Theory of the shadows / Teoría de las sombras

Hace tiempo, cuando era un estudiante de bachillerato,el profesor de física nos empezó a enseñar la teoría de la gravitación de Newton. Continuó con la explicación de las órbitas de los planetas producidas por el equilibrio entre las fuerzas centrífugas y centrípetas. Tras pensarlo un poco le pregunté. Si el universo sólo tuviera dos cuerpos, sencillamente se precipitarían el uno en el otro ¿no?. Él me contestó que dependía si estaban en movimiento y si alcanzaban un equilibrio girando uno alrededor de otro. Y... ¿Cómo saben esos cuerpos que están girando si no hay nada más alrededor para comparar su movimiento?, le apunté. A ésto él me comentó: Si se precipitan el uno sobre el otro no están girando y si no es que están girando. Le insistí, Pero. ¿cómo saben si están girando?. Ya se estaba enfadando y para salir del tema me dijo que esto se explicaba muy bien en la teoría de la relatividad de Einstein y que eso se estudia en la Universidad y no en Bachiller.

Long ago, when I was a high school student, the physics professor started teaching us the theory of gravitation Newton. He continued with the explanation of the orbits of the planets caused by the balance between the centrifugal and centripetal forces. After some thought I asked: If the universe had only two bodies, simply fall into each other, don't it?  He replied that this depended on whether they were in motion and if they reached a balance rotating around each other. And ... How do they know those corps that are spinning if there is nothing else around to compare their movement ?, I pointed it  him. He told me: If precipitate one another are not rotating and if not, they are turning. I insisted, but. How do they know if they are spinning ?. He was already getting angry and to leave the matter told me that this is very well explained in the theory of relativity and Einstein, that is studied in College and not in Bachelor.

Como me gustaba ir a la biblioteca. Allí era gratis leer la revista "Investigación y ciencia" y otras revistas de viajes, aproveché para sacar libros con la temática de la relatividad. No entendía nada pero en ningún caso observaba ninguna explicación de que pasaría en un universo reducido a dos cuerpos. Simplemente me parecía leer que el espacio se deformaba de forma radial uniforme en torno a ellos. Pero de giros... nada. Pensé que era un poco torpe y que esos conocimineto eran demasiado elevados para un chaval.

As I liked to go to the library. There it was free to read the magazine "Scientific American" and other travel magazines. I took out books with the theme of relativity. I did not understand anything but never watched any explanation that would happen in a small two corps universe. It just seemed to read that space is deformed radially uniform around them. But of the twists ... there was nothing. I thought it was a little awkward and that such knowledge was too high for a kid.

Pues no. No es cierto. Me reafirmo. Ni Newton, ni Einstein, ni ningún otro daba ninguna explicación de porqué un Universo con sólo dos cuerpos atrayéndose o deformando el espacio pudieran saber de antemano si giran.

Well no. It is not true. I reaffirm. Neither Newton nor Einstein nor any other explained why a universe with only two corps being attracted or warping space could know in advance if they rotate.

Muchos años después, y debo decir que tras una noche de copas con un amigo tan loco como yo, me di cuenta de una obviedad. Sólo pueden dos cuerpos saber que giran si no están solos. ¿Cómo de solos? ¿qué les puede hacer sombra?. Y si no era tan importante el saber que eran los otros elementos como la sombra que se hacían los cuerpos entre ellos. La noche me la pasé haciendo un montón de borrajetas en papeles. Al día siguiente, ordené los papeles. Pasé un buen rato intentando entender lo que había escrito y simplificando el problema. A continuación expondré la más simplificada teoría de unificación de fuerzas basada en las sombras que se proyectan los cuerpos debidas a "algo" que nos rodea a todos.

Many years later, and I must say it was during the period of a night of drinking with a friend as crazy as me, I noticed the obvious. They can only know that two bodies rotate if they are not alone. How alone? What can they do shadow themselves ?. And if it was not so important to know who were the other elements like the shadow that the corps were made between themselves. The night I spent doing a lot of scribbling on paper. The next day, I ordered the papers. I spent some time trying to understand what he had written and simplifying the problem. Then I will discuss the most simplified unified theory of forces based on projected shadows on corps caused by "something" that surrounds us all.

Simplified universe 

Imagine a world with only two elements. In the absence of more data imagine that there are two spheres of radius r. Are each other at a distance d. (d >> r)

This universe is being pierced by a large number of uniformly distributed particles.

Universo simplificado

Imaginemos un universo con sólo dos elementos. A falta de más datos imaginemos que se trata de dos esferas de radio r. Están uno de otro a una distancia d. (d >> r)

Este universo está siendo traspasado por un gran número de partículas uniformemente distribuidas.

Now suppose around a very large area where R >> d particles passing they are evenly distributed. The surface of this sphere is 4πR ². Each small sphere will reach all particles uniformly distributed except stops in the shade of the small sphere opposite.

Area surrounding the spheres: πd ² and the shadow of toward each other is. Πr ² (the cap resembles the equatorial circle area If passed P particles πd ² πr ² will Pr ²/d ².

