are apparent
on the water surface when their horizontal aspects
are not. The hills are not emphasized in the
reflection either.
Photo taken Feb. 18, 2003; Lake Lagunita, late morning.
Refraction through uneven panes of glass
“The window plate is obviously not a parallel
plate, but has thinner and thicker parts that
act as irregular lenses, spreading out or
collecting rays of light and giving fanciful
focal lines. Even small deviations of the rays
cause appreciable differences in brightness,
so that virtually every window of standard
glass exhibits the streaks.”
(Minnaert, 48)
This window is above the entrance to Memorial
Auditorium. Storm clouds and Hoover Tower
are reflected from the glass. The glass
seems to reflect multiple overlapping
images due to the uneven glass surface.
Like many Stanford windows, this window
has horizontally streaked glass.
Photo taken midday, Feb. 19, 2003