Convert Global Space 3D Location to Local object space?

Question

I'm trying to find a formula that could convert my 3D global coordinates into local object space.

Here are the inputs I have

A Location

B Location
B Rotation
B Scale 

I need a Location output, corresponding to my A Location in B Space, according to B Location, Rotation, and Scale. I'm working in a 3D Space

So far I didn't found any formulas online, anyone can point me in the right direction? Bests

If possible, i'd need a formula that doesn't use any matrix calculation, it's not supported by the nodal system of my engine (I'm using blender geometry node, but my problem is more generic as you can see)

Answer 1

You'll have to unpeel the transformations applied to B one by one, in reverse order. Typically we'll apply transformations in the sequence...

1. Scale
2. Euler Roll
3. Euler Pitch
4. Euler Yaw
5. Translate

So going backwards from 5 back to 1 (and assuming the {yaw, pitch, roll} axes are {z, x, y} - I'm not super familiar with Blender's conventions so you might need to swap some):

Untranslated = A.Location - B.Location

Unyawed = Yaw(Untranslated, -B.Rotation.z)

Unpitched = Pitch(Unyawed, -B.Rotation.x)

Unrolled = Roll(Unpitched, -B.Rotation.y)

Unscaled = Scale(Unrolled, (1/B.Scale.x, 1/B.Scale.y, 1/B.Scale.z) )

Now Unscaled is the point in B's local coordinate space that corresponds to the point A in the global coordinate space.

Here's how we can implement those rotation and scale functions:

(If your sine and cosine functions take their input in radians, be sure to convert your degree inputs to radians first)

Yaw(point, angle) {
    s = sin(angle)
    c = cos(angle)

    return (point.x * c - point.y * s, 
            point.y * c + point.x * s,
            point.z)
}

Pitch(point, angle) {
    s = sin(angle)
    c = cos(angle)

    return (point.x, 
            point.y * c - point.z * s, 
            point.z * c + point.y * s)
}

Roll(point, angle) {
    s = sin(angle)
    c = cos(angle)

    return (point.x * c + point.z * s,
            point.y,
            point.z * c - point.x * s)
}

Scale(point, scaleTriplet) {
    return (point.x * scaleTriplet.x,
            point.y * scaleTriplet.y,
            point.z * scaleTriplet.z)
}