Supongamos ahora que alrededor de una esfera muy grande donde R >> d las partículas que le atraviesan están distribuidas uniformemente. La superficie de esta esfera es 4πR². Cada esfera pequeña le llegarán todas las partículas uniformemente distribuidas excepto las paradas en la sombra de la esferita opuesta. 

Área en torno a las esferas:  πd² y la sombra de cada una hacia la otra es:  πr² (el casquete se asemeja al área del círculo ecuatorial. Si pasan P partículas en  πd²  en  πr² pasarán Pr²/d² .

Complicated Universe

We have two sets with M and N spheres  respectively.

Universo complicado

Tenemos dos conjuntos de M y N esferas respectivamente

Each Mi interacts with Nj. If M has m₁ elements, N has m₂, the combinatorial sum of all elements is m₁ · m₂, which our formula becomes: m₁·m₂·P·r²/d².. 

If we separate the constants elements (P·r²)·m₁·m₂/d² =  K·m₁·m₂/d². and now if we create units of measure will be: G·m₁·m₂/d². 

All this assuming that the universe is in a soup of particles we know nothing.

Cada Mi interacciona con una Nj Si M tiene m₁ elementos y N m₂ la suma combinatorio de todos los elementos es m₁ · m₂ con lo que nuestra fórmula quedaría como: m₁·m₂·P·r²/d².

Si separamos los elementos contantes  (P·r²)·m₁·m₂/d² =  K·m₁·m₂/d². y si ahora creamos unidades de medida quedará:  G·m₁·m₂/d².

Y todo esto suponiendo que el universo está en una sopa de partículas de la no conocemos nada.

There's more

• In the end I spoke of gravity. Nothing prevents the same formulation to electromagnetism (m₁ and m₂, may be q_1  and  q₂ and G becomes K).

• If the particles are not instantly move (say they do at the speed of light). Interacting Shadow is not what is in each moment ago but it was t = d / c. Consider this effect complicates the equations.

• The effects if it fails d >> r, are more complex than simple formulation above.

• In the theories of Newton or Einstein's theory in a vacuum. Both explain how two masses no are precipitated each other if they rotate around a center of mass and the centrifugal force compensates for this attraction. But in our simplified universe: How do you know that turn?. In our theory we know that rotated (or not) about the rain of particles (or "things") to which is being subjected the Universe.

• If we want to an added challenge. We suppose the particles are moving on the surface of a sphere. If this is big enough for a plane surface dweller, he will see that his "flat" universe is crossed by particles in all directions (only 2) in straight lines. Only one dimension is increasing and we, people of a universe in 3D, will see particles in our three dimensions. Only one would think that we are in the outer volume, the skin, in a four-dimensional hypersphere. And now we don't care if it turn.

Aún hay más

• Al final hablé de la gravedad. Nada impide la misma formulación para el electromagnetismo (m_{1 y} m₂, pueden ser q_{1 y} q₂ y G pasa a ser K). 

• Si las partículas no se mueven instantáneamente (supongamos que lo hacen a la velocidad de la luz) la sombra que interacciona no es la que está en cada momento sino la que estaba hace t=d/c. Tener en cuenta este efecto complica bastante las ecuaciones.

• Los efectos cuando no se cumple d >> r, son más complejos que la formulación simple anterior.

• En las teorías de Newton o Einstein hay un vacío en la teoría que es el siguiente. Ambos explican como dos masas no se precipitan una a otra si estas giran alrededor de un centro de masas y la fuerza centrífuga compensa la atracción. Pero en nuestro universo simplificado ¿Cómo se sabe que giran?. En nuestra teoría se sabe porque girarán o no respecto a la lluvia de partículas o "algo" a las que se ve sometido el Universo.

• Si queremos complicarlo más, supongamos inicialmente las partículas moviéndose en la superficie de una esfera. Si ésta es lo suficientemente grande, un habitante plano de la superficie verá que su universo "pano" es traspasado por partículas en todas direcciones (sólo 2) en líneas rectas. Sólo queda aumentar una dimensión y nosotros, habitantes de un universo en 3D, veremos partículas en nuestras tres diemensiones. Sólo cabría pensar que estamos en el volumen exterior, la pìel, de una hiperesfera de cuatro dimensiones. Y ahora, sí que nos da igual que gire.

Luis Nieto 2000

The history of fractional continuous derivatives / La historia de las derivadas continuas y fracionales

Hace mucho tiempo, tenía unos 16 años y estudiaba el bachiller. Era una época con acné y toneladas de hormonas. Cuando éstas me dejaban pensar en otras cosas tenía auténtica pasión por las matemáticas. Cuando resolvía un problema y entendía un concepto sentía como una especie de iluminación y tranquilidad interior. Me sentía bien. Un día el profesor de matemáticas nos empezo a explicar un nuevo concepto, la derivada. Todo empezó con una definición un poco simple. Primero se crea un quebrado:

A long time ago I was about 16 years old, and studied the degree. It was a time with acne and tons of hormones. When they let me think about other things I had a passion for mathematics. When solving a problem and understand a concept, I felt like a kind of enlightenment and inner tranquility. I felt good. One day the math teacher we started to explain a new concept, the derivative. It all started with a somewhat simple definition. First create a fraction:

[ f(x+h) - f(x) ] / h

La idea era hacer "h" lo más pequeña posible para así obtener la tangente en el punto x cuando h → 0. Es decir: df(x)/dx=lim (h → 0) [ f(x+h) - f(x) ] / h .

