If you want to explore the results of multiplying matrices, I've made a basic matrix multiplier that will accept simple variables, such as x and y (but not x1 - you have to use just letters). This is the reason why we have the row of ones in the node matrix. Note that this requires that the number of columns in the first matrix must equal the number of rows in the second matrix, so all our transformation matrices must have 4 rows.įor example, if we have two nodes and we multiply by the transformation matrix, the first term in the result matrix (which is the x value of the first node) is $(1 \cdot x) + (0 \cdot y) + (0 \cdot z) + (1 \cdot dx)$, which is $x + dx$. For example, if we have two nodes and we multiply by the transformation matrix, the first term in the result matrix (. This is the dot product of the vectors given by the ith row and the jth column. Then we can see how each column of the matrix actually represents the coordinates. For the 3x3 case this is particularly intuitive, as we can visualize how a certain matrix transforms standard x/y/z basis vectors, or a unit cube defined by these. Briefly, the value (i, j) in the resulting matrix is first value in the ith row times the first value in the jth column plus the second value in the ith row times the second value in the jth column and so on. Sometimes its convenient to think of matrices as transformations. If you're not familiar with matrix multiplication, then Wikipedia should help. The reason for defining a matrix like this is so that when we multiple the node matrix by this matrix, the transformation occurs.
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