For a game that I am getting started making I have had to learn vector math to calculate forces. To convert a vector from x and y to magnitude and angle I have read that I for the angle have to use the function tan^-1(y/x). Is this correct and if so how do I implement it into godots GDscript?
For a game that I am getting started making I have had to learn vector math to calculate forces.
First, this isn't necessarily true. Depending on your use case, it may be most effective to use Rigid Bodies and let Godot compute the effects of forces for you.
To convert a vector from x and y to magnitude and angle I have read that I for the angle have to use the function tan^-1(y/x).
To convert a vector from x and y to magnitude and angle, use Vector2.length() and Vector2.angle().
That being said, if you want to learn vector math, do it! Godot has its own vector math docs, and I'm sure you can find plenty of other similar lessons online.
However, it is good to be aware that the engine provides much of this functionality for you.
Writing a bunch of vector math instead of just calling a bultin function will just make your code more complicated.
As per the title, is the best way to calculate the n-dimensional cross product just using the determinant definition and using the LU Decomposition method of doing as such or could you guys suggest a better one?
Thanks
Edit: for clarity I mean http://en.wikipedia.org/wiki/Cross_product and not the Cartesian Product
Edit: It also seems that using the Leibniz Formula might help - though I don't know how that compares to LU Decomp. at the moment.
From your comment, it seems like you are looking for an operation which takes n −1 vectors as input and computes a single vector as its result, which will be orthogonal to all the input vectors and perhaps have a well-defined length as well.
With defined length
You can characterize the 3-dimensional cross product v =a ×b using the identity v ∙w =det(a,b,w). In other words, taking the cross product of the input vectors and then computing the dot product with any other vector w is the same as plugging the input vectors and that other vector into a matrix and computing its determinant.
This definition can be generalized to arbitrary dimensions. Due to the way a determinant can be computed using Laplace expansion along the last column, the resulting coordinates of that cross product will be the values of all (n −1)×(n −1) sub-determinants you can form from the input vectors, with alternating signs. So yes, Leibniz might be useful in theory, although it is hardly suitable for real-world computations. In practice, you'll soon have to figure out ways to avoid repeating computationswhile computing these n determinants. But wait for the last section of this answer…
Just the direction
Most applications however can do with a weaker requirement. They don't care about the length of the resulting vector, but only about its direction. In that case, what you are asking for is the kernel of the (n −1)×n matrix you can form by taking the input vectors as rows. Any element of that kernel will be orthogonal to the input vectors, and since computing kernels is a common task, you can build on a lot of existing implementations, e.g. Lapack. Details might depend on the language you are using.
Combining these
You can even combine the two approaches above: compute one element of the kernel, and for a non-zero entry of that vector, also compute the corresponding (n −1)×(n −1) determinant which would give you that single coordinate using the first approach. You can then simply scale the vector so that the selected coordinate reaches the computed value, and all the other coordinates will match that one.
This is over my head, can someone explain it to me better? http://mathworld.wolfram.com/Reflection.html
I'm making a 2d breakout fighting game, so I need the ball to be able to reflect when it hits a wall, paddle, or enemy (or a enemy hits it).
all their formula's are like: x_1^'-x_0=v-2(v·n^^)n^^.
And I can't fallow that. (What does ' mean or x_0? or ^^?)
The formula for reflection is easier to understand if you think to the geometric meaning of the operation of "dot product".
The dot product between two 3d vectors is mathematically defined as
<a, b> = ax*bx + ay*by + az*bz
but it has a nice geometric interpretation
The dot product between a and b is the length
of the projection of a over b taken with
a negative sign if the two vectors are pointing in
opposite directions, multiplied by the length of b.
Something that is immediately obvious using this definition and that it's not evident if you only look at the formula is for example that the dot product of two vectors doesn't change if the coordinate system is rotated, that the dot product of two perpendicular vectors is 0 (the length of the projection is clearly zero in this case) or that the dot product of a vector by itself is the square of its length.
Something that is instead less obvious using the geometric interpretation is that the dot product is commutative, i.e. that <a, b> = <b, a> (fact that is clear considering the formula).
An important point to consider is also that if the length of b is 1 then the dot product <a, b> is simply the length of the projection of a over b (taken with the proper sign).
Given this interpretation the formula for computing the reflection over a plane is quite easy to understand:
To compute the reflected vector r, given a vector a and a plane with normal n you just need to use the formula:
r = a - 2<a, n> n
the height h in the figure is in this case just <a, n> (note that n is assumed to be of unit length) and so it should be clear that you need to move twice that height in the direction of the normal.