The idea was to make "h" as small as possible to obtain the tangent at the point x when h → 0, ie: df (x) / dx = lim (h → 0) [f (x + h) - f (x)] / h.

El proceso pasó por una fase en que empezamos a derivar monomios. El proceso parecía bastante simple x ^n, pasaba a ser n·x ^n-1, sen(x) se convertía en cos(x), e ^x, quedaba inmutable. Especial fascinación me produjo la derivada del logaritmo ya que las anteriores no cambiaban el concepto. Los monomios eran otros monomios, las funciones trigonométricas se convertían en trigonométricas... pero el logaritmo... dln(x)/dx=1/x. el logaritmo pasaba a ser un monomio. 

The process went through a phase where we started to derivate monomials. The process seemed simple  x ^n, becames n·x ^n-1, sen(x) turns cos(x), e ^x,  remained unchanged. Special fascination gave me the derivative of the logarithm. The above functions did not change the concept. Monomials were other monomials, trigonometric functions were other trigonometric ... but the logarithm... dln (x) / dx = 1 / x. the logarithm he would become a monomial with negative degree.

Más increíble me pareció que df(g(x)/dx=f'·g' y que df(x)·g(x)/dx=f'·g+f·g'.

More amazing I found df(g(x)/dx=f'·g' and df(x)·g(x)/dx=f'·g+f·g'.

Ahora es muy fácil ver explicaciones a esto. Pero en 1983 uno no podía teclear en un ordenador http://es.wikipedia.org/wiki/Derivada.

Now it's easy to see explanations for this. But in 1983 you could not type on a computer http://en.wikipedia.org/wiki/Derivative.

Jugando con el tema por mi cuenta vi que si x ⁴,por ejemplo. Si se derivaba se obtenía 4·x ³. Si volvía a derivar obtenía 4·3·x ², y sucesivamente, 4·3·2·x ¹,4·3·2·1=4!, hasta acabar en 0. Le pregunté a mi profesor que qué era eso de derivar "m" veces una función como f(x)=x ⁿ y obtener n!/(n-m)!·x ^n-m. Me contestó que eso eran derivadas de oreden m y asía acabó la explicación.  

Playing with the topic on my own I saw that if x ⁴, for example. was derived was obtained 4·x ³. If again derive got 4·3·x ², and so on, 4·3·2·x ¹, 4·3·2·1=4!,, to finish in 0. I asked my teacher what it meant to derive "m" times a function like f(x)=x ⁿ and get n!/(n-m)!·x ^n-m. He replied that they were derivatives of order "m" and so ended the explanation.

En ese momento algo chispeante pasó por mi imaginación. De número simples (naturales) se había pasado a números negativos, quebrados, multiplicaciones, raíces... Del concepto número al concepto función. De escribir matemáticas con números a escribir matemáticas con letras. Y ahora, además había funciones de funciones, como la derivada y las derivadas de orden "m". (Nota: Como es costoso escribir signos raros en este blog adoptaré un nomenclatura más sencilla.) Ahora había una función "D" tal que D(m,f(x)) nos daba una transformación de una función. D(m,x ⁿ)=n!/(n-m)!·x ^n-m con  m ∈ N

At that moment something sparkling crossed my imagination. Natural number had passed to negative numbers, fractions, multiplication, roots ... from the concept number to the function concept. From writing math mathematics with numbers to write that with letters. And now, there were also functions of functions such as the derivative and derivative of order "m". (Note:. Rare signs is costly to write in this blog I will adopt a simple nomenclature). Now there was a "D" function such that D (m, f (x)) turned a transformation of a function. D(m,x ⁿ)=n!/(n-m)!·x ^n-m con  m ∈ N

f(x)=x ^n.;   f'=n·x ^n-1.;      f''=n·(n-1)·x ^n-2;      f'''=n·(n-1)·(n-2)·x ^n-3.;      f''''=n·(n-1)·(n-2)·(n-3)·x ^n-3;   

f ^(m=n!/(n-m)!·x ^n-m;   

F(x)=x ⁵ y sus sucesivas derivadas enteras.
F(x)=x ⁵  and itssuccessive integer derivatives .

