Convert global 3d coordinate system to a local one with specific direction

Question

Imagine a (x,y,z) coordinate system within a solar system, with (0,0,0) beeing the center of sun where Z axis goes thru both solar poles.

I'm standing on a planet inside a house within that solar system.

Now from my perspective Z axis is skewed in unknown direction depending on where exactly the planet is at this moment.

What would be the steps I should take to create a local coordinate system with (0,0,0) being the house and where Z axis is perpendicular to the ground/house, so that I would be able to convert coordinates from local to global.

I have the global coordinate of my house and directional vectors for where is up/forward/right for my house.

I have some limited knowledge of vector math and projecting vectors on other surfaces, but there is a gap in my understanding that keeps me from solving this.

local housePos = vec(getHouseWorldPosition())        -- (x,y,z) position of house in global cooridnates (housePos.x,housePos.y,housePos.z)

local vRight = vec(getHouseWorldOrientationRight())  -- X directional normalized vector
local vForw = vec(getHouseWorldOrientationForward()) -- Y directional normalized vector
local vUp = vec(getHouseWorldOrientationUp())        -- Z directional normalized vector

local playerPos = vec(getPlayerWorldPosition())      -- players position in global coordinates

--Now how to get players position in coordinate system relative to housePos, with Z axis going along vUp vector

Answer 1

Construct a matrix like so, where each vector is one column of the matrix:

[ House's X+ direction | House's Y+ direction | House's Z+ direction | House's position]

The fourth row should be [0, 0, 0, 1]

Since you're in Unreal with its right-handed coordinate system

• x -> right
• y -> back
• z -> up

we can write this as:

$$M = \begin{bmatrix} houseRight.x & -houseForward.x & houseUp.x & housePosition.x\\ houseRight.y & -houseForward.y & houseUp.y & housePosition.y\\ houseRight.z & -houseForward.z & houseUp.z & housePosition.z\\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Multiplying a vector \$v\$ like this:

$$v = \begin{bmatrix}v.x\\v.y\\v.z\\1\end{bmatrix}$$

by this matrix gives us...

$$M v = \begin{bmatrix} houseRight \cdot v.xyz &+& housePosition.x \\ -houseForward \cdot v.xyz &+& housePosition.y \\ houseUp \cdot v.xyz &+& housePosition.z \\ &1& \end{bmatrix}^T$$

...which takes a coordinate v.xyz from house-local coordinates to global coordinates. You can verify that...

• When v = [0, 0, 0, 1] (the origin), this matrix maps it to the center of the house - the v.w = 1 coordinate gets multiplied by the house position column in the matrix, and the rest get zero'd-out

• As we increase v.x, the result v' moves along the house's X+ (right) direction, and so on for the Y (back) & Z (up)

To convert back from world coordinates to house-local coordinates, take the inverse of this matrix.