Is the statement in Scratchapixel about transforming points correct?

Question

I am reading this lesson from scratchapixel. To my understanding (and the example given in the same website) M is the matrix that transforms from local to world.

Giong through the paragraph before figure 7, though, I get the following:

If we know the 4x4 matrix M that transforms a coordinate system A into a coordinate system B ...

Say A is the local coord system and B is the world coord system. Therefore M transforms from local to world.

... then if we transform a point whose coordinates are originally defined with respect to A with the inverse of M (...), we get the coordinates of P with respect to B.

We know, A = local coord system. The Plocal = local point coords are (-0.5, 0.5, -0.5). Then, as the quoted text says, we transform Plocal with the inverse of M to get the point w.r.t. to B = woorld coord system

However, this last operation (statement) confuses me, since the example they give the do Pworld = Plocal * M. But from the quoted text I understand that Pworld = inv(M) * Plocal

Can someone help me figuring out what am I missing? Is the wording of the text wrong?

Answer 1

Let's take a simple example with the following variables

A: Local-space coordinates system
B: World-space coordinates system
M: Transformation matrix from local-space to world-space (A -> B)

Let's say we have a point PA in local space. To get the values of PA in world coordinates, we need to apply the transformation M on it:

// Local to World: A -> B
PB = PA * M    // If you are using row-based positions (your case)
PB = M * PA    // If you are using column-based positions

You can apply the inverse transformation to get PA back from PB:

// World to local: B -> A
PA = PB * inv(M)    // If you are using row-based positions (your case)
PA = inv(M) * PB    // If you are using column-based positions

The website you linked in the question has a part on this. It talks briefly about transforming a point from world-space to local-space:

As suggested before, it is sometimes more convenient to operate on points when they are defined with respect to a local coordinate system rather than defined with respect to the world coordinate system. For instance, in the example of the cube (figure 6), defining the corners of the cube in local space is easier than in world space. But how do we convert a point or vertex from one coordinate system (such as the world coordinate space) to another (converting points from a coordinate system to another is a very common process in CG)? That's easy. If we know the 4x4 matrix M that transforms a coordinate system A into a coordinate system B, then if we transform a point whose coordinates are originally defined with respect to A [Should be B] with the inverse of M (we will explain next why we use the inverse of M rather than M), we get the coordinates of P with respect to B [Should be A].

I have added comments in square brackets ([]) for clarity.

NOTE: P with respect to A is PA, and P with respect to B is PB. X in respect to/of Y is a point X in the base Y: Coordinate vectors

But this part is slightly unclear because in this paragraph there are some typos inverting the letters. Just ignore this passage, the rest of the article seems mathematicaly accurate.