Pero en ese momento, pensé y si m fuera negativo. en nuestro viejo ejemplo  D(-1,x ⁴)=5!/6!·x ⁶=1/6·x ⁶. Volví a preguntarle a mi profesor. Éste me dijo "No te adelantes, eso son integrales y ya lo estudiaremos". Que quedé clavado. ¿Que significaba integral? Me había contestado a una duda con una palabra que carecía de significado. En aquel curso no teníamos libro de matemáticas. El profesor traía unas fotocopias y tendría que esperar a que explicase que eran eso de las "integrales". No diré a quien pregunté (aunque lo recuerdo, pero como esto es un blog no quiero que nadie se ofenda) . Pero cuando hacía la pregunta ¿Qué son las derivadas de orden negativo de una función? la respuesta siempre era la misma (o una variante similar). ESO NO EXISTE.

But then I thought what if "m" were negative. In our old example D (-1, x 4) = 5! / 6! · X 6 = 1/6 x 6 · I went back to ask my teacher. He told me "Do not get ahead, that are integrals, and we will study it." I was flabbergasted. What meant integral? I had answered a question with a word that had no meaning. In that course we had no math book. The teacher brought photocopys and would have to wait to explain that they give us abaut "integrals". I will not say who asked (although I remember, but as this is a blog not want anyone to be offended). But when asked the question What are the derivatives of negative order of a function? the answer was always the same (or a similar variant). THAT DOES NOT EXIST.

Pasaron unas semanas (o meses) y empezamos con las integrales. Si, de acuerdo nadie lo llama derivada de orden negativo pero era eso. Y claro que existía D(-1,f(x)) pero la escribían ∫f(x)dx. También existía D(-2,f(x)) pero la escribían ∫[∫f(x)dx. ]dx y así sucesivamente. Ahora nuestra función tiene valores de m pertenecientes a los números enteros.

A few weeks ago (or months) and we start with the integrals. whether, in accordance nobody called negative derivative order but they was. Clearly, there was D(-1, f (x)) but it was wrote as ∫f(x)dx. There was also D(-2,f(x)) but it was wrote as ∫[∫f(x)dx]dx and so on. Now our function has m values ​​belonging to integer numbers.

No transcurrió mucho tiempo hasta que pensé ¿y si m en lugar de un número entero fuera un número fraccionario, o mejor aún real?. Fuí de nuevo a mi profesor (a estas alturas supongo que ya lo tendría aburrido) y le dije algo parecido a ésto: ¿Qué es una derivada con un orden fraccionario o real?. Me contestó que eso no existía. Yo, creo recordar, le insistí con un ejemplo. (como ahora no recuerdo bien cual era me lo medio invento) Si tengo una función f(x)=x ⁴;  d^1/2x ⁴/(dx)^1/2=4!/3.5!·x ^3.5 . Cuando vio el 3.5! me dijo "Ves, 3.5! no existe, luego no existe una derivada de ese tipo. Qué lástima de internet. Si la hubiera tenido entonces le habría contestado con un ¿Y qué es la función Gamma (http://es.wikipedia.org/wiki/Función_gamma) . De todas maneras la contestación de "no existe" no me satisfizo en absoluto. No se desde cuando pienso que si en matemáticas algo no existe tiene una fácil solución, se inventa.

It was not long until I thought what What if "m" instead of an integer be a fraction, or better yet Real?. I went back to my teacher (at this point, I suppose he would have bored with me) and I said something like this: What is a derivative with a fractional or real order ?. He replied that it did not exist. I, as I recalls, insisted with an example. (I can not remember what it was, I made ​​it up) If I have a function f(x)=x ⁴;  d^1/2x ⁴/(dx)^1/2=4!/3.5!·x ^3.5. When he saw the 3.5! he said me: "See, 3.5! not exist, then there isn't such the derivative. What a pity Internet. If I'd had time, I would have answered with a: What is the Gamma function (http://en.wikipedia.org/wiki/Gamma_function). Anyway the answer to "don't exist" did not satisfy me at all. I don't know since when I think that if something exists in mathematics it has easy solution, it invents.

Otro punto era que no sabía es qué estaba haciendo y cual era el significado de estas cosas. Con la primera derivada se calculan tangentes, con la segunda se sabe si la función es ascendente o descendente, con la integral se calculan áreas. Pero cuando m es diferente a 1,2, y -1 ¿Qué es D(m,f(x))? la respuesta es más difícil de contestar cuando m, además no es un número entero.

Another point was that I did not know what I was doing and what was the meaning of these things. With the first derivative calculated tangentss, with the second is known if the function is ascending or descendings, with the integral are calculated areas. But when "m" is different from 1, 2 and -1 What is D(m,f(x))? the answer is more difficult when m isn't an integer number.

