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Ponendo x= π, con calcoli banali si ottiene, dalla formula di Eulero:
exp(ix)=cosx + i senx
la formula che compare nella immagine.
E' una formula che mi ha sempre affascinato, porchè contiene i 5 numeri con cui possiamo riassumere la matematica:
O e 1, con cui costruire l'aritmetica dei naturali;
π, il rapporto tra la lunghezza di una circonferenza e il suo diametro;
i, l'unità immaginaria, radice quadrata dell'unità negativa;
e, il numero di Nepero...
Sustituyendo x = π, con cálculos triviales, se obtiene, de la fórmula de Euler:
exp (ix) = cos x + i senx
la fórmula que aparece en la imagen.
Es una fórmula que siempre me encantò, ya que contiene 5 números con los que podemos resumir las matemáticas:
O y 1, con los que construir la aritmética de los numeros naturales
π, la relación entre la longitud de una circunferencia y su diámetro;
i, la unidad imaginaria, la raíz cuadrada de la unidad negativa;
e, el número de Napier
Substituting x = π, with trivial calculations we obtain, from Euler's formula:
exp (ix) = cosx + i senx
the formula which appears in the image.
It 'a formula that has always fascinated me, because it contains 5 numbers with which we can summarize the math:
O and 1, with which to build the arithmetic of natural numbers;
π, the ratio between the length of a circumference and its diameter;
i, the imaginary unit, square root of negative unity;
e, the number of Napier
Agfa 100 (50), D-76 (1+1), 13', 20°C. Expired from 2003, developed at 2015, always preserved in freezer.
Leica M6, Summicron 50mm (5th gen).
Law of cosines. The 5th of 25 mathematic Lego mini mosaics (20 inches square). When completed the entire montage will stretch over 42 feet. (THIS IS A 3-D AND DIFFICULT TO PHOTOGRAPH FROM ABOVE)
The iconic bridge of Cambridge.
from Wikipedia :- "The bridge was designed by William Etheridge, and built by James Essex in 1749. It has been rebuilt on two occasions, in 1866 and in 1905, but has kept the same overall design. Although it appears to be an arch, it is composed entirely of straight timbers built to an unusually sophisticated engineering design, hence the name."
More info here:- en.wikipedia.org/wiki/Mathematical_Bridge
Galleria Continua San Gimignano
Human Mathematics, ipaekre - roib 1982-2015, installation, mixed media
Too many variables?
I handheld the 3.0kg Nikon 500mm f/4.0P lens, to take this shot.
Imagine trying to find a target, with a 4.94° Diagonal Field of View, get the focus and exposure right, just in the very small amount of time the target has been acquired. Challenging, to say the least.
I originally had the lens mounted on a tripod, but just could not change it fast enough along a horizontal and vertical axis.
So, I do not know the angular inclination of the lens above the horizon.
If the aircraft was straight overhead, it might be simple mathematics, knowing the physical dimensions of the aircraft, comparing the number of pixels the wing or fuselage length occupy to the dimensions of the Sensor.
The Wings are nearly parallel to the Hypotenuse formed by the diagonal line on the sensor. and in reality that rectangular area would be perpendicular to the axis from the Circle of Confusion on the Sensor to the middle underside of the aircraft's fuselage. I depicted it in this way merely to give you some idea what I have been up to for the past couple days or so.
Because the aircraft is flying away at some angle and the wings is not precisely aligned with the Hypotenuse, actual pixel count seems more of a guess.
For example, knowing the wing from tip-to-tip is 36 feet (10.9728 meters) and the fuselage length is 28.25 feet (8.6106 meters) . . . I have measured between 987.5 pixels for the wing span . . . When comparing that to the length of the fuselage, the pixel count seems incorrect . . . Or, doing a pixel count of the fuselage, taking a ratio and applying that to the wing, the pixel count will be off. Frustrating.
There must be an accurate and predictable method, but I have not played with mathematics on this level in a very, very long time. Believe it or not, I designed a rocket, at university, but that was more than four decades into my past. I was smarter, then, or had many more active brain cells working for me.
What this demonstrates is that photography is more than just pretty pictures.
Possible solution:
Right Triangle ⊿, a Base, b Height, c Hypotenuse
tan = b/a
tan(2.47°) = 0.043136357952622
b = 3,634.45
a X tan(2.47°) = b
isolate a
a = 84254.9109962412 pixels
(Note: this will be from the Focal Plane to the belly of the aeroplane, between the main landing gear))
sin = b/c
sin(2.47°) = 0.043096280984403
isolate c
c X sin(2.47°) = b
c = 84333.2630329656
Taking a piece of paper, hold it parallel to the Span of the Wing, that is taken from a point in the middle of each wingtip and one gets 987.5 pixels. We know the Wing Span of this NACA 2412 type of wing on a Cessna U206G is 35 feet 10 inches (+/- 2 inches, depending on references) and is 987.5 pixels. The Wing Tip uses a NACA 0012 type of aerofoil and measures 3 feet, 8.5 inches.
The Wing Span is a known number.
So, I would want to know the number of pixels/foot of wing span.
If 987.5 pixels ÷ 35.83333 feet, then I would have 27.5581395348837 pixels/foot
Taking excerpts from the above:
a = 84254.9109962412 pixels ÷ 27.5581395348837 pixels/foot = 3,057.3512007072 feet from the Focal Plane to the belly of the Cessna, between the Main Landing Gear.