If you consider the proper dot product signs you should see that the formula applies also when the incident vector a and the plane normal n are facing in the same direction.
The prime (') indicates the second form of a number/point/structure. In this case, x₁' refers to the reflected form of x₁.
The subscript (0) shows various states of the same. In this case, x₀ is the point of reflection.
The caret notation (^) shows that something is a vector. In this case, n̂ is the normal vector.
Is this just about the equation formatting? Because I see nicely formatted equations, not the LaTeX-style markup appearing in your question. So step 1: try viewing the page in a different web browser and see if it looks clearer.
More substantively, I'd recommend a different kind of resource. Fundamentally, you're looking at collisions, which are normally better treated in a physics text than a math text. Any introductory physics textbook will have a chapter on collisions, which should be directly applicable to your game.
I have a sparse matrix that represents a 3D rectangular space. Along some of the boundaries, I know what the value is going to be (it's a constant). The other boundaries may be reflective, differential, etc.
Should I just set the problem up as if all the boundaries were say, differential, and then go back and set the nodes in the solution vector b to be the constants?
Thanks!
In the finite element method you treat Dirchelet (value constraints) and Neumann (derivative constraints) differently. Usually you assemble the matrix without consideration for boundary conditions first, then apply boundary conditions, then do LU decomposition to solve.
You apply boundary conditions by modifying both the assembled matrix and the RHS vector. I'd have to know more details to tell you exactly what you need to do.
As a programmer I think it is my job to be good at math but I am having trouble getting my head round imaginary numbers. I have tried google and wikipedia with no luck so I am hoping a programmer can explain in to me, give me an example of a number squared that is <= 0, some example usage etc...
I guess this blog entry is one good explanation:
The key word is rotation (as opposed to direction for negative numbers, which are as stranger as imaginary number when you think of them: less than nothing ?)
Like negative numbers modeling flipping, imaginary numbers can model anything that rotates between two dimensions “X” and “Y”. Or anything with a cyclic, circular relationship
Problem: not only am I a programmer, I am a mathematician.
Solution: plow ahead anyway.
There's nothing really magical to complex numbers. The idea behind their inception is that there's something wrong with real numbers. If you've got an equation x^2 + 4, this is never zero, whereas x^2 - 2 is zero twice. So mathematicians got really angry and wanted there to always be zeroes with polynomials of degree at least one (wanted an "algebraically closed" field), and created some arbitrary number j such that j = sqrt(-1). All the rules sort of fall into place from there (though they are more accurately reorganized differently-- specifically, you formally can't actually say "hey this number is the square root of negative one"). If there's that number j, you can get multiples of j. And you can add real numbers to j, so then you've got complex numbers. The operations with complex numbers are similar to operations with binomials (deliberately so).
The real problem with complexes isn't in all this, but in the fact that you can't define a system whereby you can get the ordinary rules for less-than and greater-than. So really, you get to where you don't define it at all. It doesn't make sense in a two-dimensional space. So in all honesty, I can't actually answer "give me an exaple of a number squared that is <= 0", though "j" makes sense if you treat its square as a real number instead of a complex number.
As for uses, well, I personally used them most when working with fractals. The idea behind the mandelbrot fractal is that it's a way of graphing z = z^2 + c and its divergence along the real-imaginary axes.
You might also ask why do negative numbers exist? They exist because you want to represent solutions to certain equations like: x + 5 = 0. The same thing applies for imaginary numbers, you want to compactly represent solutions to equations of the form: x^2 + 1 = 0.
Here's one way I've seen them being used in practice. In EE you are often dealing with functions that are sine waves, or that can be decomposed into sine waves. (See for example Fourier Series).
Therefore, you will often see solutions to equations of the form:
f(t) = A*cos(wt)
Furthermore, often you want to represent functions that are shifted by some phase from this function. A 90 degree phase shift will give you a sin function.
g(t) = B*sin(wt)
You can get any arbitrary phase shift by combining these two functions (called inphase and quadrature components).
h(t) = Acos(wt) + iB*sin(wt)
The key here is that in a linear system: if f(t) and g(t) solve an equation, h(t) will also solve the same equation. So, now we have a generic solution to the equation h(t).
The nice thing about h(t) is that it can be written compactly as
h(t) = Cexp(wt+theta)
Using the fact that exp(iw) = cos(w)+i*sin(w).
There is really nothing extraordinarily deep about any of this. It is merely exploiting a mathematical identity to compactly represent a common solution to a wide variety of equations.