Un día lei un artíaculo de la revista "Investigación y ciencia". No recuerdo si era de la sección de Martin Gardner en el aparecia la función Gamma. De repente todo era muy simple: Γ(n+1)=n! y admitía valores de todo el espectro de números reales. volví a mis monomios y todo resultó más simple:

One day I read an article in the magazine "Scientific American". I do not remember if it was in Martin Gardner section where it appeared in the Gamma function. Suddenly everything was very simple: Γ (n + 1) = n! and admitted values ​​of the entire spectrum of real numbers. I returned to my monomials and everything was more simple:

Para f=x ⁿ → D(m,f)=Γ(n+1)/Γ(n+1-m)·x ^n-m;   

F(x)=x ⁵ y las derivadas 0.2, 0.4, 0.6, 0.8 y 1
F(x)=x ⁵ and the drerivatives 0.2, 0.4, 0.6, 0.8 and 1

Un ejemplo sencillo sería con la / simple example with f(x)=x ⁴;

 f ^(1/2(x)=Γ(5)/Γ(4.5)·x ^3.5;   = (5/√π)·x ^3.5

Seno y coseno / Sine and cosine

f(x)=sen(x)^.;   f'=cos(x);  f''=-sen(x);   

O si se prefiere: or whether you prefer:

f(x)=sen(x)^.;   f'=sen(x+π/4);  f''=sen(x+2·π/4);   

f ^(m=sen(x-m·π/4);   

Ejemplos más complicados: / Examples more complicated:

Se toma una f(x), se obtiene la trnsformada de Taylor y se utiliza la regla de cada monomio:

We take a function f(x), we get the Taylor transform, and use the same rule for each monomial:

Véase: /  See

• http://es.wikipedia.org/wiki/Serie_de_Taylor
• http://en.wikipedia.org/wiki/Taylor_series

Inciso;Resolución de exponentes con números complejos:

Subsection; Resolution of complex numbers with complex exponents:

Antes de adentrarnos en solucionar funciones con raíces lo primeros es conocer como se resuelve un número complejo elevado a un exponente complejo:

Before going to solve functions with roots, the first is known how it is a complex number elevated a a complex exponent:

(a+bi)^(c+di)=e^(c·ln(√^{(a²+b²))-d·atn(b/a))}·[cos(c·atn(b/a)+d·ln(√(a²+b²))+i·sen(c·atn(b/a)+d·ln(√(a²+b²))]

En nuestro caso, casi siempre a<0, b=d=0 Lo cual simplifica a:
In our case, usually a<0 and b=d=0 Which simplifies to:

a^c=|a|^c·[cos(c·π)+i·sen(c·π)]

Dado que, As: tan(π) =  tan(0) = 0.

Ahora ya podemos dibujar la parte positiva y negativa de las funciones derivadas resultantes. Para ver como queda se edita el siguiente vídeo:

Now we can draw positive and negative part of the resulting derivative functions. To see how it is edited the following video:

Inciso; Subderivada de una constante / Subsection; Subderivative of a constant.

f(x)=k=k·x ^0.;

f ^(m=k·0!/(0-m)!·x ^-m;   

f ^(m=k·Γ(1)/Γ(1-m)·x ^-m;  

f ^(m=k/Γ(1-m)·x ^-m;  

Subderivada de / subderivative of e^x;

f(x)=e ^x.= 1+x+x ²/2!+x ³/3!+x ⁴/4!+x ⁵/5!+x ⁶/6! ...

f ^(m=1/Γ(1-m)·x ^0-m+Γ(2)/Γ(2-m)·x ^1-m+Γ(3)/Γ(3-m)·x ^2-m+Γ(4)/Γ(4-m)·x ^3-m+Γ(5)/Γ(5-m)·x ^3-m...

Un regalo. Las subderivadas del coseno hiperbólico:

A gift. The hiperbolic cosine subderivatives:

Todo esto ha sido más fácil hacerlo ahora con la ayuda de un ordenador e internet. Todos estos gráficos ya los hiciera hace 20 años con ayuda de una calculadora, un lápiz y un papel. Aún hoy no sé la utilidad de todo esto, pero estoy contento de saber que "no existe" es lo único que no existe en matemáticas.

All this has been easier to do now with the help of a computer and internet. All These graphs and I did 20 years ago with the help of a calculator, pencil and paper. Even today I don't know the usefulness of all this, but I am glad to know that "don't exist" is the only thing that does not exist in mathematics.

Si quieres utilizar el programa que he usado para crear estos dubujos descárgalos de aquí (hecho en VB6):
If you want use the program I have make to do this graphs, download form here (make with VB6):

Luis Nieto

Fractional continuous derivatives / Derivada continua y fracional

Partiendo del concepto de derivada:
On the basis of the concept of derivative:

Véase / See:

•       http://es.wikipedia.org/wiki/Derivada
•       http://en.wikipedia.org/wiki/Derivative 

Se puede ver fácilmente que: / You can easily see: ∃ n ∈ N;  dⁿf(x)/(dn) ^n.

El cambio importante es hacer que n pase a format parte de los numeros reales, n ∈ R.

The significant change is to get that "n" becomes a real number, n ∈ R.

Primeros ejemplos sencillos / First simple examples

Monomios: / Monomials:

f(x)=x ^n.;   f'=n·x ^n-1.;      f''=n·(n-1)·x ^n-2;      f'''=n·(n-1)·(n-2)·x ^n-3.;      f''''=n·(n-1)·(n-2)·(n-3)·x ^n-3;   

f ^(m=n!/(n-m)!·x ^n-m;   

F(x)=x ⁵ y sus sucesivas derivadas enteras.
F(x)=x ⁵  and itssuccessive integer derivatives .

