tigrou - 22 days ago 8
C# Question

# How to convert a 3D point on a plane to UV coordinates?

I have a 3d point, defined by

`[x0, y0, z0]`
.

This point belongs to a plane, defined by
`[a, b, c, d]`
.

`normal`
=
`[a, b, c]`
, and
`ax + by + cz + d = 0`

How can convert or map the 3d point to a pair of
`(u,v)`
coordinates ?

This must be something really simple, but I can't figure it out.

First of all, you need to compute your `u` and `v` vectors. There is no unique way to define them, but a convenient and fast way may be something like this:

``````n = [a, b, c]
u = normalize([b, -a, 0]) // Assuming that a != 0 and b != 0, otherwise use c.
v  = n x u // If n was normalized, v is normalized already.
``````

Now a simple dot product will do:

``````u_coord = dot(u,[x0 y0 z0])
v_coord = dot(v,[x0 y0 z0])
``````

Notice that this assumes that the origin of the u-v coordinates is the world origin (0,0,0).

This will work even if your vector `[x0 y0 z0]` doesn't exactly lies on the plane. If that is the case, it will just project it to the plane.