Well, for the programmer:
class complex {
public:
double real;
double imaginary;
complex(double a_real) : real(a_real), imaginary(0.0) { }
complex(double a_real, double a_imaginary) : real(a_real), imaginary(a_imaginary) { }
complex operator+(const complex &other) {
return complex(
real + other.real,
imaginary + other.imaginary);
}
complex operator*(const complex &other) {
return complex(
real*other.real - imaginary*other.imaginary,
real*other.imaginary + imaginary*other.real);
}
bool operator==(const complex &other) {
return (real == other.real) && (imaginary == other.imaginary);
}
};
That's basically all there is. Complex numbers are just pairs of real numbers, for which special overloads of +, * and == get defined. And these operations really just get defined like this. Then it turns out that these pairs of numbers with these operations fit in nicely with the rest of mathematics, so they get a special name.
They are not so much numbers like in "counting", but more like in "can be manipulated with +, -, *, ... and don't cause problems when mixed with 'conventional' numbers". They are important because they fill the holes left by real numbers, like that there's no number that has a square of -1. Now you have complex(0, 1) * complex(0, 1) == -1.0 which is a helpful notation, since you don't have to treat negative numbers specially anymore in these cases. (And, as it turns out, basically all other special cases are not needed anymore, when you use complex numbers)
If the question is "Do imaginary numbers exist?" or "How do imaginary numbers exist?" then it is not a question for a programmer. It might not even be a question for a mathematician, but rather a metaphysician or philosopher of mathematics, although a mathematician may feel the need to justify their existence in the field. It's useful to begin with a discussion of how numbers exist at all (quite a few mathematicians who have approached this question are Platonists, fyi). Some insist that imaginary numbers (as the early Whitehead did) are a practical convenience. But then, if imaginary numbers are merely a practical convenience, what does that say about mathematics? You can't just explain away imaginary numbers as a mere practical tool or a pair of real numbers without having to account for both pairs and the general consequences of them being "practical". Others insist in the existence of imaginary numbers, arguing that their non-existence would undermine physical theories that make heavy use of them (QM is knee-deep in complex Hilbert spaces). The problem is beyond the scope of this website, I believe.
If your question is much more down to earth e.g. how does one express imaginary numbers in software, then the answer above (a pair of reals, along with defined operations of them) is it.
I don't want to turn this site into math overflow, but for those who are interested: Check out "An Imaginary Tale: The Story of sqrt(-1)" by Paul J. Nahin. It talks about all the history and various applications of imaginary numbers in a fun and exciting way. That book is what made me decide to pursue a degree in mathematics when I read it 7 years ago (and I was thinking art). Great read!!
The main point is that you add numbers which you define to be solutions to quadratic equations like x2= -1. Name one solution to that equation i, the computation rules for i then follow from that equation.
This is similar to defining negative numbers as the solution of equations like 2 + x = 1 when you only knew positive numbers, or fractions as solutions to equations like 2x = 1 when you only knew integers.
It might be easiest to stop trying to understand how a number can be a square root of a negative number, and just carry on with the assumption that it is.
So (using the i as the square root of -1):
(3+5i)*(2-i)
= (3+5i)*2 + (3+5i)*(-i)
= 6 + 10i -3i - 5i * i
= 6 + (10 -3)*i - 5 * (-1)
= 6 + 7i + 5
= 11 + 7i
works according to the standard rules of maths (remembering that i squared equals -1 on line four).
An imaginary number is a real number multiplied by the imaginary unit i. i is defined as:
i == sqrt(-1)
So:
i * i == -1
Using this definition you can obtain the square root of a negative number like this:
sqrt(-3)
== sqrt(3 * -1)
== sqrt(3 * i * i) // Replace '-1' with 'i squared'
== sqrt(3) * i // Square root of 'i squared' is 'i' so move it out of sqrt()
And your final answer is the real number sqrt(3) multiplied by the imaginary unit i.
A short answer: Real numbers are one-dimensional, imaginary numbers add a second dimension to the equation and some weird stuff happens if you multiply...
If you're interested in finding a simple application and if you're familiar with matrices,
it's sometimes useful to use complex numbers to transform a perfectly real matrice into a triangular one in the complex space, and it makes computation on it a bit easier.
The result is of course perfectly real.
Great answers so far (really like Devin's!)
One more point:
One of the first uses of complex numbers (although they were not called that way at the time) was as an intermediate step in solving equations of the 3rd degree.
link
Again, this is purely an instrument that is used to answer real problems with real numbers having physical meaning.
In electrical engineering, the impedance Z of an inductor is jwL, where w = 2*pi*f (frequency) and j (sqrt(-1))means it leads by 90 degrees, while for a capacitor Z = 1/jwc = -j/wc which is -90deg/wc so that it lags a simple resistor by 90 deg.