Para poder pasar de [m ∈ N] a [m ∈ R]
We change [m ∈ N] to [m ∈ R]

f ^(m=Γ(n+1)/Γ(n+1-m)·x ^n-m;   

F(x)=x ⁵ y las derivadas 0.2, 0.4, 0.6, 0.8 y 1
F(x)=x ⁵ and the drerivatives 0.2, 0.4, 0.6, 0.8 and 1

Donde la función Γ(n) es la función Gámma: 
Where Γ (n) function is the gamma function:

• http://es.wikipedia.org/wiki/Función_gamma 
• http://en.wikipedia.org/wiki/Gamma_function

Un ejemplo sencillo sería con la / simple example with f(x)=x ⁴; f ^(1/2(x)=Γ(1.5)/Γ(4.5)·x ^3.5;   

Seno y coseno

f(x)=sen(x)^.;   f'=cos(x);  f''=-sen(x);   

O si se prefiere: or whether you prefer:

f(x)=sen(x)^.;   f'=sen(x+π/4);  f''=sen(x+2·π/4);   

f ^(m=sen(x-m·π/4);   

Ejemplos más complicados: / Examples more complicated:

Se toma una f(x), se obtiene la trnsformada de Taylor y se utiliza la regla de cada monomio:
We take a function f(x), we get the Taylor transform, and use the same rule for each monomial:

Véase: /  See

• http://es.wikipedia.org/wiki/Serie_de_Taylor
• http://en.wikipedia.org/wiki/Taylor_series

Inciso;Resolución de exponentes con números complejos:

Subsection; Resolution of complex numbers with complex exponents:

Antes de adentrarnos en solucionar funciones con raíces lo primeros es conocer como se resuelve un número complejo elevado a un exponente complejo:
Before going to solve functions with roots, the first is known how it is a complex number elevated a a complex exponent:

(a+bi)^(c+di)=e^(c·ln(√^{(a²+b²))-d·atn(b/a))}·[cos(c·atn(b/a)+d·ln(√(a²+b²))+i·sen(c·atn(b/a)+d·ln(√(a²+b²))]

En nuestro caso, casi siempre a<0, b=d=0 Lo cual simplifica a:
In our case, usually a<0 and b=d=0 Which simplifies to:

a^c=|a|^c·[cos(c·π)+i·sen(c·π)]

Dado que, As: tan(π) =  tan(0) = 0.

Ahora ya podemos dibujar la parte positiva y negativa de las funciones derivadas resultantes. Para ver como queda se edita el siguiente vídeo:

Now we can draw positive and negative part of the resulting derivative functions. To see how it is edited the following video:

Inciso; Subderivada de una constante / Subsection; Subderivative of a constant.

f(x)=k=k·x ^0.;

f ^(m=k·0!/(0-m)!·x ^-m;   

f ^(m=k·Γ(1)/Γ(1-m)·x ^-m;  

f ^(m=k/Γ(1-m)·x ^-m;  

Subderivada de / subderivative of e^x;

f(x)=e ^x.= 1+x+x ²/2!+x ³/3!+x ⁴/4!+x ⁵/5!+x ⁶/6! ...

f ^(m=1/Γ(1-m)·x ^0-m+Γ(2)/Γ(2-m)·x ^1-m+Γ(3)/Γ(3-m)·x ^2-m+Γ(4)/Γ(4-m)·x ^3-m+Γ(5)/Γ(5-m)·x ^3-m...

Un regalo. Las subderivadas del coseno hiperbólico:

A gift. The hiperbolic cosine subderivatives:

Si quieres utilizar el programa que he usado para crear estos dubujos descárgalos de aquí (hecho en VB6):

If you want use the program I have make to do this graphs, download form here (make with VB6):

Luis Nieto

Verification of image orthorectification techniques for low-cost geometric inspection of masonry arch bridges II/ Verificación de ortorectificación integrados para inspección geométrica de bajo costo de mampostería puentes de arco II

Artículo patrocinado por Extraco, Misturas, Lógica, Enmacosa e Ingeniería InSitu, dentro del proyecto SITEGI, cofinanciado por el CDTI. (2012). 

Article sponsored by Extraco, Misturas, Lógica, Enmacosa and Ingeniería Insitu inside the SITEGI project, cofinanced by the CDTI. (2012)

Continúa de: From: http://carreteras-laser-escaner.blogspot.com/2014/10/verification-of-image.html

2.2 Methodology
2.2.1 Data acquisition and processing

Laser scanning. The geometry of the whole structure of the three bridges was acquired using the Riegl LMS Z390i static terrestrial laser scanning system. Several scan stations are required to obtain the whole geometry of a bridges depending on its size and structural composition (12 scanner stations for the Cernadela, seven for the Carracedo, and six for the Lonia; see Fig. 6). The scanning range is lower than 100 m in all cases during the survey of the bridges. Registration of all point clouds, obtained from different scanner positions in the global coordinate systems, is performed by the accurate measurement of common control points using a total station. Registration error is lower than 1 cm for all the bridges. Therefore, all the errors involved in the data acquisition and registration are negligible compared to the errors involved in the image rectification process. This confirms that laser scanner data can be used as the ground truth.

