3D graphics are pretty much all about 4x4 and 1x4 matrix multiplication. (The last component of a coordinate's 1x4 matrix is 1, and can usually be ignored.) A typical 3D matrix will look like so:

| A B C X |
| D E F Y |
| G H I Z |
| 0 0 0 1 |

(Note that they are often mirrored across the diagonal axis of the matrix (so the last row would be X Y Z 1, instead of that being the last column. However, the above is the form I'm personally used to.)

A-I encode things like rotation and scale, and match the last 9 numbers in an LDraw type 1 line, while X, Y, and Z represent translations, and match the first 3 numbers in an LDraw type 1 line. The bottom row is typically exactly as I showed (0s and a 1). If you read the

line type 1 portion of the LDraw spec, you will see this described.

What isn't explicitly given in the spec is equations to use for models nesting inside each other, but that is done by simply multiplying all the relevant 4x4 matrices together. Repeat that process for every submodel between the source and the destination, and you'll end up with a composite matrix that represents all the nested transformations. Multiply your original coordinate by that matrix, and you'll get the final transformed coordinate.

Note that with matrix multiplication, the order is important, unlike with scalar multiplication. (With matrices, A x B x C does

**not** equal C x B x A.) Matrix multiplication is extremely simple, but very tedious, so I'm not going to write the 16 equations needed to multiply two 4x4 matrices together. (One equation for each element of the final 4x4 matrix.) But you can look it up and easily find those. Plus, Tim already gave code (or perhaps pseudo-code; I'm not sure) to do that.

Being honest, Tim answered the question fully, and my response here is perhaps redundant, but I'm trying to answer it using different phrasing that may (or may not) be easier to understand.