This is plausible. That still does not give me the height above the ground.
This may not be correct, though.
Why?
In the photo of the aeroplane, the Fuselage length appears longer than the Wing Span, but we know this is not true, as the Fuselage Length is 28.25 feet, as compared to the Wing Span of 35.83333 feet.
Measuring the Fuselage Length, I arrive at 865 pixels.
865 pixels ÷ 28.25 feet = 30.6194690265487 pixels/foot
35.833333 ÷ 28.25 = 1.26843657699115
How does this ratio compare with the pixels/foot count?
987.5 ÷ 865 = 1.14161849710983
Do you see the difference and the dilemma I have???
This is probably due to a parallax. How do I resolve that???
84254.9109962412 ÷ 30.6194690265487 = 2751.67772906799
3,057.3512007072 - 2751.67772906799 = 305.673471639207
Big difference! Which is correct? Is there a mathematical solution to know for certain?
How about this addition to the confusion? Measuring the Right Wingtip it is found to be 95 pixels and 3.70833333333333 feet or 25.6179775280899 pixels per foot.
Let's average the three pixels/foot counts. That would equal 27.9318620298408 pixels per foot
So, my best guess for the distance from the Focal Plane to the aircraft belly would be:
84254.9109962412 ÷ 27.9318620298408 = 3016.44447857533 feet
This is more plausible, though still does not give me the height above the ground of the aircraft.
3016.44447857533 feet becomes the new hypotenuse (c) to determine Height of the aircraft above the ground.
It would not be too far fetched to assume 3,000 feet above ground. And, the angle I held the big 500mm lens at could have been as much as 84° above the horizon. Maybe. All a guess, really.
I would guess 2,500 feet, considering the surrounding hilltops are approaching 1,500 feet or 457 meters.
This required two cups of strong morning coffee.
If anyone has a better solution, other than asking the pilot, I would like to learn from you.
A gentle reminder about copyright and intellectual property-
Ⓒ Cassidy Photography (All images in this Flickr portfolio)
Wooden footbridge across the River Cam, between two parts of Queens' College, Cambridge. A popular myth is that the bridge was designed and built by Sir Isaac Newton without the use of nuts or bolts.
My Palette. The 8th of 25 mathematic Lego mini mosaics (20 inches square). When completed the entire montage will stretch over 42 feet. Not so mathematical. Organizational
I wedged myself between two walls on top of a sewer grate to find this little spot. In general the building (mathematics and computer building at University of Waterloo) is very square, but it has some lines to exploit.
This is currently my favourite shot around campus.
Entry for "Chuck Norris vs." contest, resulting in winning Category 1 ^_^
Thanks a lot to the judges! :^D
The title and author of this book crack me up. This is one of my dad's textbooks from college. He graduated from Notre Dame around 1934.
Z50 with the kit lens wide open. I love how close it focuses.
Strobist: I used a EVOLV 200 in a Joe McNally mini softbox. I held the light just above the bookshelf.
From Mathematical Models, 2nd Edn, by H. M. Cundy and A.P. Rollett, Oxford University Press, 1951.
The notations are by my father, as he worked out measurements to build models.
Post is here: blog.ounodesign.com/2009/04/29/stellated-polyhedra-mathem...
This picture came out a bit too dark.. I should have better light when working.
I think the paper is from a Barbie office table set?
Messing about on the river at Cambridge's Mathematical Bridge. I felt I was lucky to be on Silver Street bridge as this punt came into sight with the chaps wearing straw boaters.
Charles Solomon: Mathematics
Hamlyn all-colour paperbacks
Paul Hamlyn, The Hamlyn Publishing Group Ltd - London, 1969
Explore Aug 1, 2012 #170
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“Happiness is when what you think, what you say, and what you do are in harmony.” by Mahatma Gandhi
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“Always aim at complete harmony of thought and word and deed. Always aim at purifying your thoughts and everything will be well.” by Mahatma Gandhi
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“In the end we shall have had enough of cynicism, skepticism and humbug, and we shall want to live more musically.” by Vincent van Gogh
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“Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.” by Deepak Chopra
Sierpinski gasket. The 10th of 25 mathematic Lego mini mosaics (20 inches square). When completed the entire montage will stretch over 42 feet. (THIS IS A 3-D AND DIFFICULT TO PHOTOGRAPH FROM ABOVE)
Mathematical and Geometrical Problems, The Iconographic Encyclopedia of Science, Literature and Art c. 1851 Copyright free.
A very cool idea in mathematics that is worth looking up is that of space filling curves or Peano curves first described by Giuseppe Peano.
The specific curve depicted here is due to David Hilbert, and is so called a Hilbert curve.
This turned out kinda wierd, kinda neat, hopefully not gruesome.
A few photos from a trip in 2015 inside the Andrew Wiles Building in Oxford, named after the man who solved Fermat's Last Theorem
From Mathematical Models, 2nd Edn, by H. M. Cundy and A.P. Rollett, Oxford University Press, 1951.
The notations are by my father, as he worked out measurements to build models.
Post is here: blog.ounodesign.com/2009/04/29/stellated-polyhedra-mathem...