Those points are used to perform a 3-D conformal transformation that finally allows us to get the complete point cloud of the bridge built. This point cloud is then processed by means of cleaning and filtering operations in order to delete noise data.

Fig. 5 Scale bars.

After point cloud depuration, a surface model is obtained from the point cloud based on the Delaunay triangulation. Surfaces texturing comes from a combination of the surfaces obtained from the Delaunay algorithm and the calibrated photographs obtained with the Nikon D200 camera, whose position and orientation is calculated relative to the coordinate system of the scanner. Geometrically, this correspondence is established through a space resection (based on colinearity condition equations), because both geodetic instruments measure a minimum of three common target points. Finally, a projection plane (best fitted to both upstream and downstream bridge walls) is selected to produce orthophotos by orthogonal projection of the texturized 3-D model.

Photogrammetric survey. Image acquisition for the photogrammetry measurements is performed only of the area of the bridge under study, using the three cameras (Canon Ixus 100 IS, Kodak M1073, and Samsumg L100) and the scale bars. The detail photographs were selected because the main elements under study are located around the arches of the bridges. All the acquisition and image processing was done by a volunteer student from the Industrial Engineering School at the University of Vigo without previous expertise in photogrammetry or photography. 

The potential market for the image rectification tool is focused on bridge inspectors, who do not typically have advanced photogrammetry training, so we tried to test for similar conditions.

One of the simplest photogrammetric procedures is based on image measurements from single-rectified photographs. As a result of the rectification process, a photograph can be used like a two-dimensional (2-D) map for measurement of distances, angles, and areas with the scale being constant everywhere. This is what happens in an ideal situation, where the coordinate transformation is performed between two perfect systems.

Fig. 6 Workflow main steps.

The core algorithm of the procedure is the plane projective transformation whose equations are shown below:

object, x0 and y0 are the pixel coordinates in the photograph (image space), and a0, a1, b0, b1, b2, c1, and c2 are the coefficients of the projective transformation matrix. Since these transformation equations involve a total of eight unknown coefficients, four plane control points are required (without having three points aligned in the same straight line). The solution that is adopted here uses four photogrammetric targets fixed on the two parallel aluminum scale bars. A specifically developed Matlab routine is used for this evaluation.

In normal conditions, the image recorded by the camera is affected by lens distortions. Radial symmetric distortion is the main source of error for most camera systems, and it is modeled through polynomial series where the independent variable is the radial distance to the principal point (orthogonal projection of perspective center on the image plane).

Symmetric radial distortion can be modeled through two different formulations that are mathematically equivalent. These are the balanced model (more often used by camera and lens manufacturers) and the unbalanced model. The following equation shows the balanced radial symmetric lens distortion:

where r is the radial distance from the principal point, dr is the radial lens distortion divided by r and A0, A1, A2, and A3 are the coefficients of the polynomial.

Once radial distortion is modeled, image coordinates can be corrected according to the following equations:

Decentering distortion is modeled through the following equations:

where dpx and dpy are the amounts of decentering lens distortion with respect to x and y, respectively, and P1 and P2 are the coefficientes of the equation.

Those models are effective for fixed focal lengths, but models vary for each different focusing distance, as well as for object distance at a constant focus.15,16 Kim and Shin17 demonstrate the influence of focusing length over the first two terms of the radial distortion in zoom-lens video cameras. As could be expected, they demonstrate that the effect of lens distortion is much higher in lenses with wider angles (shorter focal length) than in those with longer focal length.

For the ultimate accuracy, the image to be rectified requires the correction of the original photo displacement due to those distortion effects. However, depending on the particular application of the method, the errors in the final metric image can be negligible compared with the global error of the procedure (within the accuracy required for such an application). Having taken this last constraint into account, the method also opens the possibility of using any digital camera without camera calibration and using any image format.

Far from evaluating the effects of lens distortions in projective transformations, this works aims to evaluate the global geometric validity of image orthorectification for routine inspection procedures. Not only lens distortion will affect the final metric error. Many other parameters will significantly affect it, including marking errors and a lack of coplanarity in the bridge’s wall and control points (i.e., scale error). So a statistical test is used to demonstrate the metrological validation of the method in order to be implemented in routine bridge inspection programs.

Having taken into account the access limitations and obstacles in the bridges’ environment, the field data acquisition tried to reproduce the potential normal conditions that bridge inspectors would find in a real scene. This mainly consisted of data acquisition from river embankments without any access to river course. This limitation sometimes means that an image’s plane cannot be taken parallel to the bridge’s main vertical plane.

