Matrix
A matrix (pl. matrices) is a list of numbers, generally who have a relationship between them, that are ordered in a type of mathematical array. A matrix, m, may be written as either [m], or (m) in mathematics, where m is a placeholder for the numbers in the matrix.
Reading a Matrix
A matrix is ordered in rows and columns, which are counted by increasing i and j respectively. One can think of a matrix as a 2-dimensional array where each object in the 1st dimension holds an array of numbers in that row.
An entry at i=2 and j=3 is noted by either M[2,3] or m2,3. These numbers are important for some modifications involving these matrices.
Demystifying Matrices in GraalScript
If you have ever tested any of the matrix functions built into GraalScript, you may be lost as to where the return value comes from. Well, luckily there are some adept enough with math to understand these functions. These are their findings.
matrixcreate()
Let v be a vector and a consisting of a vector and a scalar.
Usage
| strMatrix = matrixcreate(v,a); |
Where v describes the translation (position) and 'a' describes the rotation (rotation axis and angle).
Return Value
Returns a matrix array of length 7 (this will be explained later).
Example
| temp.matrix = matrixcreate({1,1,1},{1,7,1,1});
// This will return 1,1,1,0.140028,0.980196058,0.140028,7.14595223 |
The Math Behind the Function
The first three numbers of the return value should be rather familiar. It is the vector you passed to the function. The other four seem to come out of nowhere. However, there is a method to this madness. The first of these three are merely a property of angles. Because of the Dot Product of vectors, we are able to calculate the cosine of the angle between two vectors. Well, we have actually created a new vector in this code!
As defined in Vectors, we know that there are three unit vectors that lay on the x, y, and z axes. These vectors have been given the letters i, j, and k, respectively. The first value defined in the "rotation" is multiplied to i, the second to j, and so forth, and this makes a new vector. However, this still does not tell us why we have these numbers returned to us.
From the definition of the dot product, we know that
cos(θ) = u•v/(|u||v|)
The first three numbers returned to us are the cosine of the angles between the new vector (in this case, <1,7,1>) and each of our unit vectors. For example:
cos(α) = <1,7,1>•<1,0,0>/(|<1,7,1>||<1,0,0>|) = 1/√51 = 0.1400280084 cos(β) = <1,7,1>•<0,1,0>/(|<1,7,1>||<0,1,0>|) = 7/√51 = 0.9801960588 cos(γ) = <1,7,1>•<0,0,1>/(|<1,7,1>||<0,0,1>|) = 1/√51 = 0.1400280084
There is an evident difference between the values given here and the return value. A computer cannot calculate cosine on its own. It merely uses a method of estimation.
cos(α)2 + cos(β)2 + cos(γ)2 = 1 (1)
(1) The returned values are accurate within ±10-5
So far, we have 6 of 7 figured out. The last is the simplest of them all. Plugging the new vector into vectorlen() or using √(x2 + y2 + z2) returns it's magnitude. This, multiplied by the fourth number of the second parameter, is the last of the returned values (and as far as anyone can tell the vector passed to the function has no bearing on the return value).
matrixcreatefromeuler()
Let r be an array describing rotation, {α,β,γ}.
Usage
| strMatrix = matrixcreatefromeuler(r); |
Where r describes the rotation (rotation axis and angle).
Return Value
Returns a matrix array of length 7.
Example
| temp.matrix = matrixcreatefromeuler({pi/2,0,pi/2});
// This will return 0,0,0,-0.577350258,-0.577350258,-0.577350258,119.999976018 |
The Math Behind the Function
Coming soon!