How do I construct a transformation matrix from 3 vertices?

Question

A question came up in IRC where a fellow wanted to calculate a position/orientation matrix based on 3 points.

Let's say that v1 should be the origin of the local coordinate space. The line from v1 to v2 should be the X axis, and v1,v2,v3 should all be in the XY plane. How do we use blender python to calculate a matrix usable as an object's transform matrix?

Answer 1

The following python code from orientation-matrix.py illustrates how to convert 3 coordinates into a transformation matrix:

import bpy
from mathutils import *

def make_matrix(v1, v2, v3):
    a = v2-v1
    b = v3-v1

    c = a.cross(b)
    if c.magnitude>0:
        c = c.normalized()
    else:
        raise BaseException("A B C are colinear")

    b2 = c.cross(a).normalized()
    a2 = a.normalized()
    m = Matrix([a2, b2, c]).transposed()
    s = a.magnitude
    m = Matrix.Translation(v1) * Matrix.Scale(s,4) * m.to_4x4()

    return m

#

obj = bpy.context.active_object
obj.matrix_world = make_matrix(Vector([1,1,1]), Vector([1,2.5,1]), Vector([0.5,1,1.5]) )

The Matrix.Scale is just thrown in for completeness and you can leave it out if you are content with scale=1. The order of the matrices in that multiplication is pretty important (I did not get it perfect on the first try).