In an image whose control point coordinates are exactly measured, the effects of obliqueness would not produce any error effect according to Eq. (1). In real computing vision systems, image coordinates are estimated through the behavior of pixel distribution. At this point, the pixel shape and size varies with distance and obliqueness between the camera and the object and thus may significantly affect the final error. Such a situation requires special care to be taken during the data acquisition in trying to find a balance between the angle and distance to the object.

The angle between the image plane and the bridge’s wall did not exceed 10 deg for the images used in this experiment. The tilt angle around the vertical axis has been computed through a spatial resection.15

2.2.2 Data processing

Orthophotos obtained from the laser scanning data and those obtained from the image rectification techniques, using three different cameras, are compared to establish the metrological quality of the photogrammetric procedure. The error of the laser scanner system is higher than that of photogrammetry, and it is assumed as the ground truth for this study.

Error is defined as the difference between the values obtained for the laser scanner and those obtained for each camera in a determined geometric parameter of the bridge. To make it more comprehensive, error was presented as a percentage.

3 Results and Discussion

The strategy for the analysis of the results was divided into four main steps. First of all, the full measurements are shown detailing the bridge element, bridge, and type of camera.

Fig. 7 Error of the different bridge elements for each bridge and camera.

Fig. 8 Number of measurements below a certain error.

Fig. 9 Error versus length for all the measurements of the study. The values are obtained from the averaging of the three camera data in each bridge element.

Second, an accumulative graph depicts the number of measurements that are under a certain value. This permits us to evaluate the possibilities of the technique without taking into account the type of element and camera. The third part establishes the relationship between the error and the length of the elements. Finally, the error results for each camera are evaluated for each element and bridge. The results for each camera come from an average.

Figure 7 shows the error values for all the elements under study, the cameras, and the bridges. The horizontal axis presents the length of the element. Four elements (rigid backfill, arch ring thickness, span, and rise), three bridges (the Carracedo, Cernadela, and Lonia) and three cameras (Canon Ixus 100 IS, Kodak M1073, and Samsumg L100) are studied. A total of 108 measurements were made of the four arches of the Carracedo Bridge, the five arches of the Cernadela Bridge, and the single arch of the Lonia Bridge. The global analysis of the data shows that, in the majority of the cases, the error is less than 4 percent. There does not appear to be a link between the length of the element and the error level. This is very important, especially to demonstrate the quality of the technique for the larger elements (for example, the span or the rise). 

Figure 8 is an accumulative graph that depicts the percentage of elements that are under a certain error level (also expressed as a percentage). This evaluation has been made individually for all the measurements. Approximately 95 percent of the measurements had an error level of less than 4 percent, 65 percent had an error level of less than 2 percent, and 50 percent had an error level of less than 1 percent.

Figure 9 shows the relationship between the error level of the measurements (regardless of camera, bridge, or element) and their length. The values are obtained from the average of the data obtained from all cameras in each element. In agreement with Fig. 7, no trend is observed.

Figure 10 represents the average error values obtained for each camera and the relationship with the type of element and bridge. No special differences are observed between the elements and bridges. This is an important observation, because the scale bars are located in only one position for the image rectification of all the elements of the arch, and it appears to be stable enough. On the other hand, the bridges are in different geographical locations, the photographs are taken in different days with different illumination conditions, and the procedure appears to be robust enough for this. In addition, it must be noted that all the photographic data acquisitions were taken by a student without previous training in photography or photogrammetry.

The comparison of the results from the different cameras shows error dependence in the function of the camera type and the quality of the camera. In fact, the results of the Canon camera are clearly better than those of the other two. An error level of around 1 percent is obtained for the majority of the dataset. The market price of the camera is not high, and it geometric bridge inspection.

Fig. 10 Error behavior of the different cameras and their relationship with the type of element and bridge.

4 Conclusions

A low-cost, easy-to-use technical procedure based of photogrammetric image rectification has been developed to obtain the overall geometry of bridges in bridge inspection programs. The procedure has been tested in three different bridges and with three different cameras by comparison with laser scanning orthophotos. The limitations of the technique related to the surveying angle and working distance must be taken into account to obtain the desirable results. The image acquisition has been done by a student without previous training in photogrammetry or photography.

The majority of the measurements (95 percent) show error values below 4 percent. The results obtained do not show dependence between the error and the length and type of element and the bridge under study. However, error values establish a clear relationship with the model of camera. The Canon camera, the novel model with better technical specifications, depicts error values around 1 percent. This technique is mainly indicated as an inexpensive tool to establish a quick and rough geometrical inventory of bridge elements by inspectors and does not try to substitute for the accurate and detailed geometries that can be obtained using total stations and laser scanners.

Acknowledgments

The authors give thanks to the financial support of the Spanish Ministry of Science and Education (Grant No. BIA2009-08012), the Spanish Centre for Technological and Industrial Development (Grant No. IDI-20101770), and the Human Resources grant IPP055—EXP44 from Xunta de Galicia.

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