Euler Rotation and Euler's Formula

EULER

It is a program for quickly and interactively computing with real and complex numbers and matrices, or with intervals, in the style of MatLab, Octave,... It can draw and animate your functions in two and three dimensions.

So, what is it for ?

Suppose you have a non-trivial function and you want to discuss it. You could use one of the plot commands of EULER to get a sketch of the function. Then there are tools to determine zeros or local extrema of the function. You could compute its integral. You could even produce plots of this function with a varying parameter (as a set of plots or as a three dimensional plot).

As another example, suppose you have data contained in a file. EULER can read these data form the file and produce plots of these data, fit polynomials to it, do further computations etc.

In a last example, we assume you have a numerical algorithm to test. You could provide a prototype of this algorithm in the EULER programming language. This is usually done much quicker than using a classical programming language. Furthermore it is interactive and you can use graphics to verify the algorithm.

Euler rotations

Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves.

These rotations are called precession, nutation, and intrinsic rotation.

Euler Angles

1. Yaw
Rotate around y-axis
2.Pitch
Rotate around (rotated) x-axis

3. Roll
Rotate around (rotated) y-axis

Intuitive Understanding Of Euler’s Formula

Can maths be beautiful? Most people understand the beauty of a painting or a piece of music, but what about the squiggles of a mathematical equation? We call great works of art “beautiful” if they are aesthetically pleasing or express fundamental ideas in a profound way, and mathematicians feel the same way about particularly elegant proofs. Many say the most beautiful result of all is Euler’s equation: ei? + 1 = 0. Leonhard Euler (pronounced “oiler”) was a Swiss mathematician in the 18th century, and is considered to be one of the greatest of all time. But to discover the beauty of Euler’s equation, you have to understand its meaning.

Euler's identity seems baffling:

It emerges from a more general formula:

Why does we call it a God's Equation?

We can only reproduce the equation and not stop to inquire into its implications. It appeals equally to the mystic, the scientists, the mathematician." This formula of Leonhard Euler (1707-1783) unites the five most important symbols of mathematics: 1, 0, pi, e and i (the square root of minus one). This union was regarded as mystic union containing representatives from each branch of the mathematical tree: arithmetic is represented by 0 and 1, algebra by the symbol i, geometry by pi, and analysis by the transcendental e. Harvard mathematician Benjamin Pierce said about the formula, "That is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

Euler's formula describes two equivalent ways to move in a circle.

That's it? This stunning equation is about spinning around? Yes -- and we can understand it by building on a few analogies: 

Starting at the number 1, see multiplication as a transformation that changes the number: 1⋅eiπ 

Regular exponential growth continuously increases 1 by some rate; imaginary exponential growth continuously rotates a number 

Growing for "pi" units of time means going pi radians around a circle 

Therefore, eiπ means starting at 1 and rotating pi (halfway around a circle) to get to -1 

That's the high-level view, let's dive into the details. By the way, if someone tries to impress you with eiπ=−1, ask them about i to the i-th power. If they can't think it through, Euler's formula is still a magic spell to them.

Understanding cos(x) + i * sin(x)

The equals sign is overloaded. Sometimes we mean "set one thing to another" (like x = 3) and others we mean "these two things describe the same concept" (like −1−−−√=i). 

Euler's formula is the latter: it gives two formulas which explain how to move in a circle. If we examine circular motion using trig, and travel x radians:

cos(x) is the x-coordinate (horizontal distance)
sin(x) is the y-coordinate (vertical distance)


The statement

is a clever way to smush the x and y coordinates into a single number. The analogy "complex numbers are 2-dimensional" helps us interpret a single complex number as a position on a circle. 

When we set x to π, we're traveling π units along the outside of the unit circle. Because the total circumference is 2π, plain old pi is halfway around, putting us at -1. 

Neato: The right side of Euler's formula (cos(x)+isin(x)) describes circular motion with imaginary numbers. Now let's figure out how the e side of the equation accomplishes it.

What is Imaginary Growth?

Combining x- and y- coordinates into a complex number is tricky, but manageable. But what does an imaginary exponent mean? 

Let's step back a bit. When I see 34, I think of it like this: 

3 is the end result of growing instantly (using e) at a rate of ln(3). In other words: 3=eln(3) 

34 is the same as growing to 3, but then growing for 4x as long. So 34=eln(3)⋅4=81 

Instead of seeing numbers on their own, you can think of them as something e had to "grow to". Real numbers, like 3, give an interest rate of ln(3) = 1.1, and that's what e "collects" as it's going along, growing continuously. 

Regular growth is simple: it keeps "pushing" a number in the same, real direction it was going. 3 × 3 pushes in the original direction, making it 3 times larger (9).

Imaginary growth is different: the "interest" we earn is in a different direction! It's like a jet engine that was strapped on sideways -- instead of going forward, we start pushing at 90 degrees. 

The neat thing about a constant orthogonal (perpendicular) push is that it doesn't speed you up or slow you down -- it rotates you! Taking any number and multiplying by i will not change its magnitude, just the direction it points. 

Intuitively, here's how I see continuous imaginary growth rate: "When I grow, don't push me forward or back in the direction I'm already going. Rotate me instead." 

EULER is an ideal tool for the tasks such as:

• Inspecting and discussing functions of one real or complex variable.
• Viewing surfaces in parameter representation.
• Linear algebra and eigenvalue computation.
• Testing numerical algorithms.
• Solving differential equations numerically.
• Computing polynomials.
• Studying interval arithmetic.
• Examining and generating sound files.

The drawback is that you will have to learn a language. It is not a difficult language. However, there is some learning involved.

Euler features :

• Real, complex and interval scalars and matrices,
• A programming language, with local variables, default values for parameters, variable parameter number, passing of functions,
• Two and three dimensional graphs,
• Marker plots,
• Density and contour plots,
• Animations,
• Numerical integration and differentiation,
• Statistical functions and tests,
• Differential equations,
• Interval methods with guaranteed inclusions,
• Function minimizers (Brent, Nelder-Mean),
• Simplex algorithm,
• Interpolation and approximation,
• Finding roots of polynomials,
• Fast Fourier transform (FFT),
• An exact scalar product using a long accumulator,
• Postscript graphics export

Refrences

• http://www.sosmath.com/complex/number/eulerformula/eulerformula.html
• http://en.wikipedia.org/wiki/Rotation
• http://euler.sourceforge.net/
• http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/
• https://answers.yahoo.com/question/index?qid=20071121062325